
Research ArticleMathematical Model of MDRTB and XDRTB with
Isolationand Lost to FollowUp
F. B. Agusto, J. Cook, P. D. Shelton, and M. G. Wickers
Department of Mathematics and Statistics, Austin Peay State
University, Clarksville, TN 37044, USA
Correspondence should be addressed to F. B. Agusto;
fbagusto@gmail.com
Received 24 February 2015; Accepted 31 May 2015
Academic Editor: JuanCarlos Cortés
Copyright © 2015 F. B. Agusto et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
We present a deterministic model with isolation and lost to
followup for the transmission dynamics of three strains
ofMycobacterium tuberculosis (TB), namely, the drug sensitive,
multidrugresistant (MDR), and extensivelydrugresistant (XDR)TB
strains. The model is analyzed to gain insights into the
qualitative features of its associated equilibria. Some of the
theoreticaland epidemiological findings indicate that the model has
locally asymptotically stable (LAS) diseasefree equilibrium when
theassociated reproduction number is less than unity. Furthermore,
the model undergoes in the presence of disease reinfection
thephenomenon of backward bifurcation, where the stable
diseasefree equilibrium of the model coexists with a stable
endemicequilibrium when the associated reproduction number is less
than unity. Further analysis of the model indicates that the
diseasefree equilibrium is globally asymptotically stable (GAS) in
the absence of disease reinfection. The result of the global
sensitivityanalysis indicates that the dominant parameters are the
disease progression rate, the recovery rate, the infectivity
parameter, theisolation rate, the rate of lost to followup, and
fraction of fast progression rates. Our results also show that
increase in isolation rateleads to a decrease in the total number
of individuals who are lost to followup.
1. Introduction
Mycobacterium tuberculosis (TB) is caused by bacteria thatare
transmitted from person to person through the air by aninfected
person’s coughing, sneezing, speaking, or singing [1].TB usually
affects the lungs, but it can also affect other parts ofthe body,
such as the brain, the kidneys, or the spine [1]. TheTB bacteria
can stay in the air for several hours, depending onthe environment.
In 2013, 9 million people were ill with TB,and
1.5millionmortalities occurred from the disease [2]; over95% of
deaths occurred in low andmiddleincome countries[2]. About
onethird of the world’s population has latent TB[2]. TB is also
among the top three causes of death in womenaged 15 to 44 [2].
Tuberculosis is second only to HIV/AIDSas the greatest killer
worldwide due to a single infectiousagent [2]. Those who have a
compromised immune system,like those who are living with HIV,
malnutrition, or diabetes,or people who use tobacco products, have
a much higherrisk of falling ill. Individuals who develop TB are
providedwith a sixmonth course of four antimicrobial drugs
alongwith supervision and support by a health worker. Improper
treatment compliance or use of poor quality medicines canall
lead to the development of drugresistant tuberculosis [2].
Multidrugresistant (MDR) TB is a form of TB causedby bacteria
that do not respond to, at least, isoniazid andrifampicin, which
are the two most powerful, standard antituberculosis drugs.MDRTB
is treatable and curable by usingsecondline drugs [3]. However,
these treatment options arelimited and recommendedmedicines are not
always available[3]. In some cases, more severe drug resistance can
develop.Extensivelydrugresistant (XDR)TB is a
formofmultidrugresistant TB that responds to even fewer available
medicines,including the secondline drugs [3]. In 2013, there were
about480,000 cases of MDRTB present in the world [4]; it
wasestimated that about 9%10% of these cases were XDRTB[2,
4–6].
The increase in drugresistant TB strains has called for
anincreased urgency for isolating individuals infected with
suchstrains of TB [7]. High priority is being placed on
identifyingand curing these individuals. With proper identification
andtreatment, about 40% of XDRTB cases could potentially becured
[6, 8]. However, only 10% of MDRTB cases are ever
Hindawi Publishing CorporationAbstract and Applied
AnalysisVolume 2015, Article ID 828461, 21
pageshttp://dx.doi.org/10.1155/2015/828461

2 Abstract and Applied Analysis
identified, leaving the potential development of XDRTB tobecome
more prevalent worldwide [2]. The need for a modern and effective
approach to curtail the rise of drugresistantTB strains is being
sought after, one of which is constructedisolation, whether it is
an athome isolation or isolation ata medical facility [6]. An
event that portrays the need forcarefully constructed isolation was
the identification of anindividual infected with MDRTB in Atlanta
InternationalAirport in 2007 [9]. The individual had flown to
Atlantaafter visiting Paris, Greece, Italy, CzechRepublic,
andCanada.The individual unknowingly was infected withMDRTB
and,after twelve days of travel, the individual was
involuntarilyisolated by the CDC in Atlanta under the Public
HealthService Act [9]. The CDC held the individual in isolationfor
one week and then moved him to a hospital in Denver,Colorado [9].
In this case, there are no reported infectionsresulting from the
travel of the individual, perhaps due tothe slightly lower
infection rate of MDRTB as comparedto drug sensitive TB strains
[9]. It should be noted herethat the isolation of this individual,
albeit brief, potentiallyhelped prevent a rise in drugresistant TB
in the UnitedStates, which is currently at admittedly low levels
[10]. Anoccurrence such as this one clearly demonstrates the need
forcarefully executed isolation procedures for individuals
withdrugresistant TB strains.
Several reports have shown the effectiveness of isolation in
reducing the number of people with TB [6, 7, 26,27]. Weis et al.
[27] showed the effectiveness of isolation,reporting lower
occurrences of primary and acquired drugresistance among
individuals with TB. Historically, to treatthe infection,
individuals were isolated in a sanatoriumwherethey would receive
proper nutrition and a constant supplyof fresh air [28]. However,
while this method is successfulin certain situations, this
treatment methodology is difficultto implement without proper
infrastructure in place [29].Locations without proper facilities
such as South Africa havepoor treatment and success rates [29];
effective isolation isessential in areas such as these which have
high treatmentfailure rates. Sutton et al. [26] studied three
hospitals inCalifornia; they found that implementing the CDC
isolationguidelines for hospitals was feasible; however, since not
everyhospital could afford the necessary equipment, the
isolationwas not the same for each hospital. Even though the
resultsvaried, it is noticeably more efficient to isolate patients
whoare infected with TB.
According to the National Committee of Fight againstTuberculosis
of Cameroon [30], about 10% of infectious individuals who start
the recommended WHO DOTS treatmenttherapy in the hospital do not
return to the hospital for therest of sputumexaminations and
checkup and are thus lost tofollowup.This can be attributed to
the long duration of treatment regimen, negligence, or lack of
information about TB[30], a brief relief from the long term
treatment [22], poverty,and so forth. As such, healthcare
personnel do not knowtheir epidemiological status, that is, if they
died, recovered,or are still infectious and this lack of
epidemiological status ofthese individuals can affect the spread of
TB in a population[30]. A number of mathematical models for
tuberculosisdeveloped account for this population by the inclusion
of
either the lost sight class [23, 30–32] or lost to followup
class[22].
The aim of this study is to develop a new
deterministictransmission model for TB to gain qualitative insight
intothe effects of isolation in the presence of individuals who
arelost to followup on TB transmission dynamics. A notablefeature
of themodel is the incorporation of isolated and lost tofollowup
classes for the three TB strains. The paper is organized as
follows.Themodel formulation and analysis is givenin Section 2.
Sensitivity analysis of the model is considered inSection 3.
Analysis of the reproduction number is carried outin Section 4.The
effect of isolation is numerically investigatedin Section 5. The
key theoretical and epidemiological resultsfrom this study are
summarized in Section 6.
2. Model Formulation
Themodel is formulated as follows: the population is dividedinto
susceptible (𝑆), latently infected (𝐸
𝑖), symptomatically
infectious with drug sensitive strain (𝑇), MDR strain (𝑀),XDR
strain (𝑋), symptomatically infectious individuals whoare lost to
followup (𝐿
𝑖), isolated (𝐽), and recovered (𝑅),
where 𝑖 = 𝑇,𝑀,𝑋. Thus, the total population is 𝑁(𝑡) =𝑆(𝑡) +
𝐸
𝑇(𝑡) + 𝑇(𝑡) + 𝐸
𝑀(𝑡) + 𝑀(𝑡) + 𝐸
𝑋(𝑡) + 𝑋(𝑡) +
𝐿𝑇(𝑡) + 𝐿
𝑀(𝑡) + 𝐿
𝑋(𝑡) + 𝐽(𝑡) + 𝑅(𝑡).
As the disease evolves individuals move from one class tothe
other with respect to their disease status. The populationof
susceptible (𝑆) is generated by new recruits (either via birthor
immigration) who enter the population at a rate 𝜋. Theparameter 𝜋
denotes the recruitment rate. It is assumed thatthere is no
vertical transmission or immigration of infectious;thus, these new
inflow does not enter the infectious classes.All individuals,
whatever their status, are subject to naturaldeath, which occurs at
a rate 𝜇. The susceptible population isreduced by infection
following effective contact with infectedindividuals with drug
sensitive, MDR, and XDRTB strainsat the rates 𝜆
𝑇, 𝜆𝑀, and 𝜆
𝑋, where
𝜆𝑇=𝛽𝑇(𝑇 + 𝜂
𝑇𝐿𝑇)
𝑁,
𝜆𝑀=𝛽𝑀(𝑀 + 𝜂
𝑀𝐿𝑀)
𝑁,
𝜆𝑋=𝛽𝑋(𝑋 + 𝜂
𝑋𝐿𝑋)
𝑁.
(1)
The parameters 𝛽𝑇, 𝛽𝑀, and 𝛽
𝑋are the effective transmission
probability per contact; we assume that 𝛽𝑋
< 𝛽𝑀
< 𝛽𝑇
[9] and the parameters 𝜂𝑇
> 1, 𝜂𝑀
> 1, and 𝜂𝑋
> 1are the modification parameter that indicates the
increasedinfectivity of individuals who are lost to followup.
A fraction 𝑙𝑇1 of the newly infected individuals with drug
sensitive strainmove into the latently infected class (𝐸𝑇),
with
𝑙𝑇2 fractionmoving into the symptomaticinfectious class (𝑇)and
the other fraction [1 − (𝑙
𝑇1 + 𝑙𝑇2)] moving into the lostto followup class (𝐿
𝑇) with 𝑙
𝑇1 + 𝑙𝑇2 < 1. The latentlyinfected individuals become
actively infectious as a result ofendogenous reactivation of the
latent bacilli at the rate 𝜎
𝑇.
Similarly a fraction 𝑙𝑀1 of the newly infected individuals
with MDR stain move into the latently infected class with

