Multiple-StrainMalariaInfectionandItsImpactson falciparum ...downloads.hindawi.com/journals/cmmm/2019/9783986.pdf · due to malaria infection. e impacts of multiple-strain malarial

Research ArticleMultiple-StrainMalaria Infection and Its Impacts onPlasmodiumfalciparum Resistance to Antimalarial Therapy A MathematicalModelling Perspective

Titus Okello Orwa Rachel Waema Mbogo and Livingstone Serwadda Luboobi

Institute of Mathematical Sciences Strathmore University PO Box 59857-00200 Nairobi Kenya

Correspondence should be addressed to Titus Okello Orwa torwastrathmoreedu

Received 14 February 2019 Accepted 15 May 2019 Published 11 June 2019

Academic Editor Konstantin Blyuss

Copyright copy 2019 Titus Okello Orwa et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

+e emergence of parasite resistance to antimalarial drugs has contributed significantly to global human mortality and morbiditydue to malaria infection +e impacts of multiple-strain malarial parasite infection have further generated a lot of scientificinterest In this paper we demonstrate using the epidemiological model the effects of parasite resistance and competitionbetween the strains on the dynamics and control of Plasmodium falciparummalaria +e analysed model has a trivial equilibriumpoint which is locally asymptotically stable when the parasitersquos effective reproduction number is less than unity Using contourplots we observed that the efficacy of antimalarial drugs used the rate of development of resistance and the rate of infection bymerozoites are the most important parameters in the multiple-strain P falciparum infection and control model Although thedrug-resistant strain is shown to be less fit the presence of both strains in the human host has a huge impact on the cost andsuccess of antimalarial treatment To reduce the emergence of resistant strains it is vital that only effective antimalarial drugs areadministered to patients in hospitals especially in malaria-endemic regions Our results emphasize the call for regular and strictsurveillance on the use and distribution of antimalarial drugs in health facilities in malaria-endemic countries

1 Introduction

+e emergence of parasite resistance [1ndash4] to antimalarialdrugs has contributed significantly to human mortality andmorbidity due to malaria infection worldwide [5ndash7] Aglobal malaria control strategy of 1992 [8] that advocated forearly diagnosis and prompt treatment has been heavilycompromised by the emergence of parasite resistance toantimalarial drugs +e evolution of parasite resistance hasbeen described in [9] as an example of a Darwinian evo-lution Parasites undergo mutations in their genome inresponse to the drug-treated human host +ese mutationsreduce the rate of parasite elimination from the host andincrease their survival chances [9] +e most extensivelyused antimalarial drugs against the deadly Plasmodiumfalciparum malaria are chloroquine (CQ) and sulfadoxine-pyrimethamine (SP) [10 11] +ese drugs are cheap easilyavailable and slowly eliminated from the human body [11]However the extensive use of CQ and SP has resulted in

P falciparum resistance +is has led to global increase inmalaria cases and mortality [12] In response the WorldHealth Organization (WHO) in 2006 recommended the useof artemisinin-based combination therapies (ACTs) asa first-line treatment for uncomplicated P falciparummalaria [13] Resistance to ACTs which are currently thestandard treatment for P falciparum is likely to cause globalhealth crisis especially in African regions where P falcipa-rum malaria is endemic [11]

+e emergence of parasite resistance to malaria therapydates back to the 19th century Quinine (1963) was the first-line antimalarial drug against P falciparum [14] Highmortality cases coupled with high parasite resistance led tothe introduction of a second drug chloroquine (CQ) in1934 [15] A decade later CQ was considered the first-lineantimalarial drug by several countries until 1957 when thefirst focus of P falciparum resistance was detected along the+ai-Cambodia border [16] In Africa P falciparum re-sistance to CQ was first discovered among travelers from

HindawiComputational and Mathematical Methods in MedicineVolume 2019 Article ID 9783986 26 pageshttpsdoiorg10115520199783986

Kenya to Tanzania [17] By 1983 CQ resistance had spreadto Sudan Uganda [18] Zambia [19] and Malawi [20]Unlike Africa CQ was replaced for the first time withsulfadoxine-pyrimethamine (SP) as a first-line antimalarialdrug in+ailand in 1967 Several other countries in Asia andSouth America followed thereafter [10] Resistance to SPwas however reported the same year [21] in the region In1988 CQ was replaced for the first time in Africa KwaZulu-Natal Province of South Africa replaced CQ with SP [22] In1993 the Malawian government changed the treatmentpolicy from CQ to SP Other African countries followedthereafter Kenya South Africa and Botswana (in 1998)Cameroon and Tanzania (in 2001) and Zimbabwe (in 2000)[23] +e effectiveness of SP was equally undermined byresistance Unlike CQ P falciparum resistance to SP wasmainly attributed to the long half-life of the drug [24]Confirmed resistance to the artemisinin derivatives was firstreported in Cambodia and Mekong regions in 2008 [25]

To leverage on parasite resistance cost of treatmentand burden of malaria infection to communities andgovernments the WHO recommends the use of artemi-sinin-based combination therapies (ACTs) as the first- andsecond-line treatment drugs for uncomplicated P falci-parum malaria [25] ACT is a combination of artemisininderivatives and a partner monotherapy drug Artemisininderivatives include artemether artesunate and dihy-droartemisinin +ese derivatives reduce the parasitebiomass within the first three days of therapy while thepartner drug with longer half-life eliminates theremaining parasites [26] +eWHO currently recommendsfive different ACTs (1) artesunate-amodiaquine (AS +AQ)(2) artesunate-mefloquine (AS +MQ) (3) artesuna-te + sulfadoxine-pyrimethamine (AS + SP) (4) artemether-lumefantrine (AM-LM) and (5) dihydroartemisinin-piperaquine (DHA+PPQ) Additionally artesunate-pyronaridine may be used in regions where ACT treatmentresponse is low [26] Access to ACT has been tremendousin the last 8 years with a recorded increase of 122 millionprocured treatment courses for the period 2010ndash2016However resistance to currently used ACTs has importantpublic health consequences especially in the African re-gion where resistant P falciparum is predominant

Numerous cross-sectional studies [27 28] have revealedthe possible impacts of multiple strains of P falciparum onthe development of resistance to ACTs In [29] and citationstherein drug-sensitive parasites are shown to stronglysuppress the growth and transmission of drug-resistant Pfalciparum parasites Although high-transmission settingssuch as sub-Saharan Africa account for about 90 of allglobal malaria deaths resistance to antimalarial drugs hasbeen shown to emerge from low-transmission settings suchas Southeast Asia and South America [29] Causes of parasiteresistance to ACTs are diverse Historical studies [30 31]indicate that antimalarial-resistant parasites could emergefrom a handful of lineages It is argued elsewhere [32 33]that recombination during sexual reproduction in themosquito vector could be responsible for the delayed ap-pearance of multilocus resistance in high-transmission re-gions Moreover owing to repeated exposure for many

years individuals in high-transmission settings are likely todevelop clinical immunity to malaria leading to strongerselection for resistance [34] Studies in [29] also support thehypothesis that in-host competition between drug-sensitiveand drug-resistant parasites could inhibit the spread ofresistance in high-transmission settings Owing to theirintegral role in the recent success of global malaria controlthe protection of efficacy of ACTs should be a global healthpriority [35]

Mathematical models of in-host malaria epidemiologyand control constitute important tools in guiding strategiesfor malaria control [36 37] and the associated financialplanning [38] While some researchers have focussed onprobabilistic models [39 40] others have investigated theeffects of drug treatment and resistance development usingdynamic models [41 42] A deterministic model by Estevaet al [43] monitored the impact of drug resistance on thetransmission dynamics of malaria in a human population In[29] the impacts of within-host parasite competition areshown to inhibit the spread of resistance [44 45] On thecontrary some models [39 46] have suggested that within-host competition is likely to speed up the spread of resistancein high-transmission settings due to a phenomenon calledldquocompetitive releaserdquo In this paper we provide theoreticalinsights using mathematical modelling of the impacts ofmultiple-strain infections on resistance dynamics andantimalarial control of P falciparum malaria

+e rest of the paper is organized as follows In Section 2we formulate the within-humanmalaria model that has boththe drug-sensitive and drug-resistant P falciparum parasitestrains subject to antimalarial therapy In Section 3 weanalyze the model based on epidemiological theoremsWithin-host competition between parasite strains and theeffects of antimalarial drug efficacy on parasite clearance arediscussed in Section 4 Sensitivity analysis and multiple-strain infection and its effects on resistance and malariadynamics are demonstrated in Section 5 We conclude thepaper in Section 6 by emphasizing the need for antimalarialtherapy with the potential to eradicate multiple-strain in-fection due to P falciparum

2 Model Formulation

We present in this paper a deterministic model that de-scribes the within-human-host competition and trans-mission dynamics of two strains of P falciparum parasitesduring malaria infection +e compartmental model con-siders the coinfection and competition between the drug-sensitive (dss) and the drug-resistant (drs) P falciparumstrains in the presence of antimalarial therapy +e drs arisepresumably from the dss +e rare mechanism here couldpossibly be due to single point mutation [47] Both drs anddss initiate immune responses that follow density-dependentkinetics

Our model is composed of eight compartments sus-ceptiblehealthyunparasitized erythrocytes (red blood cells)X(t) parasitizedinfected erythrocytes (Yr(t) and Ys(t))merozoites (Ms(t) and Mr(t)) gametocytes (Gs(t) andGr(t)) and immune cells W(t) +e healthy erythrocytes

2 Computational and Mathematical Methods in Medicine

(RBCs) make up the resource for competition between thedrug-resistant and drug-sensitive parasite strains +e in-fected red blood cells (IRBCs) and different erythrocyticparasite life cycles are categorized based on the strain of theinfecting parasite +e merozoites are therefore categorizedinto drug-sensitive and drug-resistant strains denoted byMs(t) and Mr(t) respectively +e merozoites invade thehealthy erythrocytes during the erythrocytic stage leading toformation of infected erythrocytes +e variable Ys(t) de-notes the red blood cells (RBCs) infected with drug-sensitivemerozoites whereas Yr(t) refers to the RBCs infected withdrug-resistant merozoites Similarly the variables Gs(t) andGr(t) represent drug-sensitive and drug-resistant gameto-cytes respectively Owing to saturation in cell and parasitegrowth we consider the nonlinear MichaelisndashMentedndashMonod function described in [48 49] and used in [50ndash53] tomodel the reductive effects of the immune cells on theparasite and infected-cell populations

+e density of the healthy RBCs is increased at the rate λx

per healthy RBC per unit time from the hostrsquos bone marrowand healthy RBCs die naturally at a rate μx Following parasiteinvasion by free floating merozoites the healthy erythrocytesget infected by both drug-sensitive and drug-resistant mero-zoite strains at the rates β and δrβ respectively +e parameterδr (with 0lt δr lt 1) accounts for the reduced fitness (in-fectiousness) of the resistant parasite strains in relation to thedrug-sensitive strains+e destruction of the healthy red bloodcells is however limited by the adaptive immune cellsW+is isrepresented by the term 1(1 + cW) where c is a measure ofthe efficacy of the immune cells +e equation that governs theevolution of the healthy RBCs is hence given by

dX

dt λx minus μxXminus

βX

1 + cWMs + δrMr( 1113857 (1)

+e parasitized erythrocytes are generated through massaction contact (invasion) between the susceptible healthyerythrocytes X and the blood floating merozoites (Mr and Ms)+e merozoites subdivide mitotically within the infectederythrocytes into thousands of other merozoites leading to cellburst and emergence of characteristic symptoms of malariaAdditionally a single infected erythrocyte undergoes hemolysisat the rate μys to produceP secondarymerozoites sustaining theerythrocytic cycle+e drug-sensitive IRBCs (Ys) burst open togenerate more drug-sensitive merozoites or drug-sensitivegametocytes at the rate σs Similar dynamics are observed withthe drug-resistant IRBCs where the drug-resistant gametocytesare generated at the rate σr from IRBCs Treatment with ACTisassumed to disfranchise the development of the merozoitewithin the infected erythrocyte +e drug-infested erythrocytesare hence likely to die faster +is is represented by the term(1minusωs)

minus1 where 0ltωs lt 1 represents the antimalarial-specifictreatment efficacy In this paper and for purposes of illustrationand simulations ωs corresponds to the efficacy of artemether-lumefantrine (AL) which is the recommended first-line anti-malarial ACT drug for P falciparum infection in Kenya Weassume that no treatment is available for erythrocytes infectedwith the resistant parasite strains+e time rate of change forYsand Yr takes the following form

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dYr

dtδrβXMr

1 + cWminus

kyYrW

1 + aYrminus μyrYr minus σrYr

(2)

+e drug-resistant merozoites Mr and the drug-resistantgametocytes Gr die naturally at the rates μmr and μgr re-spectively It is further assumed that drug-sensitive mero-zoites Ms and gametocytes Gs may develop into drug-resistant merozoites Mr and gametocytes Gr at the rates Ψ1and Ψ2 respectively +e cost of resistance associated withAL is represented by the parameter αs Parasite resistance toantimalarial drugs exacerbates the erythrocytic cycle andincreases the cost of treatment [54 55] +e higher theresistance to antimalarial therapy the higher the density ofmalarial parasites in blood We therefore model this declinein drug effectiveness by rescaling the density of merozoitesproduced per bursting parasitized erythrocyte P by the factor(1minus αs) where αs 1 implies no resistance that is the ACTis highly effective in eradicating the parasites If αs 0corresponds to maximum resistance the used ACT drug isleast effective in treating the infection +e converse of thesedescriptions applies to the drug-resistant P falciparumparasite strains+e equations that govern the rate of changeof the infected red blood cells and the merozoites take thefollowing formdMs

dt 1minus αs( 1113857PμysYs minus

βMsX

1 + cWminus

kmMsW

1 + aMsminus Ψ1 + μms + ζ( 1113857Ms

dMr

dt 1minus αr( 1113857PμyrYr + Ψ1Ms minus

δrβMrX

1 + cWminus

kmMrW

1 + aMrminus μmrMr

dGs

dt σsYs minus

kgWGs

1 + aGsminus Ψ2 + μgs + η1113872 1113873Gs

dGr

dt σrYr + Ψ2Gs minus

kgWGr

1 + aGrminus μgrGr

(3)

Antimalarial therapy increases the rate of elimination ofdrug-sensitive merozoites and gametocytes +is is repre-sented by the nonnegative enhancement parameters ζ and ηrespectively

