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Analysis of SIR epidemic models with nonlinear incidence rate and treatment

Zhixing Hu a,1, Wanbiao Ma a, Shigui Ruan b,⇑,2

a Department of Applied Mathematics, University of Science and Technology Beijing, Beijing 100083, PR Chinab Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA

a r t i c l e i n f o

Article history:Received 27 March 2011Received in revised form 26 March 2012Accepted 27 March 2012Available online 9 April 2012

Keywords:Epidemic modelTreatment rateNonlinear incidence rateBackward bifurcationStability

a b s t r a c t

This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic modelwith nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible andrecovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportionalto the number of infectives when it is below the capacity and constant when the number of infectivesreaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation froman endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equi-librium. In such a case, reducing the basic reproduction number less than unity is not enough to controland eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multipleendemic equilibria due to the effect of treatment, vaccination and other parameters. The existence andstability of the endemic equilibria of the model are analyzed and sufficient conditions on the existenceand stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analyticalresults.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

One of the focuses of theoretical studies in mathematical epide-miology is to understand the nonlinear dynamics of variousepidemic models. Classical susceptible-infectious-recovered (SIR)models with a bilinear incidence rate usually have a disease-freeequilibrium and at most one endemic equilibrium (see Capassoand Serio [7] and Hethcote [20]). In such a case, the basic reproduc-tion number is a threshold: when it is less than unity the disease-free equilibrium exists and is stable, which indicates that the diseasewill die out; when it is greater than unity the disease-free equilib-rium becomes unstable and an endemic equilibrium exists, whichdemonstrates that the disease will persist. The bifurcation leadingfrom a disease-free equilibrium to an endemic equilibrium is calledforward.

Recently there has been a great interest in investigating the non-linear dynamics (such as Hopf bifurcation, saddle-node bifurcation,Bogdanov–Takens bifurcation, existence of periodic and homoclinicorbits, coexistence of limit cycles and homoclinic orbits) inepidemic models with multiple endemic equilibria due to social

groups with different susceptibilities, nonlinear or nonmonotoneincidence rate, stage structure, behavioral change of susceptibles,etc. (see Alexander and Moghadas [1], Derrick and van denDriessche [12], Feng and Thieme [14], Hu et al. [21], Liu et al.[26,27], Ruan and Wang [30], Tang et al. [33], and Xiao and Ruan[39]). For instance, Liu et al. [26] proposed a nonlinear saturatedincidence function bSI‘=ð1þ aIhÞ to model the effect of behavioralchanges to certain communicable diseases, where bSI‘ describesthe infection force of the disease, 1=ð1þ aIhÞ measures the inhibi-tion effect from the behavioral change of the susceptible individualswhen the number of infectious individuals increases, ‘;h, and b areall positive constants, and a is a nonnegative constant. The casewhen ‘ ¼ h ¼ 1, i.e. the function bSI=ð1þ aIÞ, was used by Capassoand Serio [7] to represent a ‘‘crowding effect’’ or ‘‘protection mea-sure’’ in modeling the cholera epidemics in Bari in 1973. Becauseof the nonlinearity and saturation property of these incidence func-tions, it has been shown that SIR epidemic models with such non-linear incidence rates can possess multiple endemic equilibria.Moreover, such models can exhibit Hopf bifurcation, saddle-nodebifurcation and Bogdanov–Takens bifurcation [1,12,26,27,30], theexistence of two limit cycles [21], and the coexistence of a limit cy-cle and a homoclinic cycle [33]. These results demonstrate that thenonlinear dynamics of these epidemic models are very sensitive tothe model parameters and various outcomes could occur dependingon the initial population sizes. The disease may be eradicated (solu-tions approach a disease-free equilibrium), persistent (solutionsapproach an endemic equilibrium), or occurring periodically

0025-5564/$ - see front matter � 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.mbs.2012.03.010

⇑ Corresponding author.E-mail addresses: [email protected] (Z. Hu), [email protected] (S. Ruan).

1 Research was partially supported by the National Natural Science Foundation ofChina (11071013).

2 Research was partially supported by the National Science Foundation (DMS-1022728).

Mathematical Biosciences 238 (2012) 12–20

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(solutions approach a limit cycle). Understand such nonlineardynamics and identify the underlie factors is very important inthe control and prevention of spread of communicable diseases.