Abstract and Applied Analysis 3
MDR (𝐸𝑀), with 𝑙
𝑀2 fractions moving into the symptomatically infectious class
(𝑀) and the other fraction [1 − (𝑙
𝑀1 +𝑙𝑀2)] moving into the symptomatically infectious lost
tofollowup class (𝐿
𝑀) with 𝑙
𝑀1 + 𝑙𝑀2 < 1.The latently infectedindividuals with MDR strain
become actively infectious as aresult of endogenous reactivation of
the latent bacilli at therate 𝜎
𝑀.
Lastly, a fraction 𝑙𝑋1 of the newly infected individualswith
XDR stain moves into the latently infected class with XDR(𝐸𝑋),
with 𝑙
𝑋2 fractions moving into the symptomaticallyinfectious class (𝑋)
and the other fraction [1 − (𝑙
𝑋1 + 𝑙𝑋2)]moving into the symptomatically infectious lost to
followupclass (𝐿
𝑋) with 𝑙
𝑋1 + 𝑙𝑋2 < 1.The latently infected individualswith XDR strain
become actively infectious as a result ofendogenous reactivation at
the rate 𝜎
𝑋.
Members of the symptomatically infectious class withthe drug
sensitive strain (𝑇) are lost to followup at therate 𝜙
𝑇and move to the class of lost to followup with
drug sensitive strain (𝐿𝑇). They are isolated into the
isolated
class (𝐽) at the rate 𝛼𝑇. They undergo the WHO recom
mended DOTS treatment (Directly Observed Treatment,Short Course
(DOTS)); however due to treatment failure(from treatment
noncompliance) they move into the latentlyinfected class at the
rate 𝑝
𝑇1𝛾𝑇 or the latently infected classwith MDR at the rate 𝑝
𝑇2𝛾𝑇. The remaining fraction moveinto the recovered class (𝑅)
following effective treatment atthe rate (1 − 𝑝
𝑇1 − 𝑝𝑇2)𝛾𝑇 (where 𝑝𝑇1 + 𝑝𝑇2 < 1). Or they candie from the
infection at the rate 𝛿
𝑇.
Similarlymembers of the symptomatically infectiouswithMDR strain
(𝑀) are lost to followup at the rate 𝜙
𝑀and
move to the class of lost to followup with MDR strain (𝐿𝑀).
They are isolated at the rate 𝛼𝑀. And a fraction of them
move into the population of the latently infected with XDRstrain
as a result of treatment failure of the symptomaticallyinfectious
individuals with MDR strain at the rate 𝑝
𝑀1𝛾𝑀.The remaining fraction move into the recovered class at
therate (1 − 𝑝
𝑀1)𝛾𝑀 (where 𝑝𝑀1 < 1). Or they can die from theinfection at
the rate 𝛿
𝑀.
Lastly, members of the symptomatically infectious withXDR strain
(𝑋) are lost to followup at the rate 𝜙
𝑋and move
to the class of lost to followup with XDR strain (𝐿𝑋). Or
they
move into recovered class at the rate 𝛾𝑋. Or they can die
from
the infection at the rate 𝛿𝑋. We assume that 𝛾
𝑋< 𝛾𝑀< 𝛾𝑇.
The individuals who are lost to followup (𝐿𝑇) with drug
sensitive strain return at the rate𝜓𝑇andmove into the class
of
individuals with drug sensitive strain. Or they die at the
rate𝛿𝐿𝑇. Similarly, the individuals who are lost to followup
(𝐿
𝑀)
withMDR strain return at the rate𝜓𝑀andmove into the class
of individuals with MDR strain. Or they die at the rate 𝛿𝐿𝑀
.Lastly, the individuals who are lost to followup (𝐿
𝑋) with
XDR strain return at the rate 𝜓𝑋and move into the class of
individuals with XDR strain. Or they die at the rate 𝛿𝐿𝑋.
Recovered individuals (𝑅) are reinfected with drug sensitive
strain at the rate 𝜀𝜆
𝑇, with 𝑙
𝑇1𝜀𝜆𝑇 fraction movinginto the latently infected class with drug
sensitive strain,𝑙𝑇2𝜀𝜆𝑇 fraction moving into the symptomatically
infectiousclass with drug sensitive strain, and the other [1 −
(𝑙
𝑇1 +𝑙𝑀2)]𝜀𝜆𝑇 moving into the symptomatically infectious lostto
followup class with drug sensitive strain. Also, these
individuals experience reinfection with MDR and XDRstrains and
fractions of these move into the latently infectedand
symptomatically infectious classes, respectively.
It follows, from the above descriptions and assumptions, that
the model for the transmission dynamics of thetuberculosis with
isolation and lost to followup is given bythe following
deterministic system of nonlinear differentialequations (the
variables and parameters of the model aredescribed in Table 1; a
schematic diagram of the model isdepicted in Figure 1):
𝑑𝑆
𝑑𝑡= 𝜋−[
𝛽𝑇(𝑇 + 𝜂
𝑇𝐿𝑇)
𝑁+𝛽𝑀(𝑀 + 𝜂
𝑀𝐿𝑀)
𝑁
+𝛽𝑋(𝑋 + 𝜂
𝑋𝐿𝑋)
𝑁] 𝑆−𝜇𝑆,
𝑑𝐸𝑇
𝑑𝑡=𝑙𝑇1𝛽𝑇 (𝑇 + 𝜂𝑇𝐿𝑇) (𝑆 + 𝜀𝑅)
𝑁+𝑝𝑇1𝛾𝑇𝑇− (𝜎𝑇
+𝜇) 𝐸𝑇,
𝑑𝑇
𝑑𝑡=𝑙𝑇2𝛽𝑇 (𝑇 + 𝜂𝑇𝐿𝑇) (𝑆 + 𝜀𝑅)
𝑁+𝜎𝑇𝐸𝑇+𝜓𝑇𝐿𝑇
− (𝜙𝑇+𝛼𝑇+ 𝛾𝑇+𝜇+ 𝛿
𝑇) 𝑇,
𝑑𝐸𝑀
𝑑𝑡=𝑙𝑀1𝛽𝑀 (𝑀 + 𝜂𝑀𝐿𝑀) (𝑆 + 𝜀𝑅)
𝑁+𝑝𝑇2𝛾𝑇𝑇− (𝜎𝑀
+𝜇) 𝐸𝑀,
𝑑𝑀
𝑑𝑡=𝑙𝑀2𝛽𝑀 (𝑀 + 𝜂𝑀𝐿𝑀) (𝑆 + 𝜀𝑅)
𝑁+𝜎𝑀𝐸𝑀
+𝜓𝑀𝐿𝑀− (𝜙𝑀+𝛼𝑀+ 𝛾𝑀+𝜇+ 𝛿
𝑀)𝑀,
𝑑𝐸𝑋
𝑑𝑡=𝑙𝑋1𝛽𝑋 (𝑋 + 𝜂𝑋𝐿𝑋) (𝑆 + 𝜀𝑅)
𝑁+𝑝𝑀1𝛾𝑀𝑀−(𝜎𝑋
+𝜇) 𝐸𝑋,
𝑑𝑋
𝑑𝑡=𝑙𝑋2𝛽𝑋 (𝑋 + 𝜂𝑋𝐿𝑋) (𝑆 + 𝜀𝑅)
𝑁+𝜎𝑋𝐸𝑋+𝜓𝑋𝐿𝑋
− (𝜙𝑋+𝛼𝑋+ 𝛾𝑋+𝜇+ 𝛿
𝑋)𝑋,
𝑑𝐿𝑇
𝑑𝑡=(1 − 𝑙𝑇1 − 𝑙𝑇2) 𝛽𝑇 (𝑇 + 𝜂𝑇𝐿𝑇) (𝑆 + 𝜀𝑅)
𝑁+𝜙𝑇𝑇
− (𝜓𝑇+𝜇+ 𝛿
𝐿𝑇) 𝐿𝑇,
𝑑𝐿𝑀
𝑑𝑡=(1 − 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑀 (𝑀 + 𝜂𝑀𝐿𝑀) (𝑆 + 𝜀𝑅)
𝑁
+𝜙𝑀𝑀−(𝜓
𝑀+𝜇+ 𝛿
𝐿𝑀) 𝐿𝑀,
𝑑𝐿𝑋
𝑑𝑡=(1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋 (𝑋 + 𝜂𝑋𝐿𝑋) (𝑆 + 𝜀𝑅)
𝑁+𝜙𝑋𝑋
− (𝜓𝑋+𝜇+ 𝛿
𝐿𝑋) 𝐿𝑋,
𝑑𝐽
𝑑𝑡= 𝛼𝑇𝑇+𝛼𝑀𝑀+𝛼
𝑋𝑋− (𝛾
𝐽+𝜇+ 𝛿
𝐽) 𝐽,
𝑑𝑅
𝑑𝑡= (1−𝑝
𝑇1 −𝑝𝑇2) 𝛾𝑇𝑇+ (1−𝑝𝑀1) 𝛾𝑀𝑀+𝛾𝑋𝑋
+𝛾𝐽𝐽 − 𝜇𝑅− 𝜀 [
𝛽𝑇(𝑇 + 𝜂
𝑇𝐿𝑇)
𝑁+𝛽𝑀(𝑀 + 𝜂
𝑀𝐿𝑀)
𝑁
+𝛽𝑋(𝑋 + 𝜂
𝑋𝐿𝑋)
𝑁]𝑅.
(2)

4 Abstract and Applied Analysis
Table 1: Variables and parameters description of model (2).
Variable Description𝑆(𝑡) Susceptible individuals𝐸𝑇(𝑡), 𝐸𝑀(𝑡),
𝐸𝑋(𝑡) Latently infected individuals with drug sensitive, MDR, and
XDR strains
𝑇(𝑡),𝑀(𝑡), 𝑋(𝑡) Symptomatically infectious individuals with drug
sensitive, MDR, and XDR strains𝐿𝑇(𝑡), 𝐿𝑀(𝑡), 𝐿𝑋(𝑡) Symptomatically
infectious individuals who are lost to sight with drug sensitive,
MDR, and XDR strains
𝐽(𝑡) Isolated individuals𝑅(𝑡) Recovered individualsParameter
Description𝜋 Recruitment rate𝜇 Natural death rate𝜀 TB reinfection
rate𝛽𝑇, 𝛽𝑀, 𝛽𝑋
Transmission probability𝜂𝑇, 𝜂𝑀, 𝜂𝑋
Infectivity modification parameter𝑙𝑇1, 𝑙𝑇2, 𝑙𝑀1, 𝑙𝑀2, 𝑙𝑋1, 𝑙𝑋2
Fraction of fast disease progression𝑝𝑇1, 𝑝𝑇2, 𝑝𝑀1 Fraction that
failed treatment𝜎𝑇, 𝜎𝑀, 𝜎𝑋
Disease progression rate𝛾𝑇, 𝛾𝑀, 𝛾𝑋, 𝛾𝐽
Recovery rate𝛼𝑇, 𝛼𝑀, 𝛼𝑋
Isolation rate𝛿𝑇, 𝛿𝑀, 𝛿𝑋, 𝛿𝐽
Diseaseinduced death rate𝜙𝑇, 𝜙𝑀, 𝜙𝑋
Lost to followup rate𝜓𝑇, 𝜓𝑀, 𝜓𝑋
Return rate from lost to followup𝛿𝐿𝑇, 𝛿𝐿𝑀, 𝛿𝐿𝑋
Diseaseinduced death rate in individuals who are lost to
followup
lT1𝜆T
lM1𝜆T lM2𝜆T
lX1𝜆X lX2𝜆X
𝜀lM1𝜆M 𝜀lM2𝜆M
pT1𝛾T
pT2𝛾T
𝜇𝜇
𝜇𝜇 𝜇
𝜇
𝜇 𝜇
𝜇
𝜇𝜇
𝜇
ET
EM
EX
S
T
M
X
LT
LM
LX
J R
𝜆T𝜆M
𝜆X
𝜋
𝜙X
𝜙T
𝜙M
𝛿M
𝛿X
𝛿J
𝛾J
𝜎T
𝜎M
𝜎X
𝛿LM
lT2𝜆T(1 − lT1 − lT2)𝜆T 𝜀(1 − lT1 − lT2)𝜆T
(1 − pT1 − pT2)𝛾T
(1 − pM1)𝛾M
(1 − lM1 − lM2)𝜆T
𝜀(1 − lM1 − lM2)𝜆M
𝜀(1 − lX1 − lX2)𝜆X
(1 − lX1 − lX2)𝜆X
pM1𝛾M
𝜀lX1𝜆X𝜀lX2𝜆X
𝛿LT
𝛿LX
𝜀lT2𝜆T
𝛼M
𝛼T
𝛼X
𝜀lT1𝜆T
𝛿T
𝜓X
𝜓T
𝜓M
Figure 1: Systematic flow diagram of the tuberculosis model
(2).
2.1. Basic Properties
2.1.1. Positivity and Boundedness of Solutions. For TB model(2)
to be epidemiologically meaningful, it can be shown(using the
method in Appendix A of [33]) that all itsstate variables are
nonnegative for all time. In other words,
solutions of the model system (2) with nonnegative initialdata
will remain nonnegative for all time 𝑡 > 0.
Lemma 1. Let the initial data 𝑆(0) ≥ 0, 𝐸𝑇(0) ≥ 0, 𝑇(0) ≥
0, 𝐸𝑀(0) ≥ 0, 𝑀(0) ≥ 0, 𝐸
𝑋(0) ≥ 0, 𝑋(0) ≥ 0, 𝐿
𝑇(0) ≥
0, 𝐿𝑀(0) ≥ 0, 𝐿
𝑋(0) ≥ 0, 𝐽(0) ≥ 0, 𝑅(0) ≥ 0.