Although the innate immunity is faster it is often limitedby the on and off rates in its response to invading pathogens[56 57] +e adaptive immunity on the contrary is veryslower at the beginning but lasts long enough to ensure noparasite growth in subsequent infections [27] We assume animmune system that is independent of the invading parasitestrain For purposes of simplicity we only consider theadaptive immune system which is mainly composed of theCD8 + T cells [58] We adopt the assumption that thebackground recruitment of immune cells is constant (at therate λw) Additionally the production of the immune cells isassumed to be boosted by the infective and infected cells(Gr Gs) (Mr Ms) and (Yr Ys) at constant rates hg hm andhy respectively Circulating gametocytes infective

Computational and Mathematical Methods in Medicine 3

merozoites and infected erythrocytes are removed phag-ocytotically by the immune cells at the rates kgW kmW andkyW respectively +e immune cells also get depletedthrough natural death at the rate μw +e equation for theimmune cells takes the following form

dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(4)

Following invasion by the merozoites the IRBCs eitherproduce merozoites or differentiate into gametocytes uponbursting +e total erythrocyte population at any time tdenoted by C(t) is therefore given by

C(t) X(t) + Ys(t) + Yr(t) (5)

Similarly the sum total of P falciparum parasitesdenoted by P(t) within the host at any time t is described bythe following equation

P(t) Ms(t) + Mr(t) + Gs(t) + Gr(t) (6)

+e above dynamics can be represented by the schematicdiagram in Figure 1 +e list of model variables and modelparameters is provided in Tables 1 and 2 respectively

21 Model Equations Based on the above model de-scriptions and schematic diagram shown in Figure 1 themodel in this paper consists of the following nonlinearsystem of ordinary differential equations

dX


βX

1 + cWMs + δrMr( 1113857 (7)

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs (8)

dYr

dtδrβXMr

1 + cWminus

kyYrW

1 + aYrminus μyrYr minus σrYr (9)

dMs


βMsX

1 + cWminus

kmMsW

1 + aMs

minus Ψ1 + μms + ζ( 1113857Ms

(10)

dMr


δrβMrX

1 + cW

minuskmMrW

1 + aMrminus μmrMr

(11)

dGs

dt σsYs minus

kgWGs

1 + aGsminus Ψ2 + μgs + η1113872 1113873Gs (12)

dGr


kgWGr

1 + aGrminus μgrGr (13)

dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(14)

subject to the following initial conditions

X(0)gt 0

Yi(0)ge 0

Mi(0)ge 0

Gi(0)ge 0

W(0)gt 0 for i s r

(15)

3 Model Analysis

31 Positivity and Uniqueness of Solutions +e consonancebetween a formulated epidemiological model and its bi-ological reality is key to its usefulness Given that all themodel parameters and variables are nonnegative it is onlysound that the model solutions be nonnegative at any futuretime tge 0 within a given biological space

Theorem 1 8e regionR8+ with solutions of system (7)ndash(14) is

positively invariant under the flow induced by system (7)ndash(14)

Proof We need to show that every trajectory from the re-gion R8

+ will always remain within it By contradictionassume existtlowast (where tlowast refers to time) in the interval [0infin)such that X(tlowast) 0 Xprime(tlowast)lt 0 but for 0lt tlt tlowast X(t)gt 0and Yi(t)gt 0 Mi(t)gt 0 Gi(t)gt 0 and Wi(t)gt 0 fori r s Notice that at t tlowast X(t) is declining from theoriginal zero value If such an X exists then it should satisfythe differential equation (7) +at is

dX

dt λx minus μxX t

lowast( 1113857minus

βX tlowast( )

1 + cW tlowast( )Ms tlowast

( 1113857 + δrMr tlowast

( 1113857( 1113857

λx gt 0

(16)

We arrive at a contradiction ie Xprime(tlowast)gt 0 +is showsthe nonexistence of such tlowast +is argument can be extendedto all the remaining seven variables (Ys Yr Ms Mr

Gs Gr W) +e process of verification is however simplerWe can follow the steps as presented in [59 60] Let the totalerythrocyte population C(t) evolve according to the fol-lowing formulation

dC

dtle λx minus μcC (17)

where μc min μx μys μyr1113966 1113967 Similarly the total density ofmalarial parasites P(t) is described bydP

dtleP 1minus αs( 1113857μysYs + 1minus αr( 1113857μyrYr1113966 1113967 + σsYs + σrYr minus μpP

(18)


where μp min μms μmr μgs μgr1113966 1113967+e solutions of equations (14) (17) and (18) are re-

spectively given as

W(t)leλwμw

+ W(0)minusλwμw

1113888 1113889eminusμwt

C(t)leλx

μc+ C(0)minus

λx

μc1113888 1113889e

minusμct

P(t)leσs 1113938

t

0 Ys(t)ΔIFdt + σr 1113938t

0 Yr(t)ΔIFdt

ΔIF

+ P(0)minusσs + σr( 1113857μp

1minus αs( 1113857μys + 1minus αr( 1113857μyr1113888 1113889

1ΔIF

(19)

where

ΔIF exp⎧⎨

⎩minus 1minus αs( 1113857μys 1113946t

0Ys(t)dt + 1minus αr( 1113857μyr 1113946

t

0Yr(t)dt1113888 1113889

minus 1113946t

0μpdt

⎫⎬

⎭

(20)HereC(0) X(0) + Ys(0) + Yr(0) and P(0) Ms(0) +

Mr(0) + Gs(0) + Gr(0) represent the initial total pop-ulations of erythrocytes and malarial parasites respectivelyWe observe that all the solutions of equations (14) (17) and(18) remain nonnegative for all future time tge 0 Moreoverthe total populations are bounded 0leC(t)lemax C(0)

(λxμc) 0leW(t)lemax W(0) λwμw1113864 1113865 and P(t)lemax(P(0) ((σs + σr)μp)((1minus αs)μys + (1minus αr)μyr)) +us all thestate variables of model system (7)ndash(14) and all their cor-responding solutions are nonnegative and bounded in thephase space φ where

φ ⎡⎣ X Ys Yr Ms Mr Gs Gr W( 1113857 isin R8+

C(t)lemax C(0)λx

μc1113896 1113897

W(t)lemax W(0)λwμw

1113896 1113897

P(t)lemax P(0)σs + σr( 1113857μp

1minus αs( 1113857μys + 1minus αr( 1113857μyr1113888 1113889⎤⎦

(21)

X

W

μmsMs μmrMr

μyrYr

μgrGr

μxX

μwW

kmMsW(1 + aMs)

(1 ndash αs)PμysYs (1 ndash αr)PμyrYrMs

Ys

Gs

Mr

Yr

Gr

kyYsW(1 + aYs)

kgGsW(1 + aGs)

kgGrW(1 + aGr)

(1(1 ndash ωs))μysYs

kmMrW(1 + aMr)

kyYrW(1 + aYr)

βsXMs βrXMr

λx

λw

σsYs σrYr

(η + μgs)Gs

Figure 1 A model flow diagram Drug-sensitive variables are shown in green colours while the drug-resistant variables are indicated inorange colours Non-strain-specific variables like susceptible RBCs and immune cells are shown in blue colour Solid lines indicate themovement of populations from one compartment to another Dotted lines show possible interactions between the different populations

Table 1 Description of the state variables of model system(11)ndash(18)

Variable Description

X Population of uninfectedunparasitized red bloodcells (erythrocytes)

YsPopulation of red blood cells infected by drug-

sensitive merozoites

YrPopulation of red blood cells infected by drug-

resistant merozoitesMs Population of drug-sensitive merozoitesMr Population of drug-resistant merozoitesGs Population of drug-sensitive gametocytesGr Population of drug-resistant gametocytesW Population of strain-independent immune cells


It is obvious that φ is twice continuously differentiablefunction +at is φi isin C

2 +is is because its componentsφi i 1 2 8 are rational functions of state variablesthat are also continuously differentiable functions Weconclude that the domain φ is positively invariant It istherefore feasible and biological meaningful to study modelsystem (7)ndash(14)

Theorem 2 8e model system (7)ndash(14) has a uniquesolution

Proof Let x (X Ys Yr Ms Mr Gs Gr W)T isin R8+ so that

x1 X and x2 Ys as presented in system (7)ndash(14) Simi-larly let g(x) (gi(x) i 1 8)T be a vector defined inR8

+ +e model system (7)ndash(14) can hence be written asdx

dt g(x) x(0) x0 (22)

where x [0infin)⟶ R8+ denotes a column vector of state

variables and g R8+⟶ R8

+ represents the right-hand side(RHS) of system (7)ndash(14) +e result is as follows

Lemma 1 8e function g is continuously differentiable in x

Proof All the terms in g are either linear polynomials orrational functions of nonvanishing polynomials Since thestate variables (X Ys Yr Ms Mr Gs Gr W) are all contin-uously differentiable functions of t all the elements ofvector g are continuously differentiable Moreover let L(x

n θ) x + θ(nminus x) 0le θle 1 By the mean value theorem

g(n)minus g(x)infin gprime(mnminus x)

infin (23)

where m isin L(xn θ) denotes the mean value point and gprimethe directional derivative of the function g at m However

gprime(mnminus x)

infin 11139448

i1gi(m) middot (nminus x)( 1113857ei

infin

le 11139448

i1gi(m)(

infin

nminus xinfin

(24)

where ei is the ith coordinate unit in R8+ We can clearly see

that all the partial derivatives of g are bounded and that thereexists a nonnegative U such that

1113944

8

i1gi(m)(

infinleU for all m isin L (25)

+erefore there exists Ugt 0 such that

g(n)minus g(x)infin leUnminus xinfin (26)

+is shows that the function g is Lipschitz continuousSince g is Lipschitz continuous model system (7)ndash(14) hasa unique solution by the uniqueness theorem of Picard[61]

32 Stability Analysis of the Parasite-Free Equilibrium Point(PFE) +e in-host malaria dynamics are investigated bystudying the behaviour of the model at different modelequilibrium points Knowledge on model equilibrium pointsis useful in deriving parameters that drive the infection todifferent stability points +e model system (7)ndash(14) hasa parasite-free equilibrium point E0 given by

E0 Xlowast Yslowast Yrlowast Mslowast Mrlowast Gslowast Grlowast Wlowast( 1113857

λx

μx

0 0 0 0 0 0λwμw

1113888 1113889

(27)

Using the next-generation operator method by van denDriessche andWatmough [62] andmatrix notations thereinwe obtain a nonsingular matrix Q showing the terms oftransitions from one compartment to the other and a non-negative matrix F of new infection terms as follows

Table 2 Description of model parameters

Parameter Descriptionλx +e rate of recruitment of red blood cellsωs Antimalarial treatment efficacyαs αr Parasite strain-specific fitness costλw Background recruitment rate of immune cellseg em ey Hill parameters in Gi Mi and Yi dynamics (i s r)

μx

Per capita natural mortality rate of unparasitizederythrocytes

μys Natural mortality rate of drug-sensitive IRBCsμyr Natural death rate of drug-resistant IRBCs

ζ η Rate of antimalarial eradication of merozoites andgametocytes respectively

μms Death rate of drug-sensitive merozoitesμmr Mortality rate of drug-resistant merozoites

μgsPer capita mortality rate of drug-sensitive

gametocytesμgr Mortality rate of drug-resistant gametocytesμw Natural mortality rate of immune cells (CD8+Tcells)

β +e rate of infection of susceptible RBCs by bloodfloating merozoites

σr σsRate of formation of gametocytes from the infected

RBCs

P Number of merozoites produced per dying infectedRBC

hy Immune cell proliferation rate due to IRBCs

hmImmune cell proliferation rate due to asexual

merozoiteshg Immune cell proliferation rate due to gametocytesky Phagocytosis rate of IRBCs by immune cellkm Phagocytosis rate of merozoites by immune cellkg Phagocytosis rate of gametocytes by immune cell

Ψ1Rate of development of resistance by drug-sensitive

merozoites


gametocytes

δrAccounts for the reduced fitness of the resistant

parasite strains

cEfficiency of immune effector to inhibit merozoite

infection1a Half-saturation constant for Y(t) M(t) and G(t)


F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0

minusP 1minus αs( 1113857μys 0 v3 0 0 0

0 minusP 1minus αr( 1113857μyr minusΨ1 v4 0 0minusσs 0 0 0 v5 00 minusσr 0 0 minusΨ2 v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)

where v1 ((kyλwμw) + σs + (μys1minusωs)) v2 ((kyλwμw) + σr + μyr) v4 (μmr + (kmλwμw) + (δrβλxμw(cλw +

μw)μx)) v3 (ζ + μms + Ψ1 + (kmλwμw) + (βλxμw(cλw +

μw)μx)) and v5 (η + μgs + (kgλwμw) + Ψ2) v6 (μgr +

(kgλwμw))

+e effective reproduction number RE of model system(7)ndash(14) associated with the parasite-free equilibrium isthe spectral radius of the next-generation matrix FQminus1where

Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0

P 1minus αs( 1113857μysv1v3

01v3

0 0 0

P 1minus αs( 1113857μysΨ1v1v3v4

P 1minus αr( 1113857μyrv2v4

01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that

RE ρ FQminus11113872 1113873 max Rs Rr1113864 1113865 (31)

where

Rs P 1minus αs( 1113857μysβλxμw

kyλwμw1113872 1113873 + σs + μys1minusωs1113872 11138731113872 1113873 ζ + μms + kmλwμw( 1113857 + Ψ1 + βλxμx cλw + μw( 1113857μx( 1113857( 1113857 cλw + μw( 1113857μx

Rr P 1minus αr( 1113857μyrδrβλxμw

kyλwμw1113872 1113873 + σr + μyr1113872 1113873 μmr + kmλwμw( 1113857 + δrβλxμw cλw + μw( 1113857μx( 1113857( 1113857 cλw + μw( 1113857μx

(32)

From equation (31) it is evident that in a multiple-strainP falciparummalaria infection the progression of the diseasedepends on the reproduction number of different parasitestrains If the threshold quantity Rs gtRr the drug-sensitiveparasite strains will dominate the drug-resistant strains andhence the driver of the infection To manage the infection inthis case the patient should be given antimalarials that caneradicate the drug-sensitive parasites Conversely if Rr gtRsthe infection is mainly driven by the drug-resistant parasitestrains In this scenario the used antimalarial drugs should behighly efficacious and effective enough to kill both the drug-resistant and drug-sensitive parasite strains in the blood of thehuman host +is result is quite instrumental in improving

antimalarial therapy for P falciparum infections +e bestantimalarials should be sufficient enough to eradicate bothparasite strains within the human host