The phenomenon that the disease-free equilibrium coexistswith an endemic equilibrium when the basic reproduction numberis less than unity was first observed by Castillo-Chavez et al. [8,9]and Huang et al. [23] in multi-group HIV/AIDS models. It wastermed as backward bifurcation by Hadeler and Castillo-Chavez[17] and Hadeler and van den Driessche [18] and was shown to ex-ist in epidemic models that include behavioral responses (such asvia education) to perceived disease risk. The existence of backwardbifurcation in epidemic models has important qualitative implica-tions since the disease now cannot be eradicated by simply reduc-ing the basic reproduction number to be less than unity. Dushoffet al. [13] provided a general framework for the mechanisms be-hind backward bifurcations in simple epidemic models and dis-cussed the biological interpretation of the features of thesemodels that induce backward bifurcations. Therefore, it is alsoimportant to study backward bifurcation in epidemic models in or-der to seek for conditions for the control of diseases (see Arinoet al. [3], Blayneh et al. [4], Brauer [5,6], Greenhalgh and Griffiths[15], Kribs-Zaleta and Velasco-Hernandez [24], Regula et al. [29],Safan et al. [31], van den Driessche and Watmough [34], andWan and Zhu [35]). In this case, the basic reproduction numberdoes not give information on disease elimination; rather diseaseelimination is determined by the values of critical parameters atthe turning points.

Treatment is an important and effective method to prevent andcontrol the spread of various infectious diseases. In classical epi-demic models, the treatment rate of the infectives is assumed tobe proportional to the number of the infective individuals [2]. Dur-ing the SARS outbreaks in 2003, the dramatically increasing ofSARS cases in Beijing challenged the normal public-health systemand capacity in Beijing City and forced the Chinese governmentto create the first and only SARS hospital, Beijing XiaotangshanHospital, to treat the large number of SARS patients [37]. After thisexperience, researchers started to consider the capacity of thehealth-care system from both modeling and analyzing points ofview. Wang and Ruan [38] considered a SIR model in which thecapacity for the treatment of a disease in a community is a con-stant. Namely, they used the following function

TðIÞ ¼k; if I > 0;0; if I ¼ 0

�ð1:1Þ

to describe the treatment rate and studied the effect of treatmentparameter k on the dynamics of the model. They found that themodel undergoes a sequence of bifurcations including saddle-node,Hopf, and homoclinic bifurcations and exhibits homoclinic orbitseven though incidence rate is assumed to be bilinear. Moreover, itis shown that it may not be necessary to set k so high to eliminateendemic equilibria since such equilibria may be unstable. Wang[36] proposed the following piecewise linear treatment function

TðIÞ ¼kI; if 0 6 I 6 I0;

u ¼ kI0; if I > I0;

�ð1:2Þ

in a SIR model, where I0 is the infective level at which the health-care system reaches capacity; that is, treatment increases linearlywith I before the capacity is reached and is constant afterward. Itwas shown that the model has bistable endemic equilibria whenI0 is low and backward bifurcation can occur.

The treatment function (1.2) has been used by some otherresearchers. For example, Zhang and Liu [41] studied a model witha general incidence rate kðI þ SÞn�1SI ð0 6 n 6 1Þ and the treatmentfunction (1.2). Hu et al. [22] considered an epidemic model withstandard incidence rate bSI=N and the treatment function (1.2). Li

et al. [25] studied an epidemic model with nonlinear incidence ratebI=ð1þ aIÞ with the treatment function (1.2) and analyzed the sta-bility and bifurcation of the system. Other types of treatment func-tions have also been proposed. For instance, Zhang and Liu [40]used a saturated treatment function TðIÞ ¼ rI=ð1þ lIÞ; r > 0;l P 0, and found that the saturated function has the advantageof giving near-linear treatment response when I is low and ap-proaches a constant capacity as I gets large. Eckalbar and Eckalbar[11] introduced a new treatment function, TðIÞ ¼ maxfcI � gI2; 0g;c > 0; g > 0, into a SIR model with bilinear incidence rate. It wasfound that the system could have up to four equilibria with possi-ble bi-stability, backward bifurcations, and limit cycles. See also Liet al. [25], Zhou and Fan [42], etc.