Abstract and Applied Analysis 5
Then the solutions (𝑆, 𝐸𝑇, 𝑇, 𝐸𝑀,𝑀, 𝐸
𝑋, 𝑋, 𝐿𝑇, 𝐿𝑀, 𝐿𝑋, 𝐽, 𝑅)
of the tuberculosis model (2) are nonnegative for all 𝑡 >
0.Furthermore,
lim sup𝑡→∞
𝑁(𝑡) ≤𝜋
𝜇, (3)
with
𝑁 = 𝑆+𝐸𝑇+𝑇+𝐸
𝑀+𝑀+𝐸
𝑋+𝑋+𝐿
𝑇+𝐿𝑀
+𝐿𝑋+ 𝐽 +𝑅.
(4)
2.1.2. Invariant Regions. Since model (2) monitors
humanpopulations, all variables and parameters of the model
arenonnegative. Model (2) will be analyzed in a
biologicallyfeasible region as follows. Consider the feasible
region
Φ ⊂ R12+
(5)
with
Φ = {(𝑆, 𝐸𝑇, 𝑇, 𝐸𝑀,𝑀, 𝐸
𝑋, 𝑋, 𝐿𝑇, 𝐿𝑀, 𝐿𝑋, 𝐽, 𝑅)
∈R12+: 𝑁 (𝑡) ≤
𝜋
𝜇} .
(6)
The following steps are followed to establish the
positiveinvariance ofΦ (i.e., solutions inΦ remain inΦ for all 𝑡
> 0).The rate of change of the population is obtained by adding
theequations of model (2) and this gives
𝑑𝑁 (𝑡)
𝑑𝑡= 𝜋−𝜇𝑁 (𝑡) − 𝛿𝑇𝑇 (𝑡) − 𝛿𝑀𝑀(𝑡) − 𝛿𝑋𝑋(𝑡)
− 𝛿𝐿𝑇𝐿𝑇 (𝑡) − 𝛿𝐿𝑀𝐿𝑀 (𝑡) − 𝛿𝐿𝑋𝐿𝑋 (𝑡) .
(7)
And it follows that
𝑑𝑁 (𝑡)
𝑑𝑡≤ 𝜋−𝜇𝑁 (𝑡) . (8)
A standard comparison theorem [34] can then be used toshow
that
𝑁(𝑡) ≤ 𝑁 (0) 𝑒−𝜇𝑡 + 𝜋𝜇(1− 𝑒−𝜇𝑡) . (9)
In particular, 𝑁(𝑡) ≤ 𝜋/𝜇, if 𝑁(0) ≤ 𝜋/𝜇. Thus, region Φis
positively invariant. Hence, it is sufficient to consider
thedynamics of the flow generated by (2) in Φ. In this region,the
model can be considered as being epidemiologicallyand
mathematically wellposed [35]. Thus, every solution ofmodel (2)
with initial conditions in Φ remains in Φ for all𝑡 > 0.
Therefore, the 𝜔limit sets of system (2) are containedin Φ. This
result is summarized below.
Lemma 2. The region Φ ⊂ R12+× is positively invariant for
model (2) with nonnegative initial conditions in R12+.
2.2. Stability of the DiseaseFree Equilibrium (DFE).
Tuberculosis model (2) has a DFE, obtained by setting the
righthandsides of the equations in the model to zero, given by
E0
= (𝑆∗, 𝐸∗
𝑇, 𝑇∗, 𝐸∗
𝑀,𝑀∗, 𝐸∗
𝑋, 𝑋∗, 𝐿∗
𝑇, 𝐿∗
𝑀, 𝐿∗
𝑋, 𝐽∗, 𝑅∗)
= (𝜋
𝜇, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) .
(10)
The linear stability of E0 can be established using thenext
generation operator method on system (2). Using thenotations in
[36], the matrices 𝐹 and𝑉, for the new infectionterms and the
remaining transfer terms, are, respectively,given by
𝐹 = [𝐹1  𝐹2] , (11)
where
𝐹1 =
((((((((((((((((
(
0 𝑙𝑇1𝛽𝑇 0 0 0
0 𝑙𝑇2𝛽𝑇 0 0 0
0 0 0 𝑙𝑀1𝛽𝑀 0
0 0 0 𝑙𝑀2𝛽𝑀 0
0 0 0 0 00 0 0 0 00 (1 − 𝑙
𝑇1 − 𝑙𝑇2) 𝛽𝑇 0 0 00 0 0 (1 − 𝑙
𝑀1 − 𝑙𝑀2) 𝛽𝑀 00 0 0 0 00 0 0 0 0
))))))))))))))))
)
,

6 Abstract and Applied Analysis
𝐹2 =
((((((((((((((((
(
0 𝑙𝑇1𝛽𝑇𝜂𝑇 0 0 0
0 𝑙𝑇2𝛽𝑇𝜂𝑇 0 0 0
0 0 𝑙𝑀1𝛽𝑀𝜂𝑀 0 0
0 0 𝑙𝑀2𝛽𝑀𝜂𝑀 0 0
𝑙𝑋1𝛽𝑋 0 0 𝑙𝑋1𝛽𝑋𝜂𝑋 0𝑙𝑋2𝛽𝑋 0 0 𝑙𝑋2𝛽𝑋𝜂𝑋 00 (1 − 𝑙
𝑇1 − 𝑙𝑇2) 𝛽𝑇𝜂𝑇 0 0 00 0 (1 − 𝑙
𝑀1 − 𝑙𝑀2) 𝛽𝑀𝜂𝑀 0 0(1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋 0 0 (1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋𝜂𝑋
0
0 0 0 0 0
))))))))))))))))
)
,
𝑉 =
((((((((((((((((
(
𝑔1 −𝑝𝑇1𝛾𝑇 0 0 0 0 0 0 0 0−𝜎𝑇
𝑔2 0 0 0 0 −𝜓𝑇 0 0 00 −𝑝
𝑇2𝛾𝑇 𝑔3 0 0 0 0 0 0 00 0 −𝜎
𝑀𝑔4 0 0 0 −𝜓𝑀 0 0
0 0 0 −𝑝𝑀1𝛾𝑀 𝑔5 0 0 0 0 0
0 0 0 0 −𝜎𝑋
𝑔6 0 0 −𝜓𝑋 00 −𝜙
𝑇0 0 0 0 𝑔7 0 0 0
0 0 0 −𝜙𝑀
0 0 0 𝑔8 0 00 0 0 0 0 −𝜙
𝑋0 0 𝑔9 0
0 −𝛼𝑇
0 −𝛼𝑀
0 −𝛼𝑋
0 0 0 𝑔10
))))))))))))))))
)
,
(12)
where𝑔1 = 𝜎𝑇+𝜇, 𝑔2 = 𝜙𝑇+𝛾𝑇+𝛼𝑇+𝜎𝑇+𝜇, 𝑔3 = 𝜎𝑀+𝜇, 𝑔4 =𝜙𝑀+ 𝛾𝑀+ 𝛼𝑀+
𝜎𝑀+ 𝜇, 𝑔5 = 𝜎𝑋 + 𝜇, 𝑔6 = 𝜙𝑋 + 𝛾𝑋 +
𝛼𝑋+ 𝜎𝑋+ 𝜇, 𝑔7 = 𝜓𝑇 + 𝜇 + 𝛿𝐿𝑇, 𝑔8 = 𝜓𝑀 + 𝜇 + 𝛿𝐿𝑀, 𝑔9 =
𝜓𝑋+ 𝜇 + 𝛿
𝐿𝑋, 𝑔10 = 𝛾𝐽 + 𝜇 + 𝛿𝐽.
It follows that the basic reproduction number of tuberculosis
model (2), denoted byR0, is given by
R0 = 𝜌 (𝐹𝑉−1) = max (R
𝑇,R𝑀,R𝑋) , (13)
where
R𝑇=𝛽𝑇{𝜎𝑇(𝑔7 + 𝜙𝑇𝜂𝑇) 𝑙𝑇1 + (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑔1𝑙𝑇2 + [(𝑔2𝜂𝑇 + 𝜓𝑇) 𝑔1 −
𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇] (1 − 𝑙𝑇1 − 𝑙𝑇2)}
[𝑔1 (𝑔7𝑔2 − 𝜙𝑇𝜓𝑇) − 𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇],
R𝑀=𝛽𝑀[𝜎𝑀(𝑔8 + 𝜙𝑀𝜂𝑀) 𝑙𝑀1 + (𝑔8 + 𝜂𝑀𝜙𝑀) 𝑔3𝑙𝑀2 + (𝑔4𝜂𝑀 + 𝜓𝑀) 𝑔3 (1
− 𝑙𝑀1 − 𝑙𝑀2)]
𝑔3 (𝑔8𝑔4 − 𝜙𝑀𝜓𝑀),
R𝑋=𝛽𝑋[𝜎𝑋(𝑔9 + 𝜙𝑋𝜂𝑋) 𝑙𝑋1 + (𝑔9 + 𝜂𝑋𝜙𝑋) 𝑔5𝑙𝑋2 + (𝑔6𝜂𝑋 + 𝜓𝑋) 𝑔5 (1
− 𝑙𝑋1 − 𝑙𝑋2)]
𝑔5 (𝑔9𝑔6 − 𝜙𝑋𝜓𝑋).
(14)
Quantity R𝑇represents the reproduction number of TB
drug sensitiveonly population. Similarly, quantity R𝑀
isthe reproduction number for MDRonly population and
thequantity R
𝑋represents the reproduction number for XDR
only TB population.Further, using Theorem 2 in [36], the
following result is
established.
Lemma 3. TheDFE of the tuberculosis model (2), given byE0,is
locally asymptotically stable (LAS) ifR0 < 1, and unstable ifR0
> 1.
The threshold quantity (R0, i.e., the basic reproductionnumber)
measures the average number of new infectionsgenerated by a single
infected individual in a completely