Based on +eorem 2 in [63] we have the followinglemma

Lemma 2 8e parasite-free equilibrium point E0 is locallyasymptotically stable if RE lt 1 (Rs lt 1 andRr lt 1) and un-stable otherwise

+e Jacobian matrix associated with the in-host modelsystem (7)ndash(14) at E0 is given by


JE0

minusμx 0 0minusβλxμw

cλw + μw( 1113857μx

minusδrβλxμwcλw + μw( 1113857μx

0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0

0 0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 P 1minus αs( 1113857μys 0 minusv3 0 0 0 0

0 0 P 1minus αr( 1113857μyr Ψ1 minusv4 0 0 0

0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)

where the terms v1 v6 are as defined in (30) It is clear frommatrix (33) that the first four eigenvalues areminusμx (from column1) minusμw (from column 8) minus(μgr + (kgλwμw)) minusv6 (fromcolumn 7) and minus(η + μgs +(kgλwμw)) minusv5 (from column6) +ey are all negative +e remaining four eigenvalues areobtained from the roots of the following quartic equation

P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx

1minus αs( 1113857μys minus 1minus αr( 1113857μyrδr1113872 1113873(36)

p3 1K

v3 v2v4KminusP 1minus αr( 1113857μyrδrβλxμw1113872 11138731113960 1113961

minus1K

⎡⎣P 1minus αs( 1113857μysβλxμw v2 + v4( 1113857

+v1

Kv3v4( 1113857 + v2 v3 + v4( 11138571113858 1113859KminusP 1minus αr( 1113857μyrδrβλxμw⎤⎦

(37)

p4 v2v4KminusP 1minus αr( 1113857μyrδrβλxμw1113872 1113873 v1v3KminusP 1minus αs( 1113857μysβλxμw1113872 1113873

K

(38)

Due to complexity in the coefficients of the polynomial(34) we shall rely on the RouthndashHurwitz stability criterion[64] which provides sufficient condition for the existence ofthe roots of the given polynomial on the left half of the plane

Definition 1 +e solutions of the quartic equation (34)are negative or have negative real parts provided thatthe determinants of all Hurwitz matrices are positive [64]

Based on the RouthndashHurwitz criterion the system ofinequalities that describe the stability region E0 is presentedas follows

(i) p1 gt 0(ii) p3 gt 0(iii) p4 gt 0(iv) p1p2p3 gtp2

3 + p21p4

From (35) it is clear that p1 gt 0 Upon simplifying p2 in(36) we obtain

p2 v3v4 + v2v3 + v1v2 + v1v4 + v1v3 +λxμwβB1

K1113888 1113889

+ v2v4 +λxμwδrβB2

K1113888 1113889

(39)

where B1 minusP(1minus αs)μys andB2 minusP(1minus αr)μyr+us

p2 v3v4 + v2v3 + v1v2 + v1v4 + v1v3 1minusB1βλxμw

v1v3K1113890 1113891

+ v2v4 1minusB2δrβλxμw

v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859

+ v2v4 1minusRr1113858 1113859gt 0 if and only if Rs Rr lt 1

(40)


Similarly the expression for p4 can be rewritten asfollows

p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891

v1v3 1minusRs1113858 1113859v2v4 1minusRr1113858 1113859gt 0 if and only if Rs Rr lt 1

(41)

Lastly upon simplifying equation (37) we obtain

p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891

v1v3 v2 + v4( 1113857 1minusRs1113858 1113859 + v2v4 v1 + v3( 1113857 1minusRr1113858 1113859gt 0

if and only if Rs Rr lt 1

(42)

Since all the coefficients of the quartic equation (34) arenonnegative all its roots are therefore negative or havenegative real parts Hence the Jacobian matrix (33) hasnegative eigenvalues or eigenvalues with negative real parts ifand only if the effective reproduction number RE is less thanunity Equilibrium point E0 is therefore locally asymptoti-cally stable when RE lt 1 (when both Rs lt 1 and Rr lt 1) +isimplies that an effective antimalarial drug would cure thecostrain infected human host provided that the drug re-duces the effective reproduction number to less than 1

Lemma 2 shows that P falciparum malaria canbe eradicatedcontrolled within the human host if the

initial parasite and cell populations are within thebasin of attraction of the trivial equilibrium point E0To be certain to eradicatecontrol the infection irre-spective of the initial parasite and cell populations weneed to prove the global stability of the parasite-freeequilibrium point +is is presented in the followingsection

33 Global Asymptotic Stability Analysis of the Parasite-FreeEquilibrium Point Following the work by Kamgong andSallet [65] we begin by rewriting system (7)ndash(14) ina pseudotriangular form

_X1 D1(X) XminusXlowast1( 1113857 + D2(X)X2

_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)

where X1 is a vector representing the densities of non-infective population groups (unparasitized erythrocytesand immune cells) and X2 represents the densities ofinfectedinfective groups (infective P falciparum para-sites andor infected host cells) that are responsible fordisease transmissions For purposes of clarity and sim-plicity to the reader we shall represent (X1 0) with X1and (0 X2) with X2 in R8

+ times R8+ We assume the exis-

tence of a parasite-free equilibrium in φ Xlowast (Xlowast1 0)+us

X X1 X2( 1113857

X1 (X W)

X2 Ys Yr Ms Mr Gs Gr( 1113857

Xlowast1

λx

μx

λwμw

1113888 1113889

(44)

We analyze system (43) based on the assumption thatit is positively invariant and dissipative in φ Moreoverthe subsystem X1 is globally asymptotically stable at Xlowast1 onthe projection of φ on R8

+ +is implies that wheneverthere are no infective malarial parasites all cell pop-ulations will settle at the parasite-free equilibrium pointE0 Finally D2 in (43) is a Metzler matrix that is irre-ducible for any X isin φ We assume adequate interactionsbetween and among different parasites and cell com-partments in the model

+e matrices D1(X) and D2(X) are easily computedfrom subsystem _X1 in (43) so that we have


D1(X) minusμx 0

0 minusμw⎛⎝ ⎞⎠

D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)

We can easily see that the eigenvalues of matrix D1 areboth real and negative (minusμx lt 0 minusμw lt 0)+is shows that thesubsystem _X1 D1(X)(XminusXlowast1 ) + D2(X)X2 is globally

asymptotically stable at the trivial equilibrium Xlowast1 Addi-tionally from subsystem _X2 D3(X)X2 we obtain thefollowing matrix

D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0

P 1minus αs( 1113857μys 0 minusv3 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 minusv4 0 0

σs 0 0 0 minusv5 0

0 σr 0 0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)

Notice that all the off-diagonal entries of D3(X) arenonnegative (equal to or greater than zero) showing thatD3(X) is a Metzler matrix To show the global stability of theparasite-free equilibrium E0 we need to show that the squarematrix D3(X) in (46) is Metzler stable We therefore need toprove the following lemma

Lemma 3 Let K be a square Metzler matrix that is blockdecomposed

K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)

where K11 and K22 are square matrices 8e matrix K isMetzler stable if and only if K11 and K22 minusK21K

minus111K12 are

Metzler stable

Proof +ematrixK in Lemma 3 refers to D3(X) in our caseWe therefore let

K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0

P 1minus αs( 1113857μys 0 minusv3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0

βλxμwcλw + μw( 1113857μx

0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21

0 P 1minus αr( 1113857μyr Ψ1σs 0 0

0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)


Results from analytical computations based on Maplesoftware give

Kminus111

minusv3

v1v3 + Pβ αs minus 1( 1113857λxμwμys1113872 1113873 cλw + μw( 1113857μx( 11138570 minus

βλxμwv1v3 cλw + μw( 1113857μx + Pβ αs minus 1( 1113857λxμwμys

0 minus1v2

0

P αs minus 1( 1113857 cλw + μw( 1113857μxμysv1v3 cλw + μw( 1113857μx + Pβ αs minus 1( 1113857λxμwμys

0 minusv1

v1v3 + Pβ αs minus 1( 1113857λxμwμys1113872 1113873 cλw + μw( 1113857μx( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)

K22 minusK21Kminus111K12

minusv4 0 00 minusv5 00 Ψ2 minusv6

⎛⎜⎝ ⎞⎟⎠ (50)

where v4 (μmr + (kmλwμw) +(δrβλxμw(cλw + μw)μx))v5 (η + μgs + (kgλwμw) +Ψ2) and v6(μgr + (kgλwμw))

From equation (50) it is evident that all the diagonalelements of matrix K22 minusK21K

minus111K12 are negative and the

rest of the elements in the matrix are nonnegative +isshows that matrix K22 minusK21K

minus111K12 is Metzler stable and

the parasite-free equilibrium point E0 is globally asymp-totically stable in the biologically feasible region φ of modelsystem (7)ndash(14) Epidemiologically the above result impliesthat when there is no malaria infection different cell pop-ulations under consideration will stabilize at the parasite-free equilibrium However if there exists a P falciparuminfection then an appropriate control in forms of effectiveantimalarial drugs would be necessary to clear the parasitesfrom the human blood and restore the system to the stableparasite-free equilibrium state

34 Coexistence of Parasite-Persistent Equilibrium Point+e existence of a nontrivial equilibrium point representsa scenario in which the P falciparum parasites are presentwithin the host and the following conditions holdXlowast gt 0 Ylowasts ge 0 Ylowastr ge 0 Mlowasts ge 0 Mlowastr ge 0 Glowasts ge 0 Glowastr ge 0 andWlowast gt 0 Upon equating the right-hand side of system (7)ndash(14) to zero and solving for the state variables we obtainthe parasite-persistent equilibrium point E1 (Xlowast Ylowasts Ylowastr

Mlowasts Mlowastr Glowasts Glowastr Wlowast) where

Xlowast

1 + cWlowast( 1113857λx

β Mlowasts + δrMlowastr( 1113857 + 1 + cWlowast( 1113857μx

Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)

a minusa 1minusωs( 1113857σs + μys1113872 1113873 βMlowasts + βM

lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)

b minusβMlowasts minusa 1minusωs( 1113857λx minusωsσs + σs + μys1113872 1113873

minusWlowast 1minusωs( 1113857ky(βM

lowasts + βM

lowastr δr

+ cWlowastμx + μx)

(53)

c βMlowasts 1minusωs( 1113857λx gt 0 (54)

a minusa σ2 + μyr1113872 1113873 βMlowasts + βM

lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)

b βMlowastr δr aλx minus σ2 minus μyr1113872 1113873minusW

lowastky(βM

lowasts

+ βMlowastr δr + cW

lowastμx + μx)minus σ2 + μyr1113872 1113873

middot βMlowasts + cW

lowastμx + μx( 1113857

(56)

c βMlowastr δrλx gt 0 (57)

Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0

b1 aσ1Ylowasts minusW

lowastkg minus ηminus μg1 minusΨ2

c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0

b2 aG1Ψ2 + aσ2Ylowastr minusW

lowastkg minus μg2

c2 G1Ψ2 + σ2Ylowastr gt 0

(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)


a3 minus( aβMlowastr δr ζ + μms + Ψ1( 1113857 + acW

lowastμmsμx + aμmsμx

+ aβP 1minus αs( 1113857μysYlowasts + Ψ1 a cW

lowast+ 1( 1113857μx + β( 1113857

+ acζWlowastμx + aβλx + aζμx + βζ + βW

lowastkm + βμms)

(62)

b3 minusβMlowastr δr a αs minus 1( 1113857PY

lowasts μys + ζ + W

lowastkm + μms + Ψ11113872 1113873

minus αs minus 1( 1113857βPYlowasts μys minus βλx minus cW

lowast+ 1( 1113857μx(a αs minus 1( 1113857PY

lowasts μys

+ ζ + Wlowastkm + μms + Ψ1)

(63)

c3 P 1minus αs( 1113857Ylowasts μys βM

lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)

a4 minus(aβMlowasts Ψ1δr + μmr( 1113857 + μmr a cW

lowast+ 1( 1113857μx + βδr( 1113857

+ a 1minus αr( 1113857βPY2δrμy2 + βWlowastkmδr)

(65)

b4 aβMlowast2s Ψ1 + M

lowasts (minusβ aδrλx + μmr( 1113857 + a 1minus αr( 1113857βPY2μy2

+ Ψ1 a cWlowast

+ 1( 1113857μx + βδr( 1113857) + 1minus αr( 1113857PY2μy2middot a cW

lowast+ 1( 1113857μx + βδr( 1113857minusW

lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)

c4 βMlowast2s Ψ1 + M

lowasts ( 1minus αr( 1113857βPY2μy2 + βδrλx

+ Ψ1 cWlowast

+ 1( 1113857μx) + 1minus αr( 1113857PY2 cWlowast

+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ

μwΔminus hg Glowasts + Glowastr( 1113857 + hm Mlowasts + Mlowastr( 1113857 + hy Ylowasts + Ylowastr( 11138571113872 1113873

(68)

where Δ (eg + Glowasts + Glowastr )(em + Mlowasts + Mlowastr )(ey + Ylowasts + Ylowastr )Using Descartesrsquo ldquoRule of Signsrdquo [66] it is evident that

irrespective of the sign of b in (53) b in (56) b1 in (59) b2 in(60) b3 in (63) and b4 in (66) the state variablesYlowasts Ylowastr Mlowasts Mlowastr Glowasts andGlowastr can only have one real positivesolution +is shows that the model system (7)ndash(14) hasa unique parasite-persistent equilibrium point E1

35 Stability of the Coexistence of Parasite-Persistent Equi-librium Point Here we shall prove that the coexistence ofparasite-persistent equilibrium E1 is locally asymptoticallystable when RE gt 1 (orwhen Rs gt 1 andRr gt 1) We shallfollow the methodology by Esteva and Vargus presented in[67] which is based on the Krasnoselskii technique [68]+ismethodology requires that we prove that the linearization ofsystem (7)ndash(14) about the coexistence of parasite-persistentequilibrium does not have a solution of the form

S(t) S0eξt

(69)

where S0 (S1 S2 S7) (Si ξ) isin C and the real part of ξis nonnegative (Re(ξ)ge 0) Note that C is a set of complexnumbers

Next we substitute a solution of the form (69) into thelinearized system (7)ndash(14) about the coexistence of parasite-persistent equilibrium We obtain

ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +

β Clowast minusYs minusYr( 1113857

1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2

+δrβ Clowast minusYs minusYr( 1113857

1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW

1 + aMs+β Clowast minusYs minusYr( 1113857

1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2

minusδrβ Clowast minusYs minusYr( 1113857

1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5

ξS6 σrS2 + Ψ2S4 minuskgW

1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)


where (Clowast minusYs minusYr) X k1 (Ψ1 + μms + ζ) andk2 (Ψ2 + μgs + η)

Upon simplifying the equations in (70) we obtain

1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +

δrβ Clowast minusYs minusYr( 1113857

1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs

1 + cW+ P 1minus αs( 1113857μysS1 +

βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857

kgW + μgrξ⎡⎣ ⎤⎦S6

1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)

whereΔ1 βMs 1 + aYs( 1113857 1minusωs( 1113857 + kyW 1minusωs( 1113857(1 + cW)

+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)

Δ3 1 + aMs( 1113857 β Clowast minusYs minusYr( 1113857( 1113857 + kmW(1 + cW)

+ k1 1 + aMs( 1113857(1 + cW)

Δ4 1 + aMr( 1113857 δrβ Clowast minusYs minusYr( 1113857( 1113857

+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)

Separating the negative terms we obtain the followingsystem

1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0

P 1minus αs( 1113857μys 0βClowast

1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)

Note that Xlowast Clowast minusYlowasts minusYlowastr and all the elements in thesquare matrixH are nonnegative +e coordinates of E1 are allpositive and the jth coordinate of the vector H(S) is describedby the notation H(S)j for j 1 7 Additionally theequilibrium E1 (Ylowasts Ylowastr Mlowasts Mlowastr Glowasts Glowastr Wlowast) satisfiesE1 HE1 If we assume for example that system (73) has

a solution of the form S then there exists a small positive realnumber ϵ such that |S|le ϵE1 where |S| (|S1| |S2| |S7|)Note also that || is a norm in the field of complex numbers

Next we show that Re(ξ)lt 0 To do so we apply proofby contradiction We let ξ 0 and ξ ne 0 For the case whenξ 0 the determinant (nabla) of (70) is given by

nabla v5v6μw v2v4 cλx + μw( 1113857μx + Pβ 1minus αr( 1113857λxμwμyr1113966 1113967 v1v3 cλx + μw( 1113857μx + Pβ 1minus αs( 1113857λxμwμys1113966 1113967

cλx + μw( 11138572μ2x

(76)

where the positive terms v1 v6 are as defined in matrix(29)

It is clear that the above determinant is nonnegative(nablagt 0) Consequently the system (70) can only have the trivialsolution S (0 0 0 0 0 0 λwμw) On the contrary for ξ ne 0we assume Re(ξ)ge 0 and define F(ξ) min|1 + Fj(ξ)|

j 1 2 7 +is implies that F(ξ)gt 1 and ϵF(ξ)lt ϵ +eminimality of ϵ means that |S|gt ϵF(ξ)E1 While consideringthe nonnegativity property ofH if we assume the norms on thetwo sides of (73) we shall have

F(ξ)|S|leH|S|le εHE1 εE1 (77)

+is implies that |S|le ϵF(ξ)E1 le ϵE1 which is a con-tradiction +erefore Re(ξ)lt 0 and E1 is locally asymp-totically stable when RE gt 1

4 Numerical Simulations

41 Boundary Equilibrium Points In this section we showby means of numerical simulation the existence and stabilityof a positive parasite-persistent equilibrium point that in-volves only one of the parasite strains under study

411 Drug-Sensitive-Only Persistent Equilibrium Point Es+is is an equilibrium point where only the drug-sensitiveparasite strains are present in the infected human host +at

is the populations Yr Mr Gr 0 +is steady state isonly feasible if no resistant parasites emerge from infectedred blood cells and the use of antimalarial treatment does notlead to resistance development that is Ψ1 Ψ2 0 +eoriginal model (7)ndash(14) is thus reduced to

dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW

1 + aMsminus μms + ζ( 1113857Ms

dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857

Ms + em1113896 1113897Wminus μwW

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)

Numerically this equilibrium point is illustrated asshown in Figure 2


412 Drug-Resistant-Only Persistent Equilibrium Point ErIn this case the population of the drug-sensitive parasitestrains declines to zero as the density of the resistant strainsgrows and stabilizes at an optimal population size +is isalso illustrated numerically as shown in Figure 3

42 Within-Host Competition between Parasite StrainsWe investigate the competitive exclusion principle by sim-ulating the model system (7)ndash(14) under different values ofthe threshold quantities Rs and Rr in (31) Model (7)ndash(14) issimulated so that Rs 4022 and Rr 03131 and we achievea convergence to the drug-sensitive-only endemic equilib-rium point Es as shown in Figure 4(a) Again using theparameter values in Table 3 with Ψ1 09 and (Rs 0022Rr 30098) the solutions of Ys and Yr converge to the drug-resistant-only endemic equilibrium point Er (Figure 4(b))

Provided that both Rs and Rr are greater than 1 (as shownin Figure 4(c)) the parasite-infected red blood cells remainpersistent in the host +is implies that the merozoites (bothdrug-sensitive and drug-resistant) continue to multiply in theabsence of antimalarial therapy ωs 0 or in the presence ofineffective antimalarial drugs Similar results are observed inthe dynamics of merozoites (Ms and Mr) as shown inFigure 5 It should be noted that the dominant merozoitestrains are likely to drive the infection under these conditionsAs the density of one strain increases the population of theother strain is likely to decrease due to a phenomenon knownas competitive exclusion principle+emost fit parasite strainsurvives as the weaker competitor dies out as shown in

Figure 5(a) Both drug-sensitive and drug-resistant mero-zoites would remain persistent if poor-quality antimalarialdrugs are administered to P falciparum malaria patients+us in the absence of efficacious antimalarial drugs likeACTs with the potential to eradicate resistant merozoites weare likely to experience an exponential growth in the densityof drug-resistant merozoites as displayed in Figure 5(b) +ismay lead to severe malaria and eventual death of the patient

+e bifurcation analysis of both scenarios is presented inFigure 6 (with and without competition between the parasitestrains) When there is competition between the parasitestrains as shown in Figure 6(a) we observe that the strainwith a higher threshold quantity R0 would exclude the otherstrain A decrease in the population of the drug-sensitivestrain would pave way for a surge in the population of thedrug-resistant strains and vice versa +is is despite the factthat some drug-resistant strains emerge from the drug-sensitive strains as a result of mutation [77] In Figure 6(b)we observe coexistence of the strains that do not competewith each other Like the resistance strains the sensitivestrains are only present when their threshold quantity Rs isgreater than unity Both strains are however present whenRr gt 1 and Rs gt 1 Additionally when Rr lt 1 and Rs lt 1 wearrive at the parasite-free equilibrium (PFE) point as shownin Figures 6(a) and 6(b)

43 Antimalarial Drug Effects and Parasite Clearance+e effects of antimalarial drug treatment are monitored byestablishing first and foremost that

zRs

zωs

minusβμ1μ2P 1minus αs( 1113857μwλx

1minusωs( 11138572μx cλw + μw( 1113857 kyλwμw1113872 1113873 + μ2 1minusω1( 1113857( 1113857 + σs1113872 1113873

2ζ + kmλwμw( 1113857 + μms + βλxμ2x cλw + μw( 1113857( 1113857 + Ψ1( 1113857

lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11

Drug-sensitive infected erythrocytes YsDrug-resistant infected erythrocytes YrGametocytes Gs

(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)

Drug-resistant infected erythrocytes YrHealthyunparasitized erythrocytes X

(b)

Figure 2 Simulations of model system (11)ndash(18) showing the existence of drug-sensitive-only equilibrium point All parameter values are aspresented in Table 3


+us Rs is a decreasing function of ωs (the efficacy of theantimalarial drug used) +erefore using a highly efficientantimalarial drug could lead to a scenario where Rs lt 1 andRr lt 1 (disease-free state shown in Figure 7(c)) InFigure 7(a) model system (7)ndash(14) is simulated by varyingthe efficacy of the antimalarial drug ωs and other modelparameters chosen such that Rr 3221 and Rs 2221 +ehigher the efficacy of the used antimalarial the lower thedensity of infected erythrocytes +us governments andministry of health officers should only roll out or permit the

administration of antimalarials or ACTs that can eradicate(totally) both the drug-resistant and the drug-sensitivestrains of P falciparum parasites

+e rate of development of resistance by the drug-sensitive merozoites Ψ1 is shown to have very minimalimpact on the dynamics of infected red blood cells Yr as longas Rs gt 1 and Rr gt 1 (Figure 7(b)) Nevertheless analyticalresults indicate that the higher the rate of development ofresistance the lower the severity of future malaria infections+is is presented as

zRs

zΨ1 minus

βμ1P 1minus αs( 1113857μwλx

μx cλw + μw( 1113857 kyλwμw1113872 1113873 + μ2 1minusω1( 1113857( 1113857 + σs1113872 1113873 ζ + kmλwμw( 1113857 + μms + βλxμ2x cλw + μw( 1113857( 1113857 + Ψ1( 11138572 lt 0 (80)

Other parameters that have direct negative impacts onthe progression of malaria infection are the efficacy of the

immune effectors c and the rate of therapeutic eliminationof drug-sensitive merozoites ζ

zRs

zζ minus



zRr

zc minus

βμ2P 1minus αr( 1113857δrλwμ3wλx kmλw + μmrμw( 1113857

μx kyλw + μw μ2 + σr( 11138571113872 1113873 cλw + μw( 1113857 kmλw + μmrμw( 1113857 + βδrμwλxμ2x( 11138572 lt 0 (82)

Further simulations based on contour plots (see [78] fortheory on contour plots) are used to ascertain the relationaleffects of selected pairs of model parameters on the diseasethreshold quantities Rs and Rr In Figure 8(a) both β and μwincrease the reproduction number due to drug-sensitive Pfalciparum parasite strains A direct relationship exists betweenthe two parameters the higher the decay rate of the immunecells the higher the rate of infection of healthy erythrocytes

In Figure 8(b) we observe the least increase in Rs withrespect to an increase in ωs relative to μys Antimalarialtherapy is shown to be very effective in reducing the severityof P falciparum infection Conversely the number ofmerozoites produced per dying blood schizont P is shownin Figure 8(c) to have a very high positive impact on Rs andhence on the severity of malaria infection due to drug-sensitive parasite strains Clinical control should target and

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3

Drug-resistant infected erythrocytes YrDrug-sensitive infected erythrocytes Ys

(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)

Parasitized erythrocytes YsDrug-resistant merozoites Mr

(b)

Figure 3 Simulations of model system (11)ndash(18) showing the existence of drug-resistant-only equilibrium point All parameter values are aspresented in Table 3


eradicate infected red blood cells to diminish the erythro-cytic cycles of infections

We observe in Figure 9(b) that the rate at which mer-ozoites develop resistance due to treatment failure has noresultant effects on the rate of formation of gametocytes thatundergo sexual reproduction within the mosquito vector+e higher the value of Rr the higher the cost of resistanceas shown in Figure 9(a) +e higher the density of drug-resistant parasite strains the higher the level of resistanceand hence the cost of disease control Unfortunately highlyeffective antimalarial drugs (such as ACTs) that can eradicateboth parasite strains are slightly expensive in severalP falciparum malaria-endemic regions [79] Like the pa-rameter P the parasite infection rate β is shown to have

a direct positive effect on the threshold quantityRr (Figure 9(c))due to drug-resistant parasite strains Effective antimalarialsshould hence target new cell infections and eliminate re-crudescence (by killing already infected erythrocytes)

5 Effects of Multiple-Strain Infection andFitness Cost on Parasite Clearance

Numerous studies [27 80] have suggested the negative im-pacts of drug resistance on the fitness and ability of theparasite to dominate the P falciparum infection Resistance toantimalarial drugs imposes fitness cost on the drug-resistantparasite +e drug-resistant parasite strains are thought to

08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)

Infected erythrocytes YsInfected erythrocytes Yr

(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1

Drug-sensitive infected erythrocytes YsDrug-resistant infected erythrocytes Yr

(c)

Figure 4 Simulations of model system (11)ndash(18)+e figures show the dynamics of drug-sensitive and drug-resistant infected red blood cellsunder different conditions of the threshold values Rs and Rr In Figure 4a Rs gtRr In Figure 4b Rr gtRs Ψ1 0 and all other parametervalues are as presented in Table 3


experience impaired growth within the human host [29] +ecost of resistance is further exacerbated due to the compe-tition between parasite strains within an infected human hostIn Figure 10(a) the area under the curve for the drug-resistantstrain or the number of infected erythrocytes is lower thanthat of the drug-sensitive strains However in a multiple-strain infection (Figure 10(b)) the area difference is muchbigger +is implies that competition between the parasitestrains within the human host could result in elimination ofone of the parasite strains provided that both Rs and Rr areless than unity

+e presence of multiple strains of P falciparum para-sites is likely to complicate and worsen the severity ofmalaria disease infection in humans Figures 11 and 12 showthe simulated model (7)ndash(14) for single- and multiple-straininfections in the absence of preexisting immunity andantimalarial drugs +e persistence of gametocytes inFigures 11(b) and 12(b) is consistent with the actual ob-servations of human malaria infection in the absence ofantimalarial therapy [81] Acquired immunity is shown inFigure 11(c) to increase and eventually level-off at higherlevels to contain future infections

30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5

Drug-sensitive merozoites MsDrug-resistant merozoites Mr

(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)

Figure 5 Simulations of model system (11)ndash(18) +e figures show the dynamics of the merozoites under different conditions of thethreshold values Rs and Rr Competitive exclusion among the parasite strains is shown in (a) In (b) both parasite strains coexists andRr gtRs Ψ1 0 Other parameter values are available in Table 3

Table 3 Baseline values and range for parameters of model (11)ndash(18)

Parameter Value Range Units Sourceλx 3 times 103 (3 times 103 minus 3 times 108) Cellsμlminus1day [69]λw 30 (10ndash40) Cellsμlday [70]ωs 05 (0-1) Unitless Assumedαs 04 (01ndash1) Unitless Assumedαr 02 (001ndash1) Unitless Assumedeg em ey 104 (103minus105) Unitless [71]μx 1120 (005ndash01) dayminus1 [72]μys 05 (03ndash08) dayminus1 [73]μyr 03 (03ndash08) dayminus1 Assumedμms μmr 48 (46ndash50) dayminus1 [69]μgs μgr 00625 (005ndash01) dayminus1 [74]μw 005 (002ndash008) dayminus1 [74]δr 07 (001ndash099) Unitless Assumedζ η 05 (0-1) dayminus1 [73]P 16 (15ndash20) Erythrocytesday [34]β 65 times 10minus7 48 times 10minus7ndash68 times 10minus7 Merozoitesday [75]σr σs 002 (001ndash003) dayminus1 [75]hy hm hg 005 (001ndash008) mmminus3day [70]ky km kg 0000001 (0001ndash09) dayminus1 [51]Ψ1 02 (001ndash22) dayminus1 AssumedΨ2 001 (0001ndash01) dayminus1 Assumedδr 03 (0-1) Unitless Assumedc 05 (0-1) Immune cellμl Assumed1a 02 (0-1) Unitless [76]