In this paper, we consider a SIR epidemic model with the non-linear incidence rate bSI=ð1þ aIÞ, the treatment rate function(1.2), vertical transmission, and vaccination for the newborns ofthe susceptible and recovered individuals. To formulate our model,let SðtÞ; IðtÞ and RðtÞ be the number of susceptible, infective andrecovered individuals at time t, respectively. The basic assump-tions are as follows.

(i) The total population size at time t (day) is denoted byN ¼ Sþ I þ R. The newborns of S and R are susceptible indi-viduals, and the newborns of I who are not verticallyinfected are also susceptible individuals.

(ii) The positive constant b (per day) denotes the death rate andbirth rate of susceptible and recovered individuals. The posi-tive constant d (per day) denotes the death rate and birthrate of infective individuals. The positive constant c (perday) is the natural recovery rate of infective individuals.The positive constant qðq 6 1Þ (per day) is the vertical trans-mission rate, and note p ¼ 1� q (per day), then 0 6 p 6 1.Fraction m0 of all newborns with mothers in the susceptibleand recovered classes are vaccinated and appeared in therecovered class, while the remaining fraction, m ¼ 1�m0,appears in the susceptible class.

(iii) The incidence rate is described by a nonlinear functionbSI=ð1þ aIÞ, where b (per day) is a positive constant describ-ing the infection rate and a (per person) is a nonnegativeconstant represents the half saturation constant.

(iv) The treatment rate of a disease is TðIÞ given in (1.2).

Under the above assumptions, the SIR epidemic model takes thefollowing form:

dSdt ¼ bmðSþ RÞ � b SI

1þaI � bSþ pdI;dIdt ¼ b SI

1þaI þ qdI � dI � cI � TðIÞ;dRdt ¼ cI � bRþ bm0ðSþ RÞ þ TðIÞ:

8><>: ð1:3Þ

Because

dNdt¼ 0;

the total number of population N is a constant. For convenience, it isassumed that N ¼ Sþ I þ R ¼ 1, thus S; I and R are taken as the pro-portions of susceptible, infective and recovered individuals in thetotal population. By using Sþ R ¼ 1� I, the first two equations ofsystem (1.3) do not contain the variable R. Therefore, system (1.3)is equivalent to the following 2-dimensional system:

dSdt ¼ �b SI

1þaI � bSþ bmð1� IÞ þ pdI;dIdt ¼ b SI

1þaI � pdI � cI � TðIÞ:

(ð1:4Þ

It is easy to verify that the positive invariant set of system (1.4)is

D ¼ ðS; IÞjS P 0; I P 0; Sþ I 6 1f g:

Z. Hu et al. / Mathematical Biosciences 238 (2012) 12–20 13


When 0 6 I 6 I0, system (1.4) is

dSdt ¼ �b SI

1þaI � bSþ bmð1� IÞ þ pdI;

dIdt ¼ b SI

1þaI � pdI � cI � kI:

(ð1:5Þ

When I > I0, system (1.4) becomes (u ¼ kI0)

dSdt ¼ �b SI

1þaI � bSþ bmð1� IÞ þ pdI;

dIdt ¼ b SI

1þaI � pdI � cI � u:

(ð1:6Þ

The purpose of this paper is to study the nonlinear dynamics ofsystem (1.4). Under some conditions, it is shown that there exists abackward bifurcation from an endemic equilibrium, which impliesthat the disease-free equilibrium coexists with an endemic equilib-rium. In such a case, reducing the basic reproduction number lessthan unity is not enough to control and eradicate the disease, extrameasures are needed to ensure that the solutions approach the dis-ease-free equilibrium. When the basic reproduction number isgreater than unity, model (1.4) can have multiple endemic equilib-ria due to the effect of treatment, vaccination and other parame-ters. The existence and stability of the endemic equilibria of themodel are analyzed and sufficient conditions on the existenceand stability of a limit cycle are obtained. Numerical simulationsare presented to illustrate the analytical results.