Abstract and Applied Analysis 7
𝜂X𝜂M𝜂TlX2lX1lM2lM1lT2lT1
𝛿J𝛿X𝛿M𝛿T𝜙X𝜙M𝜙T𝜓X𝜓M𝜓T𝜖
pM1pT2pT1𝛼T𝛼X𝛼M𝜎X𝜎M𝜎T𝛽X𝛽M𝛽T𝛾J𝛾X𝛾M𝛾T𝜇
0 0.5 1−0.5
𝜋
𝛿LX𝛿LM𝛿LT
(a)
𝜂X𝜂M𝜂TlX2lX1lM2lM1lT2lT1
𝛿J𝛿X𝛿M𝛿T𝜙X𝜙M𝜙T𝜓X𝜓M𝜓T𝜖
pM1pT2pT1𝛼T𝛼X𝛼M𝜎X𝜎M𝜎T𝛽X𝛽M𝛽T𝛾J𝛾X𝛾M𝛾T𝜇
0 0.2 0.4 0.6 0.8 1−0.8 −0.6 −0.4 −0.2
𝜋
𝛿LX𝛿LM𝛿LT
(b)𝜂X𝜂M𝜂TlX2lX1lM2lM1lT2lT1
𝛿J𝛿X𝛿M𝛿T𝜙X𝜙M𝜙T𝜓X𝜓M𝜓T𝜖
pM1pT2pT1𝛼T𝛼X𝛼M𝜎X𝜎M𝜎T𝛽X𝛽M𝛽T𝛾J𝛾X𝛾M𝛾T𝜇
0 0.2 0.4 0.6 0.8 1−0.8 −0.6 −0.4 −0.2
𝜋
𝛿LX𝛿LM𝛿LT
(c)
Figure 2: PRCC values for model (2), using as the response
function (a) the basic reproduction number (R𝑇), (b) the basic
reproduction
number (R𝑀), and (c) the basic reproduction number (R
𝑋). Parameter values (baseline) and ranges used are as given in
Table 2.
susceptible population [35–38]. Thus, Lemma 3 implies
thattuberculosis can be eliminated from the population (whenR0 <
1) if the initial sizes of the subpopulations of model(2) are in
the basin of attraction of the DFE, E0.
3. Sensitivity Analysis
A global sensitivity analysis [39–42] is carried out, on
theparameters of model (2), to determine which of the parameters
have the most significant impact on the outcome ofthe numerical
simulations of the model. Figure 2(a) depictsthe partial rank
correlation coefficient (PRCC) values foreach parameter of the
models, using the ranges and baselinevalues tabulated in Table 2
(with the basic reproductionnumbers, R
𝑇, as the response function), from which it
follows that the parameters that have the most influence ondrug
sensitive TB transmission dynamics are the fraction offast
progression rate (𝑙
𝑇2) into the drug sensitive TB class,the infectivity
modification parameter (𝜂
𝑇), the recovery rate
(𝛾𝑇) from drug sensitive TB, disease progression rate (𝜎
𝑇),
rate of lost to followup (𝜙𝑇) of those with drug sensitive
TB, fraction of latently infected with drug sensitive TB
thatfailed treatment (𝑝
𝑇1), the fraction of fast progression rate(𝑙𝑇1) into the
latently infected with drug sensitive TB class,
and the isolation rate (𝛼𝑇) from drug sensitive TB class.
The identification of these key parameters is vital to
theformulation of effective control strategies for combating
thespread of the disease, as this study identifies the most
important parameters that drive the transmissionmechanism of
thedisease. In other words, the results of this sensitivity
analysissuggest that, to effectively control the spread of drug
sensitiveTB in the community, the effective strategy will be to
reducethe disease progression rate (reduce𝜎
𝑇), increase the recovery
rate (increase 𝛾𝑇) from drug sensitive TB, reduce the
disease
modification parameter (reduce 𝜂𝑇), increase the isolation
rate (increase 𝛼𝑇) from drug sensitive TB class, and reduce
the fraction of fast progression rates (reduce 𝑙𝑇1 and 𝑙𝑇2)
into
the drug sensitive TB class and lost to followup class
andreduce the rate of lost to followup (reduce 𝜙
𝑇) of those with
drug sensitive TB.The result of this analysis suggests that
thefraction of latently infected with drug sensitive TB that
failedtreatment (𝑝
𝑇1) be increased, since this has a negative impactof the basic
reproduction number, R
𝑇. This of course is
counter intuitive; however increasing this rate lowers the
rateof development of drugresistant TB due to treatment
failure.
The sensitivity analysis was also carried out using model(2)
with the basic reproduction number, R
𝑀, for drug
resistant TB, as the response function (see Figure 2(b)).
Thedominant parameters in this case are 𝛼
𝑀, 𝜎𝑀, 𝜂𝑀, 𝛾𝑀, 𝜙𝑀,
𝜓𝑀, 𝜇, 𝑙𝑀1, and 𝑙𝑀2. Similarly, when using as response func
tion the basic reproduction number, R𝑋, for extended drug

8 Abstract and Applied Analysis
resistance TB (see Figure 2(c)), the dominant parameters inthis
case are 𝛼
𝑋, 𝜎𝑋, 𝜂𝑋, 𝛾𝑋, 𝜙𝑋, 𝜓𝑋, 𝜇, 𝑙𝑋1, and 𝑙𝑋2.
The results from these analyses using as response functions the
basic reproduction numbers,R
𝑀andR
𝑋, suggest
that the natural death rate (𝜇) be increased, since it has
anegative impact on the basic reproduction numbers,R
𝑀and
R𝑋. However increasing this rate is not epidemiologically
relevant, as it implies reducing the population size thatwe wish
to preserve by other means aside from death bytuberculosis and
should therefore be ignored in any controlmeasures.
4. Analysis of the Reproduction Number
Following the result obtained in Section 3, we investigate
inthis section whether or not treatmentonly, isolationonlyof
individuals with tuberculosis or a combination of bothcan lead to
tuberculosis elimination in the population. Theanalysis will be
carried out using the reproduction numberfor drug sensitive TB
(R
𝑇) since R0 is the maximum of the
reproduction number of drug sensitive TB (R𝑇), MDRTB
(R𝑀), and XDRTB (R
𝑋); similar result can be obtained for
the MDR and XDRTB.In the absence of isolation (𝛼
𝑇= 0), the reproduction
number (R𝑇) reduces to
R𝑇𝛼
=𝛽𝑇[(𝜎𝑇𝑔7 + 𝜂𝑇𝜎𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝑔1𝜙𝑇 + 𝑔1𝑔7) 𝑙𝑇2 + (𝑔1𝜓𝑇 + 𝜂𝑇𝑔1 (𝜙𝑇
+ 𝛾𝑇 + 𝜇 + 𝛿𝑇) − 𝜂𝑇𝜎𝑇𝑝𝑡1𝛾𝑇) (1 − 𝑙𝑇1 − 𝑙𝑇2)]
𝑔1 (𝜙𝑇 + 𝛾𝑇 + 𝜇 + 𝛿𝑇) 𝑔7 − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7.
(15)
The reproduction number (R𝑇) can be written as
R𝑇= 𝐴𝛼R𝑇𝛼
, (16)
where
𝐴𝛼
=(𝑔1𝑔7 (𝜙𝑇 + 𝛾𝑇 + 𝜇 + 𝛿𝑇) − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7) [(𝜎𝑇𝑔7 + 𝜂𝑇𝜎𝑇𝜙𝑇)
𝑙𝑇1 + (𝜂𝑇𝑔1𝜙𝑇 + 𝑔1𝑔7) 𝑙𝑇2 + (𝑔1𝜓𝑇 + 𝜂𝑇𝑔1𝑔2 − 𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇) (1 − 𝑙𝑇1 −
𝑙𝑇2)]{(𝑔1𝑔2𝑔7 − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7) [𝜎𝑇 (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝜙𝑇 +
𝑔7) 𝑔1𝑙𝑇2 + (𝜂𝑇𝑔1 (𝜙𝑇 + 𝛾𝑇 + 𝜇 + 𝛿𝑇) + 𝑔1𝜓𝑇 − 𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇) (1 − 𝑙𝑇1 −
𝑙𝑇2)]}
.
(17)
The difference between R𝑇and R
𝑇𝛼
is in the isolation rate(𝛼𝑇); as such the factor 𝐴
𝛼compares a population with
and without isolation; however, this is in the presence
oftreatment and individuals who are lost to followup
andindividuals returning from lost to followup. If R
𝑇𝛼
< 1,then drug sensitive TB cannot develop into an epidemicin
the community. However, if R
𝑇𝛼
> 1, it is imperative
to investigate the effect of isolation on the transmission
ofdrug sensitive TB among the populace and determine thenecessary
condition for slowing down its development in thecommunity.
Following [43] we have
Δ𝛼= R𝑇𝛼
−R𝑇= (1−𝐴
𝛼)R𝑇𝛼
, (18)
where
Δ𝛼=[𝛽𝑇𝑔1 (𝑔7 + 𝜂𝑇𝜙𝑇) (𝜙𝑇 − 𝑔2 + 𝛾𝑇 + 𝜇 + 𝛿𝑇) (𝜎𝑇𝑔7𝑙𝑇1 + 𝑔1𝑔7𝑙𝑇2
+ 𝑔1𝜓𝑇 (1 − 𝑙𝑇1 − 𝑙𝑇2))]
{(𝑔1𝑔2𝑔7 − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7) [𝑔1𝑔7 (𝜙𝑇 + 𝛾𝑇 + 𝜇 + 𝛿𝑇) − 𝑔1𝜓𝑇𝜙𝑇
− 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7]}. (19)
To slow down the spread of drug sensitive tuberculosis inthe
population via effective isolation, proper treatment,
andidentification of individuals who return from lost to followup
and in the presence of individuals who are lost to followup, we
expect that Δ
𝛼> 0, and this is satisfied if 𝐴
𝛼< 1
in (18). Now, setting R𝑇= 1 and solving for 𝐴
𝛼gives the
threshold effectiveness of isolation and taking into account
treatment and identification of individuals who return fromlost
to followup and who are also lost to followup:
𝐴∗
𝛼=
1R𝑇𝛼
. (20)
Hence, drug sensitive tuberculosis can be eradicated inthe
community in the presence of isolation taking into

Abstract and Applied Analysis 9
consideration proper treatment and identification of
individuals who return from lost to followup and are lost to
followup if 𝐴
𝛼< 𝐴∗
𝛼is attained. Note that 𝐴∗
𝛼is a decreasing
function ofR𝑇𝛼
, thus indicating that higher values for𝐴∗𝛼will
result in smaller values forR𝑇𝛼
, a desired outcome. But a large
value for R𝑇𝛼
results in a small value for 𝐴∗𝛼, an indication
that eradication may not be attainable.The following limits of
𝐴
𝛼provide a further insight into
possible ways of reducing the burden of drug sensitive TB inthe
community:
lim𝐴𝛼𝛼𝑇→∞
=𝑙𝑇2𝜂𝑇 [(𝜙𝑇 + 𝜇 + 𝛿𝑇 + 𝛾𝑇) 𝑔1𝑔7 − 𝜓𝑇𝜙𝑇𝑔1 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7]
𝑔7 {𝜎𝑇 (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝜙𝑇 + 𝑔7) 𝑔1𝑙𝑇2 + [𝜓𝑇𝑔1 + 𝜂𝑇 (𝜙𝑇 + 𝛾𝑇
+ 𝜇 + 𝛿𝑇) 𝑔1 − 𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇] (1 − 𝑙𝑇1 − 𝑙𝑇2)},
(21)
lim𝐴𝛼𝜙𝑇→∞
= 1,
lim𝐴𝛼𝜓𝑇→∞
= 1,
lim𝐴𝛼𝛾𝑇→∞
= 1.
(22)
The limits in (21) will be less than unity. In practice, 𝛼𝑇→
∞
implies high rate of isolating individuals with drug
sensitiveTB, 𝜙
𝑇→ ∞ implies high rate of individuals who are lost
to followup, 𝜓𝑇
→ ∞ implies a high rate of individualswho return from lost to
followup, and 𝛾
𝑇→ ∞ implies
high treatment rate. Ideally, the results obtained from
theselimits can be pursued for the reduction of the burden of
drug sensitive TB in the community, provided of course thatit is
feasible and practicable economically. Therefore, with alook at
factor 𝐴
𝛼, one observes that an effective isolation will
lead to a reduction in the burden of drug sensitive TB in
thepopulation.
Next, we consider the case when the treatment rate is setto zero
(𝛾
𝑇= 0). The reproduction number is given as
R𝑇𝛾
=𝛽𝑇[(𝜎𝑇𝑔7 + 𝜂𝑇𝜎𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝑔1𝜙𝑇 + 𝑔1𝑔7) 𝑙𝑇2 + [𝑔1𝜓𝑇𝜂𝑇𝑔1 (𝜙𝑇 +
𝛼𝑇 + 𝜇 + 𝛿𝑇)] (1 − 𝑙𝑇1 − 𝑙𝑇2)]
𝑔1 (𝜙𝑇 + 𝛼𝑇 + 𝜇 + 𝛿𝑇) 𝑔7 − 𝑔1𝜓𝑇𝜙𝑇. (23)
The reproduction numberR𝑇can be expressed as
R𝑇= 𝐴𝛾R𝑇𝛾
, (24)
where
𝐴𝛾
={[(𝜎𝑇𝑔7 + 𝜂𝑇𝜎𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝑔1𝜙𝑇 + 𝑔1𝑔7) 𝑙𝑇2 + (𝑔1𝜓𝑇 + 𝜂𝑇𝑔1𝑔2 −
𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇) (1 − 𝑙𝑇1 − 𝑙𝑇2)] 𝑔1 [(𝜙𝑇 + 𝛼𝑇 + 𝜇 + 𝛿𝑇) 𝑔7 − 𝜓𝑇𝜙𝑇]}
{(𝑔1𝑔2𝑔7 − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑡1𝛾𝑇𝑔7) [(𝜎𝑇𝑔7 + 𝜂𝑇𝜎𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝑔1𝜙𝑇 +
𝑔1𝑔7) 𝑙𝑇2 + (𝑔1𝜓𝑇 + (𝜙𝑇 + 𝛼𝑇 + 𝜇 + 𝛿𝑇) 𝑔1𝜂𝑇) (1 − 𝑙𝑇1 − 𝑙𝑇2)]}.
(25)
The difference between R𝑇𝛾
and R𝑇is in the treatment
rate (𝛾𝑇); thus 𝐴
𝛾compares a population with and without
treatment in the presence of isolation of infected
individualswith drug sensitive TB, individuals who are lost to
followup and who return from of lost to followup. If R
𝑇𝛾
< 1,then drug sensitive TB cannot develop into an epidemic
in
the community and no control strategy will be required forits
control. Take the difference betweenR
𝑇andR
𝑇𝛾
; that is,
Δ𝛾= R𝑇−R𝑇𝛾
= (1−𝐴𝛾)R𝑇𝛾
, (26)
where
Δ𝛾={𝛽𝑇(𝑔7 + 𝜂𝑇𝜙𝑇) [𝑔1 (𝜙𝑇 + 𝜇 + 𝛿𝑇 + 𝛼𝑇 − 𝑔2) + 𝛾𝑇𝑝𝑇1𝜎𝑇]
[𝑔1𝑔7𝑙𝑇2 + 𝜎𝑇𝑔7𝑙𝑇1 + 𝑔1𝜓𝑇 (1 − 𝑙𝑇1 − 𝑙𝑇2)]}
{𝑔1 (𝑔1𝑔2𝑔7 − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7) [𝑔7 (𝜙𝑇 + 𝛼𝑇 + 𝜇 + 𝛿𝑇) −
𝜓𝑇𝜙𝑇]}. (27)