16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4

Infected erythrocytes YrInfected erythrocytes Ys

0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)

Figure 7 +e effect of varying the efficacy of antimalarial drug used ωs and the rate of development of resistance by the drug-sensitivemerozoitesΨ1 on the density of infected erythrocytes (Ys Yr)+e value of ωs ranges from 0 to 1+e rest of the parameter values are availablein Table 3 Figure (c) shows that in the absence of highly effective ACTs drug-resistant parasite would take a longer time to eradicate

5

5

1

1

Rr

Rs

PFE

Drug-resistantstrains only

Drug-sensitivestrains only

(a)

5

1

Rr

51Rs

PFE



Coexistanceof both strains

(b)

Figure 6 Bifurcation diagrams showing competitive exclusion (a) and coexistence equilibrium (b) for the drug-sensitive and drug-resistantP falciparum parasite strains under different values of threshold quantities Rs and Rr Both parasite strains coexists when Rs gt 1 and Rr gt 1(see part (b))


Although the aspect of timing is key in these multiple-strain infections we assumed here that the two strains areintroduced at the same time In the long run it is evident inFigures 10 and 12 that the sensitive strain overtakes theresistant strain We argue that this could be as a result ofstrain-specific adaptive responses that symmetrically affectthe sensitive parasites

Unlike single-strain P falciparum parasite infections dataon multiple-strain infections are not readily available Nev-ertheless amultiple-strain infection (drug-sensitive and drug-resistant) as presented in this paper is biologically reasonableand consistent with that of P Chabaudi described in [82]

51 Sensitivity Analysis In this paper the primary modeloutput of interest for the sensitivity analysis is the infectederythrocytes (Ys Yr) However the effective reproductionnumber RE is a threshold quantity which represents onoverage the number of secondary infected erythrocytes dueto merozoite invasions We can therefore measure thesensitivity indices of the effective reproduction number of

model system (7)ndash(14) relative to model parameters Forexample the sensitivity of RE relative to the parameter Ψ1 isgiven by the following formulation

ΥΨ1 zRE

zΨ1timesΨ1RE

(83)

Using the parameter values in Table 3 the expressions forsensitivity for all the parameters in RE are evaluated andpresented in Table 4 +e higher the numerical value of thesensitivity index (SI) the greater the variational impact of theparameter on the disease progression A parameter witha negative index decreases the model RE when they are in-creased On the other hand a parameter with a positive indexwould generate a proportional increase in RE when they aremagnified Results shown in Table 4 indicate that the rate ofinfection of healthy erythrocytes by the merozoites β thedensity of merozoites generated from each of the burstingschizonts P the efficacy of antimalarial drug used ωs and therate at which drug-sensitive merozoites develop resistance Ψ1are the four most influential parameters in determining thedisease dynamics as presented in model system (7)ndash(14)

0200Contour plot of Rs as a fuction of β and μw

0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)

Contour plot of Rs as a fuction of ωs and μys00200

00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)

Contour plot of Rs as a fuction of ρ and γ10

09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)

Figure 8 Contour plot of Rs as a function of (a) β and μw (b) ωs and μys (c) P and c


Results from sensitivity analysis emphasize the use ofhighly efficacious antimalarial drugs such as ACTs inmalaria-endemic regions +is would mitigate the manycases of malaria in the region and further help to reduceemerging cases of parasite resistance to existing therapiesDrugs with a higher parasite clearance rate would greatlyreduce resistance which is associated with longer parasiteexposure to antimalarial drugs It is imperative thereforethat governments and ministry of health personnel inmalaria-endemic countries enforce the use of efficient an-timalarial drugs that not only cure infected malaria patientsbut also eliminate the chance of P falciparum parasites todevelop resistance to existing therapy

6 Conclusion

In this paper a deterministic model of multiple-strain Pfalciparum malaria infection has been formulated andanalysed +e parasite strains are categorized as either drug-

sensitive or drug-resistant +e infected erythrocytes and themalaria gametocytes are similarly grouped according to thestrain of the parasite responsible for their existence +eimmune cells are incorporated to reduce the invasivecharacteristic of the malaria merozoites Antimalarialtherapy is applied to the model but only targets red bloodcells infected with drug-sensitive merozoites Based on thenext-generation matrix method we computed the effectivereproduction number RE of the formulated model Based onRE it is evident that the success of P falciparum infection inthe presence of multiple-parasite strains is directly de-pendent on the ability of the individual parasite strains todrive the infection +e parasite strain with a higherthreshold value R0 is likely to dominate the infectionPrescribed antimalarial drugs should therefore be effectiveenough to eradicate both drug-sensitive and drug-resistantparasite strains in vivo Linearization of the model at theparasite-free equilibrium reveals the local asymptotic sta-bility of the trivial equilibrium point

009

008

007

006

005

004

003

002

001

007Contour plot of Rr as a fuction of αr and μyr

006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)

Contour plot of Rr as a fuction of ψ1 and σr009

008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)

Contour plot of Rr as a fuction of ρ and β09

08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)

Figure 9 Contour plot of Rr as a function of (a) αr and μyr (b) Ψ1 and σr (c) P and β


1e7

0 25 50 75 100 125

Drug-sensitive infected erythrocytes Ys

Drug-resistant infected erythrocytes Yr

150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)

Figure 10 Dynamics of drug-sensitive (blue) and drug-resistant (orange) strains in a single infection (a) and in a multiple infection (b) ina naive human-host with no malaria therapy (ωs 0) +e density of the resistant strain is lower than that of drug-sensitive strain forRs 2123gt 1 and Rr 1912gt 1 in a multiple-strain P falciparum infection +e rest of the parameter values are as displayed in Table 3

10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

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asiti

zed

eryt

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(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

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05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)

Figure 11 Dynamics of infected erythrocytes gametocytes and the immune cells with a single-strain P falciparum infection Here we donot have preexisting immunity +e rest of the parameter values are as displayed in Table 3


By rewriting the model in the pseudotriangular form theparasite-free equilibrium is also shown to be globally as-ymptotically stable Although the parasite-persistent equi-librium exists its expression based on a single-modelvariable proved to be mathematically intractable +e use ofantimalarial treatment may eradicate one parasite strain sothat we arrive at either a drug-sensitive-only persistentequilibrium point or a drug-resistant-only persistent equi-librium point

To assess the impacts of the different parasite strains todisease dynamics the model is simulated for different valuesof the threshold quantities Rs andRrWe observed that whenRr gt 1 and Rs gt 1 then both parasite strains are persistentand the infection becomes severe If Rr gt 1 and Rs lt 1 thenthe drug-sensitive parasites would decline to zero as thedrug-resistant strains continue to multiply and remainpersistent increasing the severity of infections On the otherhand if Rs gt 1 and Rr lt 1 then the drug-resistant parasite

strains would be eradicated Moreover provided that thethreshold quantities Rs and Rr are less than unity the useof an efficacious antimalarial drug would help eradicateP falciparum infection

+e efficacy of antimalarial drug is shown to have directnegative impact on the density of infected red blood cells+e higher the efficacy of administered antimalarial drugthe lower the population of infective merozoites and thesmaller the density of infected erythrocytes +is ensuresprompt recovery from malaria infections +is result isconsistent with that in [72 83] +e efficacy of antimalarialdrug is however shown to have least effect on the populationof drug-resistant infected erythrocytes +e rate of devel-opment of resistance by drug-sensitive parasites is alsoshown to drive the infection due to resistant parasite strainsUsing contour plots and results from sensitivity analysis weobserve that the efficacy of antimalarial drug used ωs thedensity of blood floating merozoites produced per infected

10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Multiple strainsSingle strain

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)

Figure 12 Within-human dynamics of single- and multiple-strain dynamics of infected erythrocytes gametocytes and the immune cells inthe absence preexisting immunity and with no antimalarial treatment (ωs 0) +e rest of the model parameter values are in Table 3


erythrocyte P the rate of development of resistance Ψ1 andthe rate of infection by merozoites β are the most importantparameters in the disease dynamics and control

Finally although the drug-resistant strain is shown to beless fit the presence of both strains in the human host hasa huge impact on the cost and success of antimalarialtreatment To reduce the emergence of resistant strains it isvital that only effective antimalarial drugs are administeredto patients in hospitals especially in malaria-endemic re-gions To improve malaria therapy and reduce cases ofparasite resistance to existing therapy our results call forregular and strict surveillance on antimalarial drugs inclinics and hospitals in malaria-endemic countries

Data Availability

All data used in this study are included in this publishedarticle

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this article

Authorsrsquo Contributions

All authors contributed to all sections of this manuscript

Acknowledgments

+e authors are thankful to the anonymous referees for theirconstructive comments+e authors would also like to thankthe Strathmore Institute of Mathematical Sciences for itssupport in the production of this manuscript +e authorsacknowledge with gratitude the financial support from theGerman Academic Exchange Service (DAAD) (ST32-PKZ

91711149) and the Kenya National Research Fund (NRF-Kenya) (NRF-Phd Grant Titus OO) in the production ofthis manuscript

References

[1] A M Dondorp S Yeung L White et al ldquoArtemisinin re-sistance current status and scenarios for containmentrdquoNature Reviews Microbiology vol 8 no 4 pp 272ndash280 2010

[2] R J Maude W Pontavornpinyo S Saralamba et al ldquo+e lastman standing is the most resistant eliminating artemisinin-resistant malaria in CambodiardquoMalaria Journal vol 8 no 1p 31 2009

[3] A B S Sidhu D Verdier-Pinard and D A FidockldquoChloroquine resistance in plasmodium falciparum malariaparasites conferred by pfcrt mutationsrdquo Science vol 298no 5591 pp 210ndash213 2002

[4] T E Wellems and C V Plowe ldquoChloroquinemdashresistantmalariardquo Journal of Infectious Diseases vol 184 no 6pp 770ndash776 2001

[5] D L Smith E Y Klein F EMcKenzie and R LaxminarayanldquoProspective strategies to delay the evolution of anti-malarialdrug resistance weighing the uncertaintyrdquo Malaria Journalvol 9 no 217 2010

[6] R W Snow J-F Trape and K Marsh ldquo+e past present andfuture of childhood malaria mortality in africardquo TRENDS inParasitology vol 17 no 12 pp 593ndash597 2001

[7] N J White ldquoDelaying antimalarial drug resistance withcombination chemotherapyrdquo Parassitologia vol 41 no 1ndash3pp 301ndash308 1999

[8] WHO A Global Strategy for Malaria Control World HealthOrganization Geneva Switzerland 1993

[9] Y Kim and K A Schneider ldquoEvolution of drug resistance inmalaria parasite populationsrdquo Nature Education Knowledgevol 4 no 8 pp 6ndash16 2013

[10] A O Talisuna P Bloland and U DrsquoAlessandro ldquoHistorydynamics and public health importance of malaria parasiteresistancerdquo Clinical Microbiology Reviews vol 17 no 1pp 235ndash254 2004

[11] N J White and W Pongtavornpinyo ldquo+e de novo selectionof drug-resistant malaria parasitesrdquo Proceedings of the RoyalSociety of London Series B Biological Sciences vol 270no 1514 pp 545ndash554 2003

[12] J-F Trape G Pison M-P Preziosi et al ldquoImpact of chlo-roquine resistance on malaria mortalityrdquo Comptes Rendus delrsquoAcademie des SciencesmdashSeries IIImdashSciences de la Vievol 321 no 8 pp 689ndash697 1998

[13] WHO Guidelines for the Treatment of Malaria World HealthOrganization Geneva Switzerland 2018

[14] J K Baird M F S Nalim H Basri et al ldquoSurvey of resistanceto chloroquine by plasmodium vivax in Indonesiardquo Trans-actions of the Royal Society of Tropical Medicine and Hygienevol 90 no 4 pp 409ndash411 1996

[15] P E +ompson and W Leslie ldquoAntimalarial agents chem-istry and phamacologyrdquo in Medicinal ChemistryG de Stevens Ed Academic Press New York NY USA1972

[16] T Harinasuta P Suntharasamai and C Viravan ldquoChloro-quine-resistant falciparum malaria in +ailandrdquo 8e Lancetvol 286 no 7414 pp 657ndash660 1965

[17] S Fogh S Jepsen and P Effersoslashe ldquoChloroquine-resistantplasmodium falciparum malaria in Kenyardquo Transactions ofthe Royal Society of Tropical Medicine and Hygiene vol 73no 2 pp 228-229 1979

Table 4 Sensitivity indices of RE relative to the model parameters

Parameter SIβ +09988P +10000ωs minus087513λx +07199μx minus00016ky minus002701σs minus07619c minus03333μmr minus0433μms minus052123μyr minus0232Ψ1 minus077534μys minus0492λw minus03471ζ minus00041δr +00023km minus00020σr minus0541872αr minus01111αs minus009891μw 03716


[18] E Onori ldquo+e problem of plasmodium falciparum drugresistance in africa south of the saharardquo Bulletin of the WorldHealth Organization vol 62 pp 55ndash62 1984

[19] J M Ekue A M Ulrich and E K Njelesani ldquoPlasmodiummalaria resistant to chloroquine in a Zambian living inZambiardquo BMJ vol 286 no 6374 pp 1315-1316 1983

[20] D Overbosch A W van den Wall Bake P C Stuiver andH J van der Kaay ldquoChloroquine-resistant falciparum malariafrom Malawirdquo Tropical and Geographical Medicine vol 36no 1 pp 71-72 1984

[21] W H Wernsdorfer and D Payne ldquo+e dynamics of drugresistance in plasmodium falciparumrdquo Pharmacology amp8erapeutics vol 50 no 1 pp 95ndash121 1991

[22] B L Bredenkamp B L Sharp S D MthembuD N Durrheim and K I Barnes ldquoFailure of sulphadoxine-pyrimethamine in treating plasmodium falciparummalaria inKwaZulu-Natalrdquo South African Medical JournalSuid-Afri-kaanse tydskrif vir geneeskunde vol 91 no 11 pp 970ndash9722001