The organization of this paper is as follows. In next section, weanalyze the existence and bifurcations of equilibria for (1.4). In Sec-tion 3, we study the stability of various equilibria and the existenceand stability of a limit cycle in (1.4). In Section 4, we give somenumerical simulations to verify our results. A brief discussion ispresented in Section 5.

2. Existence of equilibria

In this section, we consider the equilibria of system (1.4). Obvi-ously, E0 m;0ð Þ is the disease-free equilibrium of (1.4).

For the positive equilibrium E�ðS�; I�Þ of system (1.4), when0 < I� 6 I0, let the right side of (1.5) be equal to zero. Then S� andI� satisfy the following equations:

�b S�I�

1þaI� � bS� þ bmð1� I�Þ þ pdI� ¼ 0;

b S� I�

1þaI� � ðpdþ cþ kÞI� ¼ 0:

(ð2:1Þ

When I� > I0, let the right side of (1.6) be equal to zero. Then S� andI� satisfy the following equations

�b S�I�

1þaI� � bS� þ bmð1� I�Þ þ pdI� ¼ 0;

b S� I�

1þaI� � ðpdþ cÞI� � u ¼ 0:

(ð2:2Þ

From (2.1) we obtain that

I� ¼ bðpdþ cþ kÞðR0 � 1Þbðbmþ cþ kÞ þ abðkþ cþ dpÞ ; ð2:3Þ

where

R0 ¼bm

pdþ cþ kð2:4Þ

is called the basic reproduction number of (1.4); if R0 > 1; I� in (2.3) ispositive. At the same time, I� in (2.3) must satisfy I� 6 I0, which isequivalent to

u Pkbðpdþ cþ kÞðR0 � 1Þ

bðbmþ cþ kÞ þ abðkþ cþ dpÞ ¼ u2: ð2:5Þ

Therefore, E0 m;0ð Þ is always the disease-free equilibrium of (1.4).E� S�; I�ð Þ is the endemic equilibrium of system (1.4) if and only ifR0 > 1 and u P u2, where

S� ¼ abmðpdþ cþ kÞ þ ðbmþ cþ kÞðpdþ cþ kÞbðbmþ cþ kÞ þ abðpdþ cþ kÞ : ð2:6Þ

According to (2.2), I� satisfies the following equation

b0ðI�Þ2 þ b1I� þ b2 ¼ 0; ð2:7Þ

where

b0 ¼ bðbmþ cÞ þ abðpdþ cÞ;b1 ¼ bðpdþ c� bmÞ þ uðabþ bÞ;b2 ¼ bu:

ð2:8Þ

If b1 P 0, it is clear that Eq. (2.7) does not have a positive solu-tion. Let us suppose that b1 < 0. Clearly, b1 < 0 is equivalent to

bm > pdþ c and u <bðbm� pd� cÞ

baþ b: ð2:9Þ

From (2.8), we get

D ¼ b21 � 4b0b2

¼ ðabþ bÞ2u2 � 2b ðbm� pd� cÞðabþ bÞ þ 2 bðbmþ cÞ½fþabðpdþ cÞ�guþ b2ðpdþ c� bmÞ2:

Note that

f�2b ðbm� pd� cÞðabþ bÞ þ 2 bðbmþ cÞ þ abðpdþ cÞ½ �f gg2

� 4ðabþ bÞ2b2ðpdþ c� bmÞ2

¼ 4 bðbmþ cÞ þ abðpdþ cÞ½ � bðbmþ cÞ þ abðpdþ cÞ½þ ðbm� pd� cÞðabþ bÞ� > 0:

Denote

g ¼ bðbmþ cÞ þ abðpdþ cÞ½ � bðbmþ cÞ þ abðpdþ cÞ½

þ ðbm� pd� cÞðabþ bÞ�12:

It follows that D P 0 is equivalent to

u Pb ðbm� pd� cÞðabþ bÞ þ 2 bðbmþ cÞ þ abðpdþ cÞ½ � þ 2gf g

ðabþ bÞ2

ð2:10Þor

u6b ðbm� pd� cÞðabþ bÞ þ2 bðbmþ cÞ þabðpdþ cÞ½ � �2gf g

ðabþ bÞ2¼ u0:

ð2:11Þ

As bm > pdþ c, we have

u0 <bðbm� pd� cÞ

abþ b:

Therefore, b1 < 0; D P 0 if and only if bm > pdþ c and u 6 u0.Suppose that u 6 u0 and bm > pdþ c, then (2.7) has two posi-

tive solutions I1 and I2, where

I1 ¼�b1 �

ffiffiffiffiDp

2b0; I2 ¼

�b1 þffiffiffiffiDp

2b0: ð2:12Þ

Let

Sj ¼bm� u� ðbmþ cÞIj

b:

If Ij > I0ðj ¼ 1;2Þ, then EjðSj; IjÞðj ¼ 1;2Þ is an endemic equilibrium of(1.4).

By the expression of I1, we notice that I1 > I0 is equivalent to

�ffiffiffiffiDp

> 2b0I0 þ b1: ð2:13Þ

This implies that 2b0I0 þ b1 < 0. It is easy to prove that 2b0I0 þ b1 <

0 is equivalent to

14 Z. Hu et al. / Mathematical Biosciences 238 (2012) 12–20


u <bkðbm� pd� cÞ

2½bðbmþ cÞ þ baðpdþ cÞ� þ kðbaþ bÞ ¼ u1: ð2:14Þ

Furthermore, I1 > I0 requires that

ðb1 þ 2b0I0Þ2 � D > 0:

Following some calculations, we obtain that

ðb1 þ 2b0I0Þ2 � D ¼ 4b0I0

kkb pdþ cþ k� bm½ � þ bðbmþ cþ kÞ½f

þ abðpdþ cþ kÞ�ug:

(1) If pdþ c < bm 6 pdþ cþ k, then ðb1 þ 2b0I0Þ2 � D > 0.(2) If bm > pdþ cþ k, that is, R0 > 1, then the inequalityðb1 þ 2b0I0Þ2 � D > 0 is equivalent to u > u2.

Therefore, I1 > I0 holds if and only if either u < u1; u 6 u0 andbm > pdþ c or u < u1; u > u2; u 6 u0 and R0 > 1. Similarly,I2 > I0 holds if and only if either u 6 minfu0;u1g and bm > pdþ c,or u < minfu0;u2g; u > u1 and R0 > 1.

Note that the sign of u2 � u1 is determined by

U ¼ R0 � R�0;

where

R�0 ¼ 1þ kðbðbmþ cþ kÞ þ abðpdþ cþ kÞÞðpdþ cþ kÞ bðbmþ cÞ þ abðpdþ cÞ½ � : ð2:15Þ

It follows that u1 > u2 if 1 < R0 < R�0, and u1 < u2 if R0 > R�0.Summarizing the above discussions, we have the following con-

clusions for the existence of equilibria.

Theorem 2.1. (i) For system (1.4), the disease-free equilibriumE0 m;0ð Þ always exists. (ii) E�ðS�; I�Þ is an endemic equilibrium ofsystem (1.4) if and only if R0 > 1 and u P u2. Furthermore, supposeR0 > 1; u P u2 and one of the following conditions is satisfied:

ðaÞ u > u0;ðbÞ u1 < u < u0.

Then E� is the unique endemic equilibrium of system (1.4).

Theorem 2.2. The endemic equilibria E1ðS1; I1Þ and E2ðS2; I2Þ of sys-tem (1.4) do not exist if bm 6 pdþ c or u > u0. On the other hand,suppose u 6 u0 and bm > pdþ c, we have the following results.

ðiÞ If R0 6 1 and u < u1, then the equilibria E1 and E2 of system(1.4) exist.

ðiiÞ If 1 < R0 < R�0 and u2 < u < u1, then the equilibria E1 and E2 ofsystem (1.4) exist.

ðiiiÞ If 1 < R0 < R�0 and u < u2, then the equilibrium E2 of system(1.4) exists but E1 does not exist, and the equilibrium E2 doesnot exist if u > u1.

ðivÞ If 1 < R�0 < R0, then E1 does not exist. Furthermore, E2 exists ifu < u2 and does not exist if u > u2.