10 Abstract and Applied Analysis
To slow down the spread of drug sensitive tuberculosis inthe
population using proper treatment, effective isolationand
identification of individuals who return from lost tofollowup and
in the presence of those who are of lost to
followup, we expect that Δ𝛾> 0, and this is satisfied if
𝐴𝛾< 1 in (18).Take the following limits of 𝐴
𝛾:
lim𝐴𝛾𝛾𝑇→∞
=[(𝜙𝑇+ 𝛼𝑇+ 𝜇 + 𝛿
𝑇) 𝑔7 − 𝜓𝑇𝜙𝑇] 𝜂𝑇𝑔1 (1 − 𝑙𝑇1 − 𝑙𝑇2)
𝑔7 {𝜎𝑇 (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑙𝑇1 + (𝜂𝑇𝜙𝑇 + 𝑔7) 𝑔1𝑙𝑇2 + [𝜓𝑇𝑔1 + 𝜂𝑇 (𝜙𝑇 + 𝛾𝑇
+ 𝜇 + 𝛿𝑇) 𝑔1 − 𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇] (1 − 𝑙𝑇1 − 𝑙𝑇2)},
lim𝐴𝛾𝜙𝑇→∞
= 1,
lim𝐴𝛾𝜓𝑇→∞
= 1,
lim𝐴𝛾𝛼𝑇→∞
= 1.
(28)
From the limit of𝐴𝛾, one observes that an effective
treatment
will lead to a reduction in the burden of drug sensitive TB
inthe population.
Comparing the quantities 𝐴𝛼and 𝐴
𝛾shows that 𝐴
𝛼<
𝐴𝛾; that is, size
𝐴𝛼−𝐴𝛾= − {[(𝑔7 + 𝜂𝑇𝜙𝑇) + (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑙𝑇2𝑔1 + (𝜓𝑇𝑔1 + 𝜂𝑇𝑔1𝑔2 −
𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇) (1− 𝑙𝑇1 − 𝑙𝑇2)] [𝑔1 (𝛼𝑇 − 𝛾𝑇)
+ 𝜎𝑇𝑝𝑇1𝛾𝑇] (𝑔7 + 𝜂𝑇𝜙𝑇) [𝜎𝑇𝑔7𝑙𝑇1 +𝑔1𝑔7𝑙𝑇2 +𝑔1𝜓𝑇 (1− 𝑙𝑇1 − 𝑙𝑇2)]}
({(𝑔1𝑔2𝑔7 −𝑔1𝜓𝑇𝜙𝑇 −𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7)
⋅ [𝜎𝑇(𝑔7𝜂𝑇𝜙𝑇) 𝑙𝑇1 + (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑔1𝑙𝑇2 + (𝜓𝑇𝑔1 + 𝜂𝑇𝑔1 (𝜙𝑇 + 𝛼𝑇 +
𝜇 + 𝛿𝑇)) (1 − 𝑙𝑇1 − 𝑙𝑇2)]
⋅ [𝜎𝑇(𝑔7 + 𝜂𝑇𝜙𝑇) 𝑙𝑇1 + (𝑔7 + 𝜂𝑇𝜙𝑇) 𝑔1𝑙𝑇2 + (𝜓𝑇𝑔1 + 𝜂𝑇𝑔1 (𝜙𝑇 + 𝛾𝑇
+ 𝜇 + 𝛿𝑇) − 𝜂𝑇𝜎𝑇𝑝𝑇1𝛾𝑇) (1 − 𝑙𝑇1 − 𝑙𝑇2)]})
−1< 0.
(29)
This implies that 𝐴𝛾will provide better results in slowing
down drug sensitive tuberculosis spread, using reduction inthe
prevalence, than using𝐴
𝛼. On the other hand if𝐴
𝛼−𝐴𝛾>
0, this means that 𝐴𝛼will give better outcome in slowing
down drug sensitive tuberculosis spread compared to
using𝐴𝛾.Thus, from the above discussions, to slow down the
spread of the disease and reduce the number of
secondaryinfections in the population, we require control
strategieswith parameter values that would make 𝐴
𝛼< 1 or 𝐴
𝛾<
1. Hence, the necessary condition for slowing down
thedevelopment of drug sensitive TB at the population level isthat
Δ
𝛼> 0 or Δ
𝛾> 0. However, Δ
𝛼gives a better result
in terms of reduction in the prevalence of the disease
overΔ𝛾provided Δ
𝛾> Δ𝛼; otherwise if Δ
𝛼> Δ𝛾, then Δ
𝛾
gives a better result over Δ𝛼. Using parameters in Table 1,
we
have that 𝐴𝛼= 0.8524, 𝐴
𝛾= 0.8802 and Δ
𝛼= 0.1061,
Δ𝛾= 0.0833; thus 𝐴
𝛼− 𝐴𝛾= −0.02789. It follows that, the
isolationonly strategy provides more effective control
measures in curtailing the disease transmission in the
community.
Using the threshold quantity, R𝑇, we determine how
isolation and treatment rates could lead to
tuberculosiselimination in the population. Thus
limR𝑇𝛼𝑇→∞
=𝛽𝑇(1 − 𝑙𝑇1 − 𝑙𝑇2) 𝜂𝑇
(𝜓𝑇+ 𝜇 + 𝛿
𝑇)
> 0,
limR𝑇𝛾𝑇→∞
=𝛽𝑇(1 − 𝑙𝑇1 − 𝑙𝑇2) 𝜂𝑇
(𝜓𝑇+ 𝜇 + 𝛿
𝑇)
> 0.
(30)
Thus a sufficient effective TB control program that isolates
(ortreats) the identified cases at a high rate 𝛼
𝑇→ ∞ (or 𝛾
𝑇→
∞) can lead to effective disease control if it results in
makingthe respective righthand side of (30) less than unity.
Differentiating partially the reproduction number of 𝜕R𝑇
with respect to the key parameters (𝛼𝑇and 𝛾𝑇), this gives
𝜕R𝑇
𝜕𝛼𝑇
=−𝛽𝑇𝑔1 (𝑔7 + 𝜂𝑇𝜙𝑇) [𝑙𝑇1𝜎𝑇𝑔7 + 𝑙𝑇2𝑔1𝑔7 + (1 − 𝑙𝑇1 − 𝑙𝑇2)
𝑔1𝜓𝑇]
[𝑔1𝑔7 (𝜙𝑇 + 𝛼𝑇 + 𝛾𝑇 + 𝜇 + 𝛿𝑇) − 𝑔1𝜓𝑇𝜙𝑇 − 𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7]2 , (31)
𝜕R𝑇
𝜕𝛾𝑇
=−𝛽𝑇(𝑔1 − 𝜎𝑇𝑝𝑡1) (𝑔7 + 𝜂𝑇𝜙𝑇) [𝑙𝑇1𝜎𝑇𝑔7 + 𝑙𝑇2𝑔1𝑔7 + (1 − 𝑙𝑇1 −
𝑙𝑇2) 𝑔1𝜓𝑇][𝑔1𝑔7 (𝜙𝑇 + 𝛼𝑇 + 𝛾𝑇 + 𝑔1𝑔7𝜇 + 𝛿𝑇) − 𝑔1𝜓𝑇𝜙𝑇 −
𝜎𝑇𝑝𝑇1𝛾𝑇𝑔7]
2 . (32)