[23] P B Bloland E M Lackritz P N Kazembe J B O WereR Steketee and C C Campbell ldquoBeyond chloroquine im-plications of drug resistance for evaluating malaria therapyefficacy and treatment policy in africardquo Journal of InfectiousDiseases vol 167 no 4 pp 932ndash937 1993

[24] S J Foote and A F Cowman ldquo+e mode of action and themechanism of resistance to antimalarial drugsrdquo Acta Tropicavol 56 no 2-3 pp 157ndash171 1994

[25] WHO Artemisinin and Artemisinin-Based Combination8erapy Resistance Global Malaria Programme WorldHealth Organization Geneva Switzerland 2016

[26] WHO Q amp A on Artemisinin Resistance World HealthOrganization Geneva Switzerland 2018

[27] M Bushman L Morton N Duah et al ldquoWithin-hostcompetition and drug resistance in the human malaria par-asite plasmodium falciparumrdquo Proceedings of the Royal So-ciety B Biological Sciences vol 283 no 1826 article 201530382016

[28] W E Harrington T K Mutabingwa A Muehlenbachs et alldquoCompetitive facilitation of drug-resistant plasmodium fal-ciparum malaria parasites in pregnant women who receivepreventive treatmentrdquo Proceedings of the National Academy ofSciences vol 106 no 22 pp 9027ndash9032 2009

[29] M Bushman R Antia V Udhayakumar and J C de RoodeldquoWithin-host competition can delay evolution of drug re-sistance in malariardquo PLoS biology vol 16 no 8 2018

[30] T Mita K Tanabe and K Kita ldquoSpread and evolution ofplasmodium falciparum drug resistancerdquo Parasitology In-ternational vol 58 no 3 pp 201ndash209 2009

[31] S Vinayak M T Alam T Mixson-Hayden et al ldquoOrigin andevolution of sulfadoxine resistant plasmodium falciparumrdquoPLoS Pathogens vol 6 no 3 article e1000830 2010

[32] I M Hastings and W M Watkins ldquoIntensity of malariatransmission and the evolution of drug resistancerdquo ActaTropica vol 94 no 3 pp 218ndash229 2005

[33] A O Talisuna P Langi T K Mutabingwa et al ldquoIntensity oftransmission and spread of gene mutations linked to chlo-roquine and sulphadoxine-pyrimethamine resistance in fal-ciparum malariardquo International Journal for Parasitologyvol 33 no 10 pp 1051ndash1058 2003

[34] E Y Klein D L Smith M F Boni and R LaxminarayanldquoClinically immune hosts as a refuge for drug-sensitivemalaria parasitesrdquo Malaria Journal vol 7 no 67 2008

[35] WHO World Malaria Report 2017 World Health Organi-zation Geneva Switzerland 2017

[36] F E McKenzie ldquoWhy model malariardquo Parasitology Todayvol 16 no 12 pp 511ndash516 2000

[37] F E McKenzie and E M Samba ldquo+e role of mathematicalmodeling in evidence-based malaria controlrdquo AmericanJournal of Tropical Medicine and Hygiene vol 71 pp 94ndash962004

[38] P G Coleman S Shillcutt C Morel A J Mills andC Goodman ldquoA threshold analysis of the cost-effectiveness ofartemisinin-based combination therapies in sub-SaharanAfricardquo American Journal of Tropical Medicine and Hygienevol 71 pp 196ndash204 2004

[39] I M Hastings ldquoA model for the origins and spread of drug-resistant malariardquo Parasitology vol 115 no 2 pp 133ndash1411997

[40] M J Mackinnon ldquoDrug resistance models for malariardquo ActaTropica vol 94 no 3 pp 207ndash217 2005

[41] S J Aneke ldquoMathematical modelling of drug resistantmalariaparasites and vector populationsrdquo Mathematical Methods inthe Applied Sciences vol 25 no 4 pp 335ndash346 2002

[42] J Koella and R Antia ldquoEpidemiological models for the spreadof anti-malarial resistancerdquoMalaria Journal vol 2 no 1 p 32003

[43] L Esteva A B Gumel and C V De LeoN ldquoQualitative studyof transmission dynamics of drug-resistant malariardquo Math-ematical and Computer Modelling vol 50 no 3-4 pp 611ndash630 2009

[44] E Y Klein D L Smith R Laxminarayan and S LevinldquoSuperinfection and the evolution of resistance to antimalarialdrugsrdquo Proceedings of the Royal Society B Biological Sciencesvol 279 no 1743 pp 3834ndash3842 2012

[45] M Legros and S Bonhoeffer ldquoA combined within-host andbetween-hosts modelling framework for the evolution ofresistance to antimalarial drugsrdquo Journal of the Royal SocietyInterface vol 13 no 117 article 20160148 2016

[46] A R Wargo S Huijben J C De Roode J Shepherd andA F Read ldquoCompetitive release and facilitation of drug-resistant parasites after therapeutic chemotherapy in a rodentmalaria modelrdquo Proceedings of the National Academy ofSciences vol 104 no 50 pp 19914ndash19919 2007

[47] P A zurWiesch R Kouyos J Engelstadter R R Regoes andS Bonhoeffer ldquoPopulation biological principles of drug-re-sistance evolution in infectious diseasesrdquo 8e Lancet In-fectious Diseases vol 11 no 3 pp 236ndash247 2011

[48] Z Agur D Abiri and L H Van der Ploeg ldquoOrdered ap-pearance of antigenic variants of african trypanosomesexplained in a mathematical model based on a stochasticswitch process and immune-selection against putative switchintermediatesrdquo Proceedings of the National Academy of Sci-ences vol 86 no 23 pp 9626ndash9630 1989

[49] R Antia B R Levin and R MMay ldquoWithin-host populationdynamics and the evolution and maintenance of micro-parasite virulencerdquo American Naturalist vol 144 no 3pp 457ndash472 1994

[50] L Cai N Tuncer and M Martcheva ldquoHow does within-hostdynamics affect population-level dynamics insights from animmuno-epidemiological model of malariardquo MathematicalMethods in the Applied Sciences vol 40 no 18 pp 6424ndash64502017

[51] C Chiyaka W Garira and S Dube ldquoModelling immuneresponse and drug therapy in human malaria infectionrdquoComputational andMathematical Methods inMedicine vol 9no 2 pp 143ndash163 2008


[52] S Pilyugin and R Antia ldquoModeling immune responses withhandling timerdquo Bulletin of Mathematical Biology vol 62no 5 pp 869ndash890 2000

[53] M A Selemani L S Luboobi and Y Nkansah-Gyekye ldquo+ein-human host and in-mosquito dynamics of malaria para-sites with immune responsesrdquo New Trends in MathematicalSciences vol 5 no 3 pp 182ndash207 2017

[54] J C de Roode R Culleton A S Bell and A F ReadldquoCompetitive release of drug resistance following drugtreatment of mixed plasmodium chabaudi infectionsrdquoMalaria Journal vol 3 no 1 2004

[55] R Hayward K J Saliba and K Kirk ldquopfmdr1 mutationsassociated with chloroquine resistance incur a fitness cost inplasmodium falciparumrdquo Molecular Microbiology vol 55no 4 pp 1285ndash1295 2005

[56] D L Doolan C Dobano and J K Baird ldquoAcquired im-munity to malariardquo Clinical Microbiology Reviews vol 22no 1 pp 13ndash36 2009

[57] P Liehl P Meireles I S Albuquerque et al ldquoInnate im-munity induced by plasmodium liver infection inhibitsmalaria reinfectionsrdquo Infection and Immunity vol 83 no 3pp 1172ndash1180 2015

[58] N Villarino and N W Schmidt ldquoCD8+ T cell responses toplasmodium and intracellular parasitesrdquo Current ImmunologyReviews vol 9 no 3 pp 169ndash178 2013

[59] T O Orwa R W Mbogo and L S Luboobi ldquoMathematicalmodel for hepatocytic-erythrocytic dynamics of malariardquoInternational Journal of Mathematics and MathematicalSciences vol 2018 Article ID 7019868 18 pages 2018

[60] T O Orwa R W Mbogo and L S Luboobi ldquoMathematicalmodel for the in-host malaria dynamics subject to malariavaccinesrdquo Letters in Biomathematics vol 5 no 1 pp 222ndash2512018

[61] J K Hale Ordinary Differential Equations John Wiley ampSons New York NY USA 1969

[62] P van den Driessche and J Watmough ldquoReproductionnumbers and sub-threshold endemic equilibria for com-partmental models of disease transmissionrdquo MathematicalBiosciences vol 180 no 1-2 pp 29ndash48 2002

[63] K S Vannice G V Brown B D Akanmori andV S Moorthy ldquoMALVAC 2012 scientific forum acceleratingdevelopment of second-generationmalaria vaccinesrdquoMalariaJournal vol 11 no 1 p 372 2012

[64] L J Allen Introduction to Mathematical Biology PearsonPrentice Hall Upper Saddle River NJ USA 2007

[65] J C Kamgang and G Sallet ldquoGlobal asymptotic stability forthe disease free equilibrium for epidemiological modelsrdquoComptes Rendus Mathematique vol 341 no 7 pp 433ndash4382005

[66] X Wang ldquoA simple proof of Descartesrsquos rule of signsrdquo 8eAmerican Mathematical Monthly vol 111 no 6 pp 525-5262004

[67] L Esteva and C Vargas ldquoInfluence of vertical and mechanicaltransmission on the dynamics of dengue diseaserdquo Mathe-matical Biosciences vol 167 no 1 pp 51ndash64 2000

[68] M Krasnoselskii Positive Solutions of Operator EquationsNoordhoff Groningen Netherlands 1964

[69] Y Li S Ruan and D Xiao ldquo+e within-host dynamics ofmalaria infection with immune responserdquo MathematicalBiosciences and Engineering vol 8 no 4 pp 999ndash1018 2011

[70] C Chiyaka ldquoUsing mathematics to understand malaria in-fection during erythrocytic stagesrdquo Zimbabwe Journal ofScience and Technology vol 5 pp 1ndash11 2010

[71] C Colijn and T Cohen ldquoHow competition governs whethermoderate or aggressive treatment minimizes antibiotic re-sistancerdquo Elife vol 4 2015

[72] R M Anderson C A Facer and D Rollinson ldquoResearchdevelopments in the study of parasitic infectionsrdquo Parasi-tology vol 99 no S1 p S1 1989

[73] A Mohammed A Ndaro A Kalinga et al ldquoTrends inchloroquine resistance marker Pfcrt-K76Tmutation ten yearsafter chloroquine withdrawal in Tanzaniardquo Malaria Journalvol 12 no 1 p 415 2013

[74] A Ofosu-Okyere M J Mackinnon M P K Sowa et alldquoNovel plasmodium falciparum clones and rising clonemultiplicities are associated with the increase in malariamorbidity in ghanaian children during the transition into thehigh transmission seasonrdquo Parasitology vol 123 no 2pp 113ndash123 2001

[75] B Hellriegel ldquoModelling the immune response to malariawith ecological concepts short-term behaviour against long-term equilibriumrdquo Proceedings of the Royal Society of LondonSeries B vol 250 no 1329 pp 249ndash256 1992

[76] A M Niger and A B Gumel ldquoImmune response and im-perfect vaccine in malaria dynamicsrdquo Mathematical Pop-ulation Studies vol 18 no 2 pp 55ndash86 2011

[77] E A Ashley M Dhorda R M Fairhurst et al ldquoSpread ofartemisinin resistance in plasmodium falciparum malariardquoNew England Journal of Medicine vol 371 no 5 pp 411ndash4232014

[78] D Lane ldquoOnline statistics education a multimedia course ofstudyrdquo in Proceedings of the EdMedia World Conference onEducational Media and Technology pp 1317ndash1320 Associa-tion for the Advancement of Computing in Education(AACE) Rice Univesity Houston TX USA 2003

[79] H Gelband C B Panosian K J Arrow et al Saving LivesBuying Time Economics of Malaria Drugs in An Age of Re-sistance National Academies Press Washington DC USA2004

[80] H A Babiker I M Hastings and G Swedberg ldquoImpairedfitness of drug-resistant malaria parasites evidence and im-plication on drug-deployment policiesrdquo Expert Review ofAnti-Infective 8erapy vol 7 no 5 pp 581ndash593 2009

[81] B Teun O Lucy S Seif et al ldquoRevisiting the circulation timeof plasmodium falciparum gametocytes molecular detectionmethods to estimate the duration of gametocyte carriage andthe effect of gametocytocidal drugsrdquo Malaria Journal vol 2no 9 p 136 2010

[82] J C de Roode M E H Helinski M A Anwar andA F Read ldquoDynamics of multiple infection and withinmdashhostcompetition in genetically diverse malaria infectionsrdquoAmerican Naturalist vol 166 no 5 pp 531ndash542 2005

[83] J A N Filipe E M Riley C J Drakeley C J Sutherland andA C Ghani ldquoDetermination of the processes driving theacquisition of immunity to malaria using a mathematicaltransmission modelrdquo PLoS Computational Biology vol 3no 12 p e255 2007


Stem Cells International

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EndocrinologyInternational Journal of



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OncologyJournal of



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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Journal of

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Computational and Mathematical Methods in Medicine


Behavioural Neurology

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Research and TreatmentAIDS


Gastroenterology Research and Practice


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Evidence-Based Complementary andAlternative Medicine

Volume 2018Hindawiwwwhindawicom

Submit your manuscripts atwwwhindawicom

Kenya to Tanzania [17] By 1983 CQ resistance had spreadto Sudan Uganda [18] Zambia [19] and Malawi [20]Unlike Africa CQ was replaced for the first time withsulfadoxine-pyrimethamine (SP) as a first-line antimalarialdrug in+ailand in 1967 Several other countries in Asia andSouth America followed thereafter [10] Resistance to SPwas however reported the same year [21] in the region In1988 CQ was replaced for the first time in Africa KwaZulu-Natal Province of South Africa replaced CQ with SP [22] In1993 the Malawian government changed the treatmentpolicy from CQ to SP Other African countries followedthereafter Kenya South Africa and Botswana (in 1998)Cameroon and Tanzania (in 2001) and Zimbabwe (in 2000)[23] +e effectiveness of SP was equally undermined byresistance Unlike CQ P falciparum resistance to SP wasmainly attributed to the long half-life of the drug [24]Confirmed resistance to the artemisinin derivatives was firstreported in Cambodia and Mekong regions in 2008 [25]