Abstract and Applied Analysis 11
Table 2: Values of the parameters of the model (2).
Parameter Baselinevalues Range Reference
𝜋 𝜇 × 105 (0.0143 ×105, 0.04 × 105) [11]
𝜇 0.0159 (0.0143, 0.04) [12]𝜀 0.06 (0.01, 0.5) [13]𝛽𝑇 9.75 (4.5,
15.0) [14]𝛽𝑀 1.5 (1.5, 3.5) [13, 15]𝛽𝑋 0.0000085 (0.01, 0.1)
[13]𝜂𝑇, 𝜂𝑀, 𝜂𝑋 0.5 (0, 1) Assumed
𝑙𝑇1, 𝑙𝑇2, 𝑙𝑀1, 𝑙𝑀2, 𝑙𝑋1, 𝑙𝑋2 0.14 (0.02, 0.3) [12, 16, 17]𝑝𝑇1
0.3 (0.1, 0.5) [14, 18]𝑝𝑇2 0.03 (0.01, 0.05) [14, 18]𝑝𝑀1 0.03
(0.01, 0.1) [14, 18]
𝜎𝑇, 𝜎𝑀, 𝜎𝑋
0.05, 0.0018,0.013 (0.005, 0.05) [12, 19, 20]
𝛾𝑇, 𝛾𝐽 1.5 (1.5, 2.5) [11, 16, 21]
𝛾𝑀, 𝛾𝑋 0.75 (0.5, 1.0) [13]
𝛼𝑇, 𝛼𝑀, 𝛼𝑋 0.6 (0.2, 1.0) Assumed
𝜙𝑇, 𝜙𝑀, 𝜙𝑋 0.2511 (0.0022, 0.5) [22]
𝜓𝑇, 𝜓𝑀, 𝜓𝑋 0.1 (0.5, 1.0) [23]
𝛿𝑇, 𝛿𝐽 0.365 (0.22, 0.39) [20, 24, 25]
𝛿𝑀, 𝛿𝑋 0.028 (0.01, 0.03) [13]
𝛿𝐿𝑇, 𝛿𝐿𝑀
, 𝛿𝐿𝑋 0.02 (0.01, 0.039) [22, 23]
Thus, it follows from (31) that 𝜕R𝑇/𝜕𝛼𝑇
< 0, henceshowing further the effectiveness of the control
measures.Thus, isolation (𝛼
𝑇) of drug sensitive tuberculosis will have
a positive impact in reducing the drug sensitive TB burdenin the
community, regardless of the values of the otherparameters. This
result is stated in the following lemma.
Lemma 4. The use of isolation (𝛼𝑇) will have a positive
impact on the reduction of the drug sensitive TB burden in
acommunity regardless of the values of other parameters in thebasic
reproduction number under isolation.
Similarly, from (32), we have that 𝜕R𝑇/𝜕𝛾𝑇
< 0.Thus, effective treatment (𝛾
𝑇) of drug sensitive tuberculosis
will have a positive impact in reducing the drug
sensitivetuberculosis burden in the community, irrespective of
thevalues of the other parameters. This result is
summarizedbelow.
Lemma 5. The use of effective treatment (𝛾𝑇) will have a
positive impact on the reduction of the drug sensitive TB
burdenin a community irrespective of the values of other parameters
inthe basic reproduction number under treatment.
A contour plot of the reproduction number R𝑇, as a
function of the effective treatment rate (𝛾𝑇) and isolation
rate (𝛼𝑇), is depicted in Figure 3(a). As expected, the plot
shows a decrease in R𝑇values with increasing values of
the treatment and isolation rates. For instance, if the useof
effective treatment result in 𝛾
𝑇= 0.8 and 𝛼
𝑇= 0.8,
drug sensitive TB burden will be reduced considerably in
thepopulation. Similarly in Figure 3(b) the plot shows a decreasein
R𝑇values with decreasing values of the return rate from
lost to followup (𝜓𝑇) and lost to followup rate (𝜙
𝑇).
A contour plot of the reproduction number R𝑇, as a
function of return rate from lost to followup (𝜓𝑇) and the
effective treatment rate (𝛾𝑇), is depicted in Figure 4(a).
The
plot shows a decrease inR𝑇values with increasing values of
the treatment rate. Similarly the plot in Figure 4(b) shows
adecrease inR
𝑇values with increasing values of the isolation
rate (𝛼𝑇).
4.1. Backward Bifurcation Analysis. Model (2) is now
investigated for the possibility of the existence of the
phenomenonof backward bifurcation (where a stable DFE coexists with
astable endemic equilibrium when the reproduction number,R0, is
less than unity) [44–53]. The epidemiological implication of
backward bifurcation is that the elimination (oreffective control)
of the TB (and various strains) in the systemis no longer
guaranteed when the reproduction number isless than unity but is
dependent on the initial sizes of thesubpopulations. The
possibility of backward bifurcation inmodel (2) is explored using
the centre manifold theory [47],as described in [54] (Theorem
4.1).
Theorem 6. Model (2) undergoes a backward bifurcation atR𝑇=
1whenever inequality (A.9), given inAppendix A, holds.
Theproof ofTheorem 6 is given in Appendix A (the proofcan be
similarly given for the case whenR
𝑀= 1 orR
𝑋= 1).
The backward bifurcation property of model (2) is illustratedby
simulating themodel using a set of parameter values givenin Table 2
(such that the bifurcation parameters, 𝑎 and 𝑏,given in Appendix A,
take the values 𝑎 = 558.61 > 0 and 𝑏 =1.59 > 0, resp.). The
backward bifurcation phenomenon ofmodel (2) makes the effective
control of the TB strains in thepopulation difficult, since, in
this case, disease control whenR0 < 1 is dependent on the
initial sizes of the subpopulationsof model (2). This phenomenon is
illustrated numerically inFigures 5 and 6 for individuals with drug
sensitive, MDR,and XDRTB, as well as individuals who are lost to
followupwith drug sensitive, MDR, and XDRTB, respectively.
It is worth mentioning that when the reinfection parameter of
model (2), for the recovered individuals, is set to zero(i.e., 𝜀 =
0), the bifurcation parameter, 𝑎, becomes negative(see Appendix A).
This rules out backward bifurcation (inline with Item (iv) of
Theorem 4.1 of [54]) in this case. Thus,this study shows that the
reinfection of recovered individualscauses backward bifurcation in
the transmission dynamicsof TB in the system. To further confirm
the absence of thebackward bifurcation phenomenon in model (2) for
thiscase, the global asymptotic stability of the DFE of the modelis
established below for the case when no reinfection ofrecovered
individuals occurs.

12 Abstract and Applied Analysis
4
4
6
6
6
6
8
8
8
10
10
1214
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1𝛾T
𝛼T
(a)
2.5
33
33
3.53.5
3.5
4
4
4.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
𝜙T
𝜓T
(b)
Figure 3: Contour plot of the reproduction number (R𝑇) of model
(2) as: (a) a function of isolation rate (𝛼
𝑇) and treatment rate (𝛾
𝑇); (b) a
function of return rate from lost to followup (𝜓𝑇) and lost to
followup rate (𝜙
𝑇). Parameter values used are as given in Table 2.
4.5 4.5 4.55
5 5 55.5 5.5 5.56
6 6 66.5 6.5 6.57
7 7 77.5 7.5 7.58 8 8 88.5 8.5 8.590 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
𝛾T
𝜓T
(a)
2.6
2.8 2.82.8
33 3 3
3.2 3.23.2
3.4 3.43.4
3.6 3.63.6
3.8 3.83.8
4 40 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1𝛼T
𝜓T
(b)
Figure 4: Contour plot of the reproduction number (R𝑇) of model
(2) as: (a) a function of return rate from lost to followup (𝜓
𝑇) and
treatment rate (𝛾𝑇); (b) a function of return rate from lost to
followup (𝜓
𝑇) and treatment rate (𝛼
𝑇). Parameter values used are as given in
Table 2.
4.2. Global Stability of the DFE: Special Case. Consider
thespecial case of model (2) where the reinfection parametersare
set to zero (i.e., 𝜀 = 0). It is convenient to define
thereproduction threshold R̃0 = R0𝜀=0.
Theorem 7. The DFE of model (2), with 𝜀 = 0, is GAS in Φwhenever
R̃0 < 1.
The proof of Theorem 7 is given in Appendix B.The
epidemiological significance ofTheorem 7 is that, for
the special case of model (2) with 𝜀 = 0, TB will be
eliminatedfrom the community if the reproduction number (R̃0) can
bebrought to (and maintained at) a value less than unity.
5. The Effects of Isolation
Following the result obtained from the sensitivity analysis,
weinvestigate the impact of the isolation parameters 𝛼
𝑇, 𝛼𝑀, and
𝛼𝑋, which are one of the dominant parameters of model (2).
We start by individually varying these parameters for (say)𝛼𝑇=
0.2, 0.4, 0.6, 0.8, 1.0, with the other parameters given
in Table 2 kept constant. We observed that (see Figure 7) asthe
isolation rate, 𝛼
𝑇, for drug sensitive TB increases, the
total number of individuals (with drug sensitive, MDR, andXDR)
isolatedwith each strain of TB increases, while the totalnumber of
individuals who are lost to followup decreases.We observed similar
result for the isolation rate 𝛼
𝑀(see
Figure 8). However, for isolation rate 𝛼𝑋, negligible change
was observed and the plots are not shown.
6. Conclusion
In this paper, we have developed and analyzed a system
ofordinary differential equations for the transmission dynamics of
drugresistant tuberculosis with isolation. From ouranalysis, we
have the following results which are summarizedbelow:

Abstract and Applied Analysis 13
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T
800
700
500
300
100
600
400
200
0
(a)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M
0
20
40
60
80
100
120
140
160
(b)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
0
5
10
15
20
25
(c)
Figure 5: Backward bifurcation plot of model (2) as a function
of time. (a) Individuals with drug sensitive TB (𝑇). (b)
Individuals with MDR(𝑀). (c) Individuals with XDR (𝑋). Parameter
values used are as given in Table 2.
(i) Themodel is locally asymptotically stable (LAS)R0 <1 and
unstable whenR0 > 1.
(ii) The model exhibits in the presence of disease reinfection
the phenomenon of backward bifurcation, wherethe stable
diseasefree coexists with a stable endemicequilibrium, when the
associated reproduction number is less than unity.
(iii) As the isolation rate for each strain of TB increases,the
total number of individuals infected with theparticular strain of
TB decreases.
(iv) Model (2) in the absence of disease reinfection isglobally
asymptotically stable (GAS)R0 < 1.
(v) The sensitivity analysis of the model shows that thedominant
parameters for the drug sensitive TB are
the disease progression rate (𝜎𝑇), the recovery rate
(𝛾𝑇) from drug sensitive TB, the infectivity parameter
(𝜂𝑇), the isolation rate (𝛼
𝑇) from drug sensitive TB
class, fraction of fast progression rates and (𝑙𝑇1 and
𝑙𝑇2) into the drug sensitive TB class and lost to followup
class, and the rate of lost to followup (𝜙
𝑇). Similar
parameters and return rates from lost to followup(𝜓𝑀
and 𝜓𝑋) are dominant for MDR and XDR
TB. The natural death rate (𝜇), although dominantin MDR and
XDRTB, is however epidemiologicallyirrelevant.
(vi) Increase in isolation rate leads to increase in totalnumber
of individuals isolated with each TB strainresulting in decreases
in the total number of individuals who are lost to followup.

14 Abstract and Applied Analysis
Appendices
A. Proof of Theorem 6
Proof. Theproof is based on using the centremanifold theory[47],
as described in [54]. It is convenient to make thefollowing
simplification and change of variables.
Let 𝑆 = 𝑥1, 𝐸𝑇 = 𝑥2, 𝑇 = 𝑥3, 𝐸𝑀 = 𝑥4, 𝑀 = 𝑥5, 𝐸𝑋 =𝑥6, 𝑋 = 𝑥7, 𝐿𝑇
= 𝑥8, 𝐿𝑀 = 𝑥9, 𝐿𝑋 = 𝑥10, 𝐽 = 𝑥11and 𝑅 = 𝑥12, so that 𝑁 = 𝑥1 + 𝑥2 +
𝑥3 + 𝑥4 + 𝑥5 +𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12. Using the
vectornotation x = (𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑥7, 𝑥8, 𝑥9, 𝑥10, 𝑥11,
𝑥12)
𝑇,model (2) can be written in the form 𝑑x/𝑑𝑡 = m(x), wherem =
(𝑚1, 𝑚2, 𝑚3, 𝑚4, 𝑚5, 𝑚6, 𝑚7, 𝑚8, 𝑚9, 𝑚10, 𝑚11, 𝑚12)
𝑇, asfollows:
𝑑𝑥1𝑑𝑡
= 𝑚1 = 𝜋−[𝛽𝑇(𝑥3 + 𝜂𝑇𝑥8) + 𝛽𝑀 (𝑥5 + 𝜂𝑀𝑥9) + 𝛽𝑋𝑥7 (𝑥7 + 𝜂𝑋𝑥10)]
𝑥1
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)−
𝜇𝑥1,
𝑑𝑥2𝑑𝑡
= 𝑚2 =(1 − 𝑙𝑇1 − 𝑙𝑇2) 𝛽𝑇 (𝑥3 + 𝜂𝑇𝑥8) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)−
𝑔1𝑥2,
𝑑𝑥3𝑑𝑡
= 𝑚3 =𝑙𝑇1𝛽𝑇 (𝑥3 + 𝜂𝑇𝑥8) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)+
𝜎𝑇𝑥2 +𝜓𝑇𝑥8 −𝑔2𝑥3,
𝑑𝑥4𝑑𝑡
= 𝑚4 =(1 − 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑇 (𝑥5 + 𝜂𝑀𝑥9) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)−
𝑔3𝑥4,
𝑑𝑥5𝑑𝑡
= 𝑚5 =𝑙𝑀1𝛽𝑇 (𝑥5 + 𝜂𝑀𝑥9) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)+
𝜎𝑚𝑥4 +𝜌𝑇𝑥3 +𝜓𝑚𝑥9 −𝑔4𝑥5,
𝑑𝑥6𝑑𝑡
= 𝑚6 =(1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋 (𝑥7 + 𝜂𝑋𝑥10) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)−
𝑔5𝑥6,
𝑑𝑥7𝑑𝑡
= 𝑚7 =𝑙𝑋1𝛽𝑋 (𝑥7 + 𝜂𝑋𝑥10) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)+
𝜎𝑋𝑥6 +𝜌𝑀𝑥5 +𝜓𝑋𝑥10 −𝑔6𝑥7,
𝑑𝑥8𝑑𝑡
= 𝑚8 =𝑙𝑇2𝛽𝑇 (𝑥3 + 𝜂𝑇𝑥8) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)+
𝜙𝑇𝑥3 −𝑔7𝑥8,
𝑑𝑥9𝑑𝑡
= 𝑚9 =𝑙𝑀2𝛽𝑀 (𝑥5 + 𝜂𝑀𝑥6) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)+
𝜙𝑀𝑥5 −𝑔8𝑥9,
𝑑𝑥10𝑑𝑡
= 𝑚10 =𝑙𝑋2𝛽𝑋 (𝑥7 + 𝜂𝑇𝑥10) (𝑥1 + 𝜀𝑥12)
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 + 𝑥12)+
𝜙𝑋𝑥7 −𝑔9𝑥10,
𝑑𝑥11𝑑𝑡
= 𝑚11 = 𝛼𝑇𝑥3 +𝛼𝑀𝑥5 +𝛼𝑀𝑥7 −𝑔10𝑥11,
𝑑𝑥12𝑑𝑡
= 𝑚12 = 𝛾𝑇𝑥3 + 𝛾𝑀𝑥5 + 𝛾𝑋𝑥7 −𝑔11𝑥12 −𝜀 [𝛽𝑇(𝑥3 + 𝜂𝑇𝑥8) + 𝛽𝑀 (𝑥5 +
𝜂𝑀𝑥9) + 𝛽𝑋 (𝑥7 + 𝜂𝑋𝑥10)] 𝑥12
(𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10 + 𝑥11 +
𝑥12).
(A.1)
The Jacobian of the transformed system (A.1), at the
diseasefree equilibriumE1, is given by
𝐽 (E1) = (𝐽1  𝐽2) , (A.2)

Abstract and Applied Analysis 15
where
𝐽1 =
((((((((((((((((((((((
(
−𝜇 0 −𝛽𝑇
0 −𝛽𝑀
00 −𝑔1 𝑙𝑇1𝛽𝑇 + 𝑝𝑇1𝛾𝑇 0 0 00 𝜎𝑇
𝑙𝑇2𝛽𝑇 − 𝑔2 0 0 0
0 0 0 −𝑔3 𝑙𝑀1𝛽𝑀 00 0 𝑝
𝑇2𝛾𝑇 𝜎𝑀 𝑙𝑀2𝛽𝑀 − 𝑔4 00 0 0 0 𝑝
𝑀1𝛾𝑀 −𝑔5
0 0 0 0 0 𝜎𝑋
0 0 (1 − 𝑙𝑇1 − 𝑙𝑇2) 𝛽𝑇 + 𝜙𝑇 0 0 0
0 0 0 0 (1 − 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑀 + 𝜙𝑀 0
0 0 0 0 0 00 0 𝛼
𝑇0 𝛼
𝑀0
0 0 (1 − 𝑝𝑇1 − 𝑝𝑇2) 𝛾𝑇 0 (1 − 𝑝𝑀1) 𝛾𝑀 0
))))))))))))))))))))))
)
,
𝐽2
=
((((((((((((((((((((((
(
−𝛽𝑋
−𝛽𝑇𝜂𝑇
−𝛽𝑀𝜂𝑀
−𝛽𝑋𝜂𝑋
0 00 𝑙
𝑇1𝛽𝑇𝜂𝑇 0 0 0 00 𝑙
𝑇2𝛽𝑇𝜂𝑇 + 𝜓𝑇 0 0 0 00 0 𝑙
𝑀1𝛽𝑀𝜂𝑀 0 0 00 0 𝑙
𝑀2𝛽𝑀𝜂𝑀 + 𝜓𝑀 0 0 0𝑙𝑋1𝛽𝑋 0 0 𝑙𝑋1𝛽𝑋𝜂𝑋 0 0
𝑙𝑋2𝛽𝑋 − 𝑔6 0 0 𝑙𝑋2𝛽𝑋𝜂𝑋 + 𝜓𝑋 0 0
0 (1 − 𝑙𝑇1 − 𝑙𝑇2) 𝛽𝑇𝜂𝑇 − 𝑔7 0 0 0 0
0 0 (1 − 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑀𝜂𝑀 − 𝑔8 0 0 0
(1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋 + 𝜙𝑋 0 0 (1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋𝜂𝑋 − 𝑔9 0 0
𝛼𝑋
0 0 0 −𝑔10 0𝛾𝑋
0 0 0 0 −𝑔11
))))))))))))))))))))))
)
.
(A.3)
Consider the case when R0 = 1. Suppose, further, that 𝛽𝑇is
chosen as a bifurcation parameter. Solving (2) for 𝛽
𝑇from
R0 = 1 gives 𝛽𝑇 = 𝛽∗
𝑇. The transformed system (A.1) at
the DFE evaluated at 𝛽𝑇= 𝛽∗
𝑇has a simple zero eigenvalue
(and all other eigenvalues having negative real parts).
Hence,the centre manifold theory [47] can be used to analyze
thedynamics of (A.1) near 𝛽
𝑝= 𝛽∗
𝑇. In particular, the theorem
in [54] (see also [36, 47, 48]) is used (it is reproduced inthe
Appendix for convenience). To apply the theorem, thefollowing
computations are necessary (it should be noted thatwe are using
𝛽
𝑇instead of 𝜙 for the bifurcation parameter).
Eigenvectors of 𝐽(E1)𝛽𝑇=𝛽∗
𝑇
. The Jacobian of (A.1) at 𝛽𝑇= 𝛽∗
𝑇,
denoted by 𝐽(E1)𝛽𝑇=𝛽∗
𝑇
, has a right eigenvector (associatedwith the zero eigenvalue)
given by
w = (𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6, 𝑤7, 𝑤8, 𝑤9, 𝑤10, 𝑤11,
𝑤12)𝑇,
(A.4)
where
𝑤1 = −1𝜇[𝛽𝑇𝑤3 +𝛽𝑀𝑤5 +𝛽𝑋𝑤7 +𝛽𝑇𝜂𝑇𝑤8
+𝛽𝑀𝜂𝑀𝑤9 +𝛽𝑋𝜂𝑋𝑤10] ,
𝑤2 =1𝑔1
{[𝑝𝑇1𝛾𝑇 + 𝑙𝑇1𝛽𝑇] 𝑤3 + 𝑙𝑇1𝛽𝑇𝜂𝑇𝑤8} ,
𝑤3 > 0,
𝑤5 > 0,
𝑤4 =1𝑔3
[𝑝𝑇2𝛾𝑇𝑤3 + 𝑙𝑀1𝛽𝑀𝑤5 + 𝑙𝑀1𝛽𝑀𝜂𝑀𝑤9] ,
𝑤6 =1𝑔5
[𝑝𝑀1𝛾𝑀𝑤5 + 𝑙𝑋1𝛽𝑋𝑤7 + 𝑙𝑋1𝛽𝑋𝜂𝑋𝑤10] ,
𝑤7 > 0,
𝑤8 =[(1 − 𝑙
𝑇1 − 𝑙𝑇2) 𝛽𝑇 + 𝜙𝑇] 𝑤3
[𝑔7 − (1 − 𝑙𝑇1 − 𝑙𝑇2) 𝛽𝑇𝜂𝑇],

16 Abstract and Applied Analysis
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LT
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1000
800
600
400
200
0
(a)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LM
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
80
70
60
50
40
30
20
10
0
(b)
Stable EEP
Unstable EEP
Unstable DFEStable DFE
LX
R0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
15
10
5
0
(c)
Figure 6: Backward bifurcation plot of model (2) as a function
of time. (a) Individuals who are lost to followup with drug
sensitive TB (𝑇).(b) Individuals who are lost to followup with MDR
(𝑀). (c) Individuals who are lost to followup with XDR (𝑋).
Parameter values used areas given in Table 2.
𝑤9 =[(1 − 𝑙
𝑀1 − 𝑙𝑀2) 𝛽𝑀 + 𝜙𝑀] 𝑤5
[𝑔8 − (1 − 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑀𝜂𝑀],
𝑤10 =[(1 − 𝑙
𝑋1 − 𝑙𝑋2) 𝛽𝑋 + 𝜙𝑋] 𝑤7
(𝑔9 − (1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋𝜂𝑋),
𝑤11 =(𝛼𝑀𝑤5 + 𝛼𝑋𝑤7)
𝑔10,
𝑤12 =1𝑔11
[(1−𝑝𝑇1 −𝑝𝑇2) 𝛾𝑇𝑤3 + (1−𝑝𝑀1) 𝛾𝑀𝑤5
+ 𝛾𝑋𝑤7] .
(A.5)
Also, 𝐽(E1)𝛽𝑇=𝛽∗
𝑇
has a left eigenvector k = (V1, V2, V3,V4, V5, V6, V7, V8, V9,
V10, V11, V12) (associated with the zeroeigenvalue), where
V1 = 0,
V2 =𝜎𝑇V3𝑔1
,
V4 =𝜎𝑀V4
𝑔3,
V6 =𝜎𝑋V7
𝑔5,
V3 > 0,

Abstract and Applied Analysis 17
𝛼T = 0.20
𝛼T = 0.40
𝛼T = 0.60
𝛼T = 0.80
𝛼T = 1.0
0 2 4 6 8 10Time (years)
0
200
400
600
800
1000
1200
1400
1600To
tal n
umbe
r of i
sola
ted
indi
vidu
als
(a)
𝛼T = 0.20
𝛼T = 0.40
𝛼T = 0.60
𝛼T = 0.80
𝛼T = 1.0
0 2 4 6 8 10Time (years)
0
500
1000
1500
2000
2500
Tota
l num
ber o
f los
t to
sight
indi
vidu
als
(b)
Figure 7: Simulation of model (2) as a function of time varying
𝛼𝑇for (a) total number of isolated individuals and (b) total number
of
individuals who are lost to followup. Parameter values used are
as given in Table 2.
𝛼M = 0.20
𝛼M = 0.40
𝛼M = 0.60
𝛼M = 0.80
𝛼M = 1.0
0 2 4 6 8 10Time (years)
0
200
400
600
800
1000
Tota
l num
ber o
f iso
late
d in
divi
dual
s
(a)
𝛼M = 0.20
𝛼M = 0.40
𝛼M = 0.60
𝛼M = 0.80
𝛼M = 1.0
0 2 4 6 8 10Time (years)
0
500
1000
1500
2000
Tota
l num
ber o
f los
t to
sight
indi
vidu
als
(b)
Figure 8: Simulation of model (2) as a function of time varying
𝛼𝑀
for (a) total number of isolated individuals and (b) total
number ofindividuals who are lost to followup. Parameter values
used are as given in Table 2.
V7 > 0,
V5 =1
(𝑔4 − 𝑙𝑀2𝛽𝑀)[𝑙𝑀1𝛽𝑀V4 +𝑝𝑀1𝛾𝑀V6
+ [(1− 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑀 +𝜙𝑀] V9] ,
V8 =1
[𝑔7 − (1 − 𝑙𝑇1 − 𝑙𝑇2) 𝛽𝑇𝜂𝑇][𝑙𝑇1𝛽𝑇𝜂𝑇V2
+ (𝑙𝑇2𝛽𝑇𝜂𝑇 +𝜓𝑇) V3] ,
V9 =1
(𝑔8 − (1 − 𝑙𝑀1 − 𝑙𝑀2) 𝛽𝑀𝜂𝑀)[𝑙𝑀1𝛽𝑀𝜂𝑀V4
+ (𝑙𝑀2𝛽𝑀𝜂𝑀 +𝜓𝑀) V5] ,
V10 =1
(𝑔9 − (1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋𝜂𝑋)[𝑙𝑋1𝛽𝑋𝜂𝑋V6
+ (𝑙𝑋2𝛽𝑋𝜂𝑋 +𝜓𝑋) V7] ,
V11 = 0,
V12 = 0.(A.6)
Computations of Bifurcation Coefficients 𝑎 and 𝑏. The
application of the theorem (given in the Appendix) entails the