To leverage on parasite resistance cost of treatmentand burden of malaria infection to communities andgovernments the WHO recommends the use of artemi-sinin-based combination therapies (ACTs) as the first- andsecond-line treatment drugs for uncomplicated P falci-parum malaria [25] ACT is a combination of artemisininderivatives and a partner monotherapy drug Artemisininderivatives include artemether artesunate and dihy-droartemisinin +ese derivatives reduce the parasitebiomass within the first three days of therapy while thepartner drug with longer half-life eliminates theremaining parasites [26] +eWHO currently recommendsfive different ACTs (1) artesunate-amodiaquine (AS +AQ)(2) artesunate-mefloquine (AS +MQ) (3) artesuna-te + sulfadoxine-pyrimethamine (AS + SP) (4) artemether-lumefantrine (AM-LM) and (5) dihydroartemisinin-piperaquine (DHA+PPQ) Additionally artesunate-pyronaridine may be used in regions where ACT treatmentresponse is low [26] Access to ACT has been tremendousin the last 8 years with a recorded increase of 122 millionprocured treatment courses for the period 2010ndash2016However resistance to currently used ACTs has importantpublic health consequences especially in the African re-gion where resistant P falciparum is predominant

Numerous cross-sectional studies [27 28] have revealedthe possible impacts of multiple strains of P falciparum onthe development of resistance to ACTs In [29] and citationstherein drug-sensitive parasites are shown to stronglysuppress the growth and transmission of drug-resistant Pfalciparum parasites Although high-transmission settingssuch as sub-Saharan Africa account for about 90 of allglobal malaria deaths resistance to antimalarial drugs hasbeen shown to emerge from low-transmission settings suchas Southeast Asia and South America [29] Causes of parasiteresistance to ACTs are diverse Historical studies [30 31]indicate that antimalarial-resistant parasites could emergefrom a handful of lineages It is argued elsewhere [32 33]that recombination during sexual reproduction in themosquito vector could be responsible for the delayed ap-pearance of multilocus resistance in high-transmission re-gions Moreover owing to repeated exposure for many

years individuals in high-transmission settings are likely todevelop clinical immunity to malaria leading to strongerselection for resistance [34] Studies in [29] also support thehypothesis that in-host competition between drug-sensitiveand drug-resistant parasites could inhibit the spread ofresistance in high-transmission settings Owing to theirintegral role in the recent success of global malaria controlthe protection of efficacy of ACTs should be a global healthpriority [35]

Mathematical models of in-host malaria epidemiologyand control constitute important tools in guiding strategiesfor malaria control [36 37] and the associated financialplanning [38] While some researchers have focussed onprobabilistic models [39 40] others have investigated theeffects of drug treatment and resistance development usingdynamic models [41 42] A deterministic model by Estevaet al [43] monitored the impact of drug resistance on thetransmission dynamics of malaria in a human population In[29] the impacts of within-host parasite competition areshown to inhibit the spread of resistance [44 45] On thecontrary some models [39 46] have suggested that within-host competition is likely to speed up the spread of resistancein high-transmission settings due to a phenomenon calledldquocompetitive releaserdquo In this paper we provide theoreticalinsights using mathematical modelling of the impacts ofmultiple-strain infections on resistance dynamics andantimalarial control of P falciparum malaria

+e rest of the paper is organized as follows In Section 2we formulate the within-humanmalaria model that has boththe drug-sensitive and drug-resistant P falciparum parasitestrains subject to antimalarial therapy In Section 3 weanalyze the model based on epidemiological theoremsWithin-host competition between parasite strains and theeffects of antimalarial drug efficacy on parasite clearance arediscussed in Section 4 Sensitivity analysis and multiple-strain infection and its effects on resistance and malariadynamics are demonstrated in Section 5 We conclude thepaper in Section 6 by emphasizing the need for antimalarialtherapy with the potential to eradicate multiple-strain in-fection due to P falciparum

2 Model Formulation

We present in this paper a deterministic model that de-scribes the within-human-host competition and trans-mission dynamics of two strains of P falciparum parasitesduring malaria infection +e compartmental model con-siders the coinfection and competition between the drug-sensitive (dss) and the drug-resistant (drs) P falciparumstrains in the presence of antimalarial therapy +e drs arisepresumably from the dss +e rare mechanism here couldpossibly be due to single point mutation [47] Both drs anddss initiate immune responses that follow density-dependentkinetics

Our model is composed of eight compartments sus-ceptiblehealthyunparasitized erythrocytes (red blood cells)X(t) parasitizedinfected erythrocytes (Yr(t) and Ys(t))merozoites (Ms(t) and Mr(t)) gametocytes (Gs(t) andGr(t)) and immune cells W(t) +e healthy erythrocytes





dX


βX

1 + cWMs + δrMr( 1113857 (1)



dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dYr

dtδrβXMr

1 + cWminus

kyYrW


(2)



βMsX

1 + cWminus

kmMsW


dMr


δrβMrX

1 + cWminus

kmMrW

1 + aMrminus μmrMr

dGs

dt σsYs minus

kgWGs


dGr


kgWGr

1 + aGrminus μgrGr

(3)





dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(4)


C(t) X(t) + Ys(t) + Yr(t) (5)


P(t) Ms(t) + Mr(t) + Gs(t) + Gr(t) (6)



dX


βX

1 + cWMs + δrMr( 1113857 (7)

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs


dYr

dtδrβXMr

1 + cWminus

kyYrW


dMs


βMsX

1 + cWminus

kmMsW

1 + aMs


(10)

dMr


δrβMrX

1 + cW

minuskmMrW

1 + aMrminus μmrMr

(11)

dGs

dt σsYs minus

kgWGs


dGr


kgWGr


dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(14)


X(0)gt 0

Yi(0)ge 0

Mi(0)ge 0

Gi(0)ge 0

W(0)gt 0 for i s r

(15)

3 Model Analysis






dX

dt λx minus μxX t


βX tlowast( )



( 1113857( 1113857

λx gt 0

(16)



dC




(18)



spectively given as

W(t)leλwμw

+ W(0)minusλwμw

1113888 1113889eminusμwt

C(t)leλx

μc+ C(0)minus

λx

μc1113888 1113889e

minusμct

P(t)leσs 1113938

t


0 Yr(t)ΔIFdt

ΔIF



1ΔIF

(19)

where

ΔIF exp⎧⎨



t

0Yr(t)dt1113888 1113889

minus 1113946t

0μpdt

⎫⎬

⎭





C(t)lemax C(0)λx

μc1113896 1113897


1113896 1113897



(21)

X

W

μmsMs μmrMr

μyrYr

μgrGr

μxX

μwW

kmMsW(1 + aMs)


Ys

Gs

Mr

Yr

Gr

kyYsW(1 + aYs)

kgGsW(1 + aGs)

kgGrW(1 + aGr)


kmMrW(1 + aMr)

kyYrW(1 + aYr)

βsXMs βrXMr

λx

λw

σsYs σrYr

(η + μgs)Gs
















dt g(x) x(0) x0 (22)








infin (23)


gprime(mnminus x)

infin 11139448


infin

le 11139448

i1gi(m)(

infin

nminus xinfin

(24)



1113944

8

i1gi(m)(







λx

μx

0 0 0 0 0 0λwμw

1113888 1113889

(27)




μx









RBCs






merozoites


gametocytes


parasite strains




F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)




(kgλwμw))


Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0


01v3

0 0 0



01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that


where





(32)







JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















dX


βX

1 + cWMs + δrMr( 1113857 (1)



dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dYr

dtδrβXMr

1 + cWminus

kyYrW


(2)



βMsX

1 + cWminus

kmMsW


dMr


δrβMrX

1 + cWminus

kmMrW

1 + aMrminus μmrMr

dGs

dt σsYs minus

kgWGs


dGr


kgWGr

1 + aGrminus μgrGr

(3)





dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(4)


C(t) X(t) + Ys(t) + Yr(t) (5)


P(t) Ms(t) + Mr(t) + Gs(t) + Gr(t) (6)



dX


βX

1 + cWMs + δrMr( 1113857 (7)

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs


dYr

dtδrβXMr

1 + cWminus

kyYrW


dMs


βMsX

1 + cWminus

kmMsW

1 + aMs


(10)

dMr


δrβMrX

1 + cW

minuskmMrW

1 + aMrminus μmrMr

(11)

dGs

dt σsYs minus

kgWGs


dGr


kgWGr


dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(14)


X(0)gt 0

Yi(0)ge 0

Mi(0)ge 0

Gi(0)ge 0

W(0)gt 0 for i s r

(15)

3 Model Analysis






dX

dt λx minus μxX t


βX tlowast( )



( 1113857( 1113857

λx gt 0

(16)



dC




(18)



spectively given as

W(t)leλwμw

+ W(0)minusλwμw

1113888 1113889eminusμwt

C(t)leλx

μc+ C(0)minus

λx

μc1113888 1113889e

minusμct

P(t)leσs 1113938

t


0 Yr(t)ΔIFdt

ΔIF



1ΔIF

(19)

where

ΔIF exp⎧⎨



t

0Yr(t)dt1113888 1113889

minus 1113946t

0μpdt

⎫⎬

⎭





C(t)lemax C(0)λx

μc1113896 1113897


1113896 1113897



(21)

X

W

μmsMs μmrMr

μyrYr

μgrGr

μxX

μwW

kmMsW(1 + aMs)


Ys

Gs

Mr

Yr

Gr

kyYsW(1 + aYs)

kgGsW(1 + aGs)

kgGrW(1 + aGr)


kmMrW(1 + aMr)

kyYrW(1 + aYr)

βsXMs βrXMr

λx

λw

σsYs σrYr

(η + μgs)Gs
















dt g(x) x(0) x0 (22)








infin (23)


gprime(mnminus x)

infin 11139448


infin

le 11139448

i1gi(m)(

infin

nminus xinfin

(24)



1113944

8

i1gi(m)(







λx

μx

0 0 0 0 0 0λwμw

1113888 1113889

(27)




μx









RBCs






merozoites


gametocytes


parasite strains




F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)




(kgλwμw))


Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0


01v3

0 0 0



01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that


where





(32)







JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(4)


C(t) X(t) + Ys(t) + Yr(t) (5)


P(t) Ms(t) + Mr(t) + Gs(t) + Gr(t) (6)



dX


βX

1 + cWMs + δrMr( 1113857 (7)

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs


dYr

dtδrβXMr

1 + cWminus

kyYrW


dMs


βMsX

1 + cWminus

kmMsW

1 + aMs


(10)

dMr


δrβMrX

1 + cW

minuskmMrW

1 + aMrminus μmrMr

(11)

dGs

dt σsYs minus

kgWGs


dGr


kgWGr


dW

dt λw +

hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113896 1113897W

minus μwW

(14)


X(0)gt 0

Yi(0)ge 0

Mi(0)ge 0

Gi(0)ge 0

W(0)gt 0 for i s r

(15)

3 Model Analysis






dX

dt λx minus μxX t


βX tlowast( )



( 1113857( 1113857

λx gt 0

(16)



dC




(18)



spectively given as

W(t)leλwμw

+ W(0)minusλwμw

1113888 1113889eminusμwt

C(t)leλx

μc+ C(0)minus

λx

μc1113888 1113889e

minusμct

P(t)leσs 1113938

t


0 Yr(t)ΔIFdt

ΔIF



1ΔIF

(19)

where

ΔIF exp⎧⎨



t

0Yr(t)dt1113888 1113889

minus 1113946t

0μpdt

⎫⎬

⎭





C(t)lemax C(0)λx

μc1113896 1113897


1113896 1113897



(21)

X

W

μmsMs μmrMr

μyrYr

μgrGr

μxX

μwW

kmMsW(1 + aMs)


Ys

Gs

Mr

Yr

Gr

kyYsW(1 + aYs)

kgGsW(1 + aGs)

kgGrW(1 + aGr)


kmMrW(1 + aMr)

kyYrW(1 + aYr)

βsXMs βrXMr

λx

λw

σsYs σrYr

(η + μgs)Gs
















dt g(x) x(0) x0 (22)








infin (23)


gprime(mnminus x)

infin 11139448


infin

le 11139448

i1gi(m)(

infin

nminus xinfin

(24)



1113944

8

i1gi(m)(







λx

μx

0 0 0 0 0 0λwμw

1113888 1113889

(27)




μx









RBCs






merozoites


gametocytes


parasite strains




F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)




(kgλwμw))


Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0


01v3

0 0 0



01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that


where





(32)







JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















spectively given as

W(t)leλwμw

+ W(0)minusλwμw

1113888 1113889eminusμwt

C(t)leλx

μc+ C(0)minus

λx

μc1113888 1113889e

minusμct

P(t)leσs 1113938

t


0 Yr(t)ΔIFdt

ΔIF



1ΔIF

(19)

where

ΔIF exp⎧⎨



t

0Yr(t)dt1113888 1113889

minus 1113946t

0μpdt

⎫⎬

⎭





C(t)lemax C(0)λx

μc1113896 1113897


1113896 1113897



(21)

X

W

μmsMs μmrMr

μyrYr

μgrGr

μxX

μwW

kmMsW(1 + aMs)


Ys

Gs

Mr

Yr

Gr

kyYsW(1 + aYs)

kgGsW(1 + aGs)

kgGrW(1 + aGr)


kmMrW(1 + aMr)

kyYrW(1 + aYr)

βsXMs βrXMr

λx

λw

σsYs σrYr

(η + μgs)Gs
















dt g(x) x(0) x0 (22)








infin (23)


gprime(mnminus x)

infin 11139448


infin

le 11139448

i1gi(m)(

infin

nminus xinfin

(24)



1113944

8

i1gi(m)(







λx

μx

0 0 0 0 0 0λwμw

1113888 1113889

(27)




μx









RBCs






merozoites


gametocytes


parasite strains




F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)




(kgλwμw))


Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0


01v3

0 0 0



01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that


where





(32)







JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of

























dt g(x) x(0) x0 (22)








infin (23)


gprime(mnminus x)

infin 11139448


infin

le 11139448

i1gi(m)(

infin

nminus xinfin

(24)



1113944

8

i1gi(m)(







λx

μx

0 0 0 0 0 0λwμw

1113888 1113889

(27)




μx









RBCs






merozoites


gametocytes


parasite strains




F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)




(kgλwμw))


Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0


01v3

0 0 0



01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that


where





(32)







JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















F

0 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 0 0δrβλxμw

cλw + μw( 1113857μx

0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(28)

Q

v1 0 0 0 0 00 v2 0 0 0 0



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(29)




(kgλwμw))


Qminus1

1v1

0 0 0 0 0

01v2

0 0 0 0


01v3

0 0 0



01v4

0 0

σsv1v5 0 0 01v5

0

σsΨ2v1v5v6σr

v2v60 0 Ψ2v5v6

1v6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(30)

It follows that


where





(32)







JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















JE0


cλw + μw( 1113857μx


0 0 0

0 minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0 0


cλw + μw( 1113857μx

0 0 0



0 σs 0 0 0 minusv5 0 0

0 0 σr 0 0 Ψ2 minusv6 0

0hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμwminusμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(33)


P(λ) λ4 + p1λ3

+ p2λ2

+ p3λ + p4 (34)

where

p1 v1 + v2 + v3 + v4( 1113857gt 0 (35)

p2 v3v4 + v2 v3 + v4( 1113857 + v1 v2 + v3 + v4( 1113857

minusPβλxμw

cλw + μw( 1113857μx


p3 1K


minus1K


+v1


(37)


K

(38)





3 + p21p4



K1113888 1113889


K1113888 1113889

(39)



v1v3K1113890 1113891


v2v4K1113890 1113891

v1 + v3( 1113857 v2 + v4( 1113857 + v1v3 1minusRs1113858 1113859


(40)



p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















p4 v1v3 +B1βλxμx

K1113890 1113891 v2v4 +

B2δrβλxμwK

1113890 1113891

v1v3 1 +B1βλxμw

v1v3K1113890 1113891v2v4 1 +

B2δrβλxμwv2v4K

1113890 1113891


(41)


p3 v2v3v4 + v1v3v4 + v1v2 v3 + v4( 1113857

+βB1λxμw v2 + v4( 1113857

K

+δrβB2λxμw v1 + v3( 1113857

K

v1v2v3v4⎡⎣ 1

v41 +

βB1λxμwv1v3K

1113888 1113889 +1v2

1 +βB1λxμw

v1v3K1113888 1113889

+1v1

1 +δrβB2λxμw

v2v4K1113888 1113889 +

1v3

1 +δrβB2λxμw

v2v4K1113888 1113889⎤⎦

v1v2v3v4v2 + v4

v2v41minusRs( 1113857 +

v1 + v3

v1v31minusRr( 11138571113890 1113891



(42)






_X2 D3(X)X2

⎫⎪⎬

⎪⎭ (43)




X X1 X2( 1113857

X1 (X W)


Xlowast1

λx

μx

λwμw

1113888 1113889

(44)





D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















D1(X) minusμx 0


D2(X)

0 0minusβλxμw

cλw + μw( 1113857μx


0 0

hyλweyμw

hyλweyμw

hmλwemμw

hmλwemμw

hgλwegμw

hgλwegμw

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(45)



D3(X)

minusv1 0βλxμw

cλw + μw( 1113857μx

0 0 0

0 minusv2 0βλxμw

cλw + μw( 1113857μx

0 0



σs 0 0 0 minusv5 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(46)



K

K11 K12

K21 K22

⎛⎜⎝ ⎞⎟⎠ (47)


minus111K12 are

Metzler stable


K11

minusv1 0βλxμw

cλw + μw( 1113857μx

0 minusv2 0


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K12

0 0 0


0 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K21


0 σr 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

K22

minusv4 0 0

0 minusv5 0

0 Ψ2 minusv6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(48)



Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















Kminus111

minusv3



0 minus1v2

0


0 minusv1


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(49)



⎛⎜⎝ ⎞⎟⎠ (50)









Xlowast



Ylowasts

b +

b2 minus 4ac

1113969

minus2a

Ylowastr

b +

b2 minus 4 a c

1113969

minus2 a

(51)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(52)



lowasts + βM

lowastr δr


(53)



lowastr δr + cW

lowast+ 1( 1113857μx( 1113857lt 0

(55)


lowastky(βM

lowasts





(56)


Glowasts

b1 +

b21 minus 4a1c1

1113969

minus2a1

Glowastr

b2 +

b22 minus 4a2c2

1113969

minus2a2

(58)

a1 minusa η + μg1 + Ψ21113872 1113873lt 0



c1 σ1Ylowasts gt 0

(59)

a2 minusaμg2 lt 0


lowastkg minus μg2


(60)

Mlowasts

b3 +

b23 minus 4a3c3

1113969

minus2a3

Mlowastr

b4 +

b24 minus 4a4c4

1113969

minus2a4

(61)





lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















lowast+ 1( 1113857μx + β( 1113857


lowastkm + βμms)

(62)



lowastkm + μms + Ψ11113872 1113873



lowasts μys


(63)


lowastr δr + cW

lowast+ 1( 1113857μx( 1113857gt 0 (64)


lowast+ 1( 1113857μx + βδr( 1113857


(65)



+ Ψ1 a cWlowast



lowastkm βM

lowasts + cW

lowast+ 1( 1113857μx( 1113857

minus μmr cWlowast

+ 1( 1113857μx

(66)



+ Ψ1 cWlowast


+ 1( 1113857μxμy2 gt 0

(67)

Wlowast

Δ


(68)




S(t) S0eξt

(69)



ξS1 minusβMs

1 + cW+

kyW

1 + cW+

μys1minusωs

+ σs1113888 1113889 S1

minusβMs

1 + cWS2 +


1 + cWS3

ξS2 minusδrβMr

1 + cWS1 minus

δrβMr

1 + cW+

kyW

1 + aYr+ μyr + σr1113888 1113889S2


1 + cWS4

ξS3 minusβMs

1 + cW+ P 1minus αs( 1113857μys1113888 1113889S1 +

βMs

1 + cWS2

minuskmW


1 + cW+ k11113888 1113889S3

ξS4 Ψ1S3 +δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2


1 + cW+

kmW

1 + aMr+ μmr1113888 1113889S4 +

δrβMr

1 + cWS1

ξS5 σsS1 minuskgW

1 + aGs+ k21113888 1113889S5


1 + aGr+ μgr1113888 1113889S5

ξS7 λ +hg Gs + Gr( 1113857

Gs + Gr + eg+

hy Ys + Yr( 1113857

Ys + Yr + ey+

hm Ms + Mr( 1113857

Ms + Mr + em1113888 1113889S7

minus μwS7

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(70)




1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















1 +(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ1113890 1113891S1

(1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1minus

βMs

1 + cWS2 +


1 + cWS31113888 1113889

1 +ξ(1 + cW) 1 + aYr( 1113857

Δ21113890 1113891S2

(1 + cW) 1 + aYr( 1113857

Δ2minusδrβMr

1 + cWS1 +


1 + cWS41113888 1113889

1 +(1 + cW) 1 + aMs( 1113857

Δ3ξ1113890 1113891S3

(1 + cW) 1 + aMs( 1113857

Δ3βMs


βMs

1 + cWS21113888 1113889

1 +ξ(1 + cW) 1 + aMr( 1113857

Δ41113890 1113891S4

(1 + cW) 1 + aMr( 1113857

Δ4Ψ1S3 +

δrβMr

1 + cW+ P 1minus αr( 1113857μyr1113888 1113889S2 +

δrβMr

1 + cWS41113888 1113889

1 +1 + aGs( 1113857

kgW + k2ξ1113890 1113891S5

σs 1 + aGs( 1113857

kgW + k2S1

1 +1 + aGr( 1113857


1 + aGr( 1113857

kgW + μgrσrS2 + Ψ2S41113864 1113865

1 +1μw

ξ1113890 1113891S7 λwμw

+W

μw

hg S5 + S6( 1113857

Gs + Gr + eg+

hy S1 + S2( 1113857

Ys + Yr + ey+

hm S3 + S4( 1113857

Ms + Mr + em1113888 1113889

(71)


+ μys 1 + aYs( 1113857(1 + cW) + σs 1 + aYs( 1113857

1minusωs( 1113857(1 + cW)

Δ2 δrβMr 1 + aYr( 1113857 + kyW(1 + cW)

+ μyr + σr1113872 1113873 1 + aYr( 1113857(1 + cW)


+ k1 1 + aMs( 1113857(1 + cW)


+ kmW(1 + cW) + μmr 1 + aMr( 1113857(1 + cW)

(72)


1 + Fj(ξ)1113960 1113961Sj (HS)j for j 1 2 7 (73)

where

F1(ξ) (1 + cW) 1 + aYs( 1113857 1minusωs( 1113857

Δ1ξ

F2(ξ) ξ(1 + cW) 1 + aYr( 1113857

Δ2

F3(ξ) (1 + cW) 1 + aMs( 1113857

Δ3ξ

F4(ξ) ξ(1 + cW) 1 + aMr( 1113857

Δ4

F5(ξ) 1 + aGs( 1113857

kgW + k2ξ

F6(ξ) 1 + aGr( 1113857

kgW + μgrξ

F7(ξ) 1μw

ξ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(74)


with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















with

H

0 0βClowast

1 + cw0 0 0 0

0 0 0δrβClowast

1 + cW0 0 0


1 + cW+ k1 0 0 0 0

0 P 1minus αr( 1113857μyr Ψ1 0 0 0 0

σs 0 0 0 0 0 0

0 σr 0 0 0 0 0

0 0 λw 0 0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(75)





cλx + μw( 11138572μ2x

(76)










dX


βXMs

1 + cW

dYs

dt

βXMs

1 + cWminus

kyYsW

1 + aYsminus

11minusωs

μysYs minus σsYs

dMs


βMsX

1 + cWminus

kmMsW


dGs

dt σsYs minus

kgWGs

1 + aGsminus μgs + η1113872 1113873Gs

dW

dt λw +

hg Gs( 1113857

Gs + eg+

hy Ys( 1113857

Ys + ey+

hm Ms( 1113857


⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(78)









zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of

























zRs

zωs




lt 0

(79)

0 10 20 30 40 50 60 70 80Time (days)

16

14

12

10

08

06

04

02

00

Popu

latio

n

1e11


(a)

6

5

4

3

2

1

0

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)






zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of






















zRs

zΨ1 minus





zRs

zζ minus



zRr

zc minus





0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n

1e3


(a)

25

20

15

10

05

00

Popu

latio

n

1e6

0 10 20 30 40 50 60 70 80Time (days)


(b)








08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of
























08

06

04

02

00

1e4

0 10 20

Den

sity

of in

fect

ed er

ythr

ocyt

es

30 40 50 60 70 80Time (days)


(a)

25

20

15

10

05

00Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

1e4

0 10 20 30 40 50 60 70 80Time (days)


(b)

0 10 20 30 40 50 60 70 80Time (days)

5

4

3

2

1

0

Popu

latio

n of

infe

cted

red

bloo

d ce

lls

1e5

Rr = 32221 gt Rs

Rs = 1066 gt 1


(c)





30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















30

25

20

15

10

05

000 10 20 30 40 50

Time (days)

Popu

latio

n of

mer

ozoi

tes

1e5


(a)

0 10 20 30 40 50Time (days)

5

4

3

2

1

0

Popu

latio

n of

mer

ozoi

tes

1e5


Rr = 32221 gt Rs

Rs = 1066 gt 1

(b)





16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

with

resis

tant

stra

ins Y r

ωs = 0ωs = 025ωs = 05

ωs = 075ωs = 095

(a)

16

14

12

10

08

06

04

00

02

1e5

0 10 20 30 40 50 60 70 80Time (days)

Popu

latio

n of

infe

cted

eryt

hroc

ytes

ψ1 = 075ψ1 = 095ψ1 = 025

ψ1 = 05

ψ1 = 01

Rr = 2361 Yr Rs = 1102 Ys

(b)

1e4


0 10 20 30 40 50 60 70 80Time (days)

08

06

04

00

02

Popu

latio

n of

par

asite

-infe

cted

eryt

hroc

ytes

(c)


5

5

1

1

Rr

Rs

PFE



(a)

5

1

Rr

51Rs

PFE




(b)







ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























ΥΨ1 zRE

zΨ1timesΨ1RE

(83)



0175

0150

0125

0100

0075

0050

0025

02 04 06 08 10

0007

0006

0005

0004

0003

0002

0001

0000

μw

β

0002

000

3 0006

0004

000

5

(a)


00175

00150

00125

00100

00075

00050

00025

00000

010

008

006

004

002

01 02 03 04 05 06 07 08 09 10

ωs

μys

0003 0005

0007

0010

001

3

001

5

001

8

(b)


09

08

07

06

05

04

03

02

01140 145 150 155 160 165 170 175 180

0232

0224

0216

0208

0200

0192

0184

0176

0224

0216

0208

0200

0192

0184

0176 0168

ρ

γ

(c)




6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of




















6 Conclusion



009

008

007

006

005

004

003

002

001


006

005

004

003

002

001

00001 02 03 04 05 06 07 08

μyr

αr

0010

0020

0030

0040

0050

(a)


008

007

006

005

004

003

002

001

10

09

08

07

06

05

04

03

02

0101 02 03 04 05 06 07 08 09 10

ψ1

σr

002

0

003

0

004

0

005

00

060

007

00

080

(b)


08

07

06

05

04

03

02

01

140 145 150 155 160 165 170 175 180

120

105

090

075

060

045

030

β

ρ

1050

07500900

06000150

(c)



1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















1e7

0 25 50 75 100 125



150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(a)

1e7



0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n

(b)


10

09

08

07

06

05

04

03

02

1e7

0 25 50 75 100 125 150 175 200Time (days)

Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200Time (days)

20

15

10

05

00

Popu

latio

n of

gam

etoc

ytes

(b)

1e4

0 25 50 75 100 125 150 175 200Time (days)

10

08

06

04

00

02

Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)







10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























10

08

06

04

02

1e7

0 25 50 75 100 125 150 175 200Time (days)


Popu

latio

n of

infe

cted

par

asiti

zed

eryt

hroc

ytes

(a)

1e9

0 25 50 75 100 125 150 175 200

8

7

6

5

4

3

2

1

0

Time (days)


Tota

l pop

ulat

ion

of g

amet

ocyt

es

(b)

1e4

0 25 50 75 100 125 150 175 200

14

12

10

08

06

04

02

00

Time (days)


Popu

latio

n of

imm

une c

ells

(acq

uire

d)

(c)





Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of





















Data Availability






Acknowledgments



References





























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



























































































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of
























































of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of























of




Disease Markers



OncologyJournal of





PPAR Research



Volume 2018


Journal of

ObesityJournal of



















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