18 Abstract and Applied Analysis
computation of two bifurcation coefficients 𝑎 and 𝑏. It can
beshown, after some algebraic manipulations, that
𝑎 = V212∑
𝑖,𝑗=1𝑤𝑖𝑤𝑗
𝜕2𝑓2
𝜕𝑥𝑖𝜕𝑥𝑗
+ V312∑
𝑖,𝑗=1𝑤𝑖𝑤𝑗
𝜕2𝑓3
𝜕𝑥𝑖𝜕𝑥𝑗
+ V412∑
𝑖,𝑗=1𝑤𝑖𝑤𝑗
𝜕2𝑓4
𝜕𝑥𝑖𝜕𝑥𝑗
+ V512∑
𝑖,𝑗=1𝑤𝑖𝑤𝑗
𝜕2𝑓5
𝜕𝑥𝑖𝜕𝑥𝑗
+ V612∑
𝑖,𝑗=1𝑤𝑖𝑤𝑗
𝜕2𝑓6
𝜕𝑥𝑖𝜕𝑥𝑗
+ V712∑
𝑖,𝑗=1𝑤𝑖𝑤𝑗
𝜕2𝑓7
𝜕𝑥𝑖𝜕𝑥𝑗
=2𝑥1
(−𝑤2 −𝑤3 −𝑤4 −𝑤5 −𝑤6 −𝑤7 −𝑤8 −𝑤9
−𝑤10 −𝑤11 −𝑤12 +𝑤12𝜀) {𝛽𝑇 (𝑤3 +𝑤8𝜂𝑇)
⋅ [𝑙𝑇1V2 + 𝑙𝑇2V3 + (1− 𝑙𝑇1 − 𝑙𝑇2) V8]
+ 𝛽𝑀(𝑤5 +𝑤9𝜂𝑀)
⋅ [𝑙𝑀1V4 + 𝑙𝑀2V5 + (1− 𝑙𝑀1 − 𝑙𝑀2) V9]
+ 𝛽𝑋(𝑤7 +𝑤10𝜂𝑋)
⋅ [𝑙𝑋1V6 + 𝑙𝑋2V7 + (1− 𝑙𝑋1 − 𝑙𝑋2) V10]} .
(A.7)
Furthermore,
𝑏 = V212∑
𝑖=1𝑤𝑖
𝜕2𝑓2
𝜕𝑥𝑖𝜕𝛽∗𝑝
+ V312∑
𝑖=1𝑤𝑖
𝜕2𝑓3
𝜕𝑥𝑖𝜕𝛽∗𝑝
= (𝑤3 +𝑤8𝜂𝑇) [𝑙𝑇1V2 + 𝑙𝑇2V3 + (1− 𝑙𝑇1 − 𝑙𝑇2) V8]
> 0.
(A.8)
Hence, it follows from Theorem 4.1 of [54] that the transformed
model (A.1) (or, equivalently, (2)) undergoes backward bifurcation
atR0 = 1whenever the following inequalityholds:
𝑎 > 0. (A.9)
It is worth noting that if 𝜀 = 0 (i.e., reinfectionof recovered
individuals does not occur), the bifurcationcoefficient, 𝑎, given
in (A.7), reduces to
𝑎 = −2𝑥1
(𝑤2 +𝑤3 +𝑤4 +𝑤5 +𝑤6 +𝑤7 +𝑤8 +𝑤9
+𝑤10 +𝑤11) {𝛽𝑇 (𝑤3 +𝑤8𝜂𝑇)
⋅ [𝑙𝑇1V2 + 𝑙𝑇2V3 + (1− 𝑙𝑇1 − 𝑙𝑇2) V8]
+ 𝛽𝑀(𝑤5 +𝑤9𝜂𝑀)
⋅ [𝑙𝑀1V4 + 𝑙𝑀2V5 + (1− 𝑙𝑀1 − 𝑙𝑀2) V9]
+ 𝛽𝑋(𝑤7 +𝑤10𝜂𝑋)
⋅ [𝑙𝑋1V6 + 𝑙𝑋2V7 + (1− 𝑙𝑋1 − 𝑙𝑋2) V10]} .
(A.10)
It follows from (A.10) that the bifurcation coefficient 𝑎 <
0(ruling out backward bifurcation in this case, in line withTheorem
4.1 in [54]). Thus, this study shows that the backward bifurcation
phenomenon ofmodel (A.1) is caused by thereinfection of the
recovered individuals in the population.
B. Proof of Theorem 7
Proof. The proof is based on using a comparison theorem.The
equations for the infected components of model (2), with𝜀 = 0, can
be rewritten as
((((((((((((((((((((((((((((((((((((
(
𝑑𝐸𝑇 (𝑡)
𝑑𝑡
𝑑𝑇 (𝑡)
𝑑𝑡
𝑑𝐸𝑀 (𝑡)
𝑑𝑡
𝑑𝑀 (𝑡)
𝑑𝑡
𝑑𝐸𝑋 (𝑡)
𝑑𝑡
𝑑𝑋 (𝑡)
𝑑𝑡
𝑑𝐿𝑇 (𝑡)
𝑑𝑡
𝑑𝐿𝑀 (𝑡)
𝑑𝑡
𝑑𝐿𝑋 (𝑡)
𝑑𝑡
𝑑𝐽 (𝑡)
𝑑𝑡
))))))))))))))))))))))))))))))))))))
)
= (𝐹−𝑉)
(((((((((((((((((((
(
𝐸𝑇 (𝑡)
𝑇 (𝑡)
𝐸𝑀 (𝑡)
𝑀 (𝑡)
𝐸𝑋 (𝑡)
𝑋 (𝑡)
𝐿𝑇 (𝑡)
𝐿𝑀 (𝑡)
𝐿𝑋 (𝑡)
𝐽 (𝑡)
)))))))))))))))))))
)
−𝑃𝑄
(((((((((((((((((((
(
𝐸𝑇 (𝑡)
𝑇 (𝑡)
𝐸𝑀 (𝑡)
𝑀 (𝑡)
𝐸𝑋 (𝑡)
𝑋 (𝑡)
𝐿𝑇 (𝑡)
𝐿𝑀 (𝑡)
𝐿𝑋 (𝑡)
𝐽 (𝑡)
)))))))))))))))))))
)
,
(B.1)

Abstract and Applied Analysis 19
where 𝑃 = 1 − 𝑆, the matrices 𝐹 and 𝑉 are as given inSection
2.2, and𝑄 is nonnegativematrices given, respectively,by
𝑄 = [𝑄1  𝑄2] , (B.2)
where
𝑄1 =
(((((((((((((
(
0 𝑙𝑇1𝛽𝑇 0 0 0
0 𝑙𝑇2𝛽𝑇 0 0 0
0 0 0 𝑙𝑀1𝛽𝑀 0
0 0 0 𝑙𝑀2𝛽𝑀 0
0 0 0 0 00 0 0 0 00 (1 − 𝑙
𝑇1 − 𝑙𝑇2) 𝛽𝑇 0 0 00 0 0 (1 − 𝑙
𝑀1 − 𝑙𝑀2) 𝛽𝑀 00 0 0 0 00 0 0 0 0
)))))))))))))
)
,
𝑄2 =
(((((((((((((
(
0 𝑙𝑇1𝛽𝑇𝜂𝑇 0 0 0
0 𝑙𝑇2𝛽𝑇𝜂𝑇 0 0 0
0 0 𝑙𝑀1𝛽𝑀𝜂𝑀 0 0
0 0 𝑙𝑀2𝛽𝑀𝜂𝑀 0 0
𝑙𝑋1𝛽𝑋 0 0 𝑙𝑋1𝛽𝑋𝜂𝑋 0𝑙𝑋2𝛽𝑋 0 0 𝑙𝑋2𝛽𝑋𝜂𝑋 00 (1 − 𝑙
𝑇1 − 𝑙𝑇2) 𝛽𝑇𝜂𝑇 0 0 00 0 (1 − 𝑙
𝑀1 − 𝑙𝑀2) 𝛽𝑀𝜂𝑀 0 0(1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋 0 0 (1 − 𝑙𝑋1 − 𝑙𝑋2) 𝛽𝑋𝜂𝑋
0
0 0 0 0 0
)))))))))))))
)
.
(B.3)
Thus, since 𝑆(𝑡) ≤ 𝑁(𝑡) in Φ for all 𝑡 ≥ 0, it follows from(B.1)
that
((((((((((((((((((((((((((((((((
(
𝑑𝐸𝑇 (𝑡)
𝑑𝑡
𝑑𝑇 (𝑡)
𝑑𝑡
𝑑𝐸𝑀 (𝑡)
𝑑𝑡
𝑑𝑀 (𝑡)
𝑑𝑡
𝑑𝐸𝑋 (𝑡)
𝑑𝑡
𝑑𝑋 (𝑡)
𝑑𝑡
𝑑𝐿𝑇 (𝑡)
𝑑𝑡
𝑑𝐿𝑀 (𝑡)
𝑑𝑡
𝑑𝐿𝑋 (𝑡)
𝑑𝑡
𝑑𝐽 (𝑡)
𝑑𝑡
))))))))))))))))))))))))))))))))
)
≤ (𝐹−𝑉)
(((((((((((((((((((
(
𝐸𝑇 (𝑡)
𝑇 (𝑡)
𝐸𝑀 (𝑡)
𝑀 (𝑡)
𝐸𝑋 (𝑡)
𝑋 (𝑡)
𝐿𝑇 (𝑡)
𝐿𝑀 (𝑡)
𝐿𝑋 (𝑡)
𝐽 (𝑡)
)))))))))))))))))))
)
. (B.4)
Using the fact that the eigenvalues of the matrix 𝐹 − 𝑉 allhave
negative real parts (see the local stability result givenin Lemma
3, where 𝜌(𝐹𝑉−1) < 1 if R0 < 1, which isequivalent to 𝐹 − 𝑉
having eigenvalues with negative realparts when R0 < 1 [36]), it
follows that the linearizeddifferential inequality system (B.4) is
stable whenever R0 <1. Consequently, by comparison of theorem
[34] (Theorem1.5.2, p. 31),
(𝐸𝑇 (𝑡) , 𝑇 (𝑡) , 𝐿𝑇 (𝑡) , 𝐸𝑀 (𝑡) ,𝑀 (𝑡) , 𝐿𝑀 (𝑡) , 𝐸𝑋 (𝑡) ,
𝑋 (𝑡) , 𝐿𝑋 (𝑡) , 𝐽 (𝑡)) → (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) ,
as 𝑡 → ∞.
(B.5)
Substituting 𝐸𝑇= 𝑇 = 𝐿
𝑇= 𝐸𝑀= 𝑀 = 𝐿
𝑀= 𝐸𝑋= 𝑋 =
𝐿𝑋= 𝐽 = 0 into the equations of 𝑆 and 𝑅 in model (2), and
noting that 𝜀 = 0, gives 𝑆(𝑡) → 𝑆∗, 𝑅(𝑡) → 0 as 𝑡 → ∞.Thus, in
summary,
(𝑆 (𝑡) , 𝐸𝑇 (𝑡) , 𝑇 (𝑡) , 𝐿𝑇 (𝑡) , 𝐸𝑀 (𝑡) ,𝑀 (𝑡) , 𝐿𝑀 (𝑡) ,
𝐸𝑋 (𝑡) , 𝑋 (𝑡) , 𝐿𝑋 (𝑡) , 𝐽 (𝑡) , 𝑅 (𝑡)) → (𝑆
∗, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0) ,
(B.6)

20 Abstract and Applied Analysis
as 𝑡 → ∞. Hence, the DFE (E0) of model (2), with 𝜀 = 0, isGAS in
Φ if R̃0 < 1.
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
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