We can see that under the conditions in Theorem 2.1 (i), the dis-ease-free equilibrium E0 coexists with two endemic equilibria E1

and E2. In fact, we have the following result.

Corollary 2.3. If R0 < 1; bm > pdþ c, and u < min u0;u1f g, thensystem (1.4) has a backward bifurcation of endemic equilibria.

We now present examples to show that, for various parametervalues, system (1.4) has a forward bifurcation from one endemicequilibrium to another endemic equilibrium (see Example 2.4)

and a backward bifurcation with a disease-free equilibrium andtwo endemic equilibria (Example 2.5). Note that by Theorem 2.1(ii) and Theorem 2.2 (ii), there are conditions that guarantee theexistence of all three endemic equilibria E1; E2, and E� (Example 2.6).

Example 2.4. We choose the parameter values as follows:b ¼ 0:34; a ¼ 0:4; c ¼ 0:01; d ¼ 0:01; b ¼ 0:2; p ¼ 0:02; k ¼ 0:03and m ¼ 0:3. By calculations, R0 = 2.53731, R�0 ¼ 2:12826;R0 > R�0and u2 ¼ 0:00996346, case (iv) of Theorem 2.2 holds. When u 6 u2,a bifurcation diagram is illustrated in Fig. 1, the bifurcation atu ¼ u2 is forward when the parameter u decreases, and system(1.4) has a unique endemic equilibrium for all u > 0.

Example 2.5. We choose the parameter values as follows: b ¼0:2;a ¼ 0:4; c ¼ 0:01; d ¼ 0:01; b ¼ 0:2; p ¼ 0:02; k ¼ 0:03 and m ¼0:1. By calculations, we have R0 ¼ 0:497512< 1;u0 ¼ 0:000590643;u1 ¼ 0:00266885, so case (i) of Theorem 2.2 holds. A backward bifur-cation diagram is illustrated in Fig. 2, where the horizontal linedenotes the disease-free equilibrium E0. Two endemic equilibriaappear simultaneously at u¼ u0 when the parameter u decreases.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

I

I2

*

u2

u

I

Fig. 1. The forward bifurcation diagram from I� to I2 versus u for (1.4).

0 1 2 3 4 5 6 7 8x 10−4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

I 2

I

I=0

1

I

u

u0

Fig. 2. The backward bifurcation diagram of I1 and I2 versus u.



Example 2.6. Choose b ¼ 0:34; a ¼ 0:4; c ¼ 0:01; d ¼ 0:01;b ¼ 0:2; p ¼ 0:02; k ¼ 0:03 and m ¼ 0:2. We have R0 = 1.69154,R�0 = 2.274 05, u0 = 0.00582652, u1 = 0.00719025 and u2 =0.00548396, so case (ii) of Theorem 2.2 holds. A bifurcation dia-gram is illustrated in Fig. 3, where the horizontal line denotesthe endemic equilibrium E�. It shows that there is a bifurcationat u ¼ u0 when the parameter u decreases, which gives rise tothe existence of multiple endemic equilibria E�; E1, and E2.

3. Stability of equilibria

For the stability of the disease-free equilibrium E0 m;0ð Þ, wehave the following theorem.

Theorem 3.1. The disease-free equilibrium E0 m;0ð Þ is locally asymp-totically stable if R0 < 1 and unstable if R0 > 1. Moreover, E0 m;0ð Þ isglobally asymptotically stable in D if R0 < 1 and u > u0.

Proof. It is easy to obtain that the characteristic roots to the line-arized equation of system (1.4) at E0 m;0ð Þ are k1 ¼ �b < 0 andk2 ¼ ðpdþ cþ kÞðR0 � 1Þ. Thus, E0 is locally asymptotically stableif R0 < 1 and unstable if R0 > 1.

Next, if R0 < 1; E� does not exist by Theorem 2.1. By Theorem2.2, E1 and E2 do not exist if u > u0. Therefore, E0 is the uniqueequilibrium of system (1.4). Since D is the invariant set of system(1.4) and E0 is locally asymptotically stable, it follows fromBendixson Theorem that every solution of system (1.4) in Dapproaches E0 when t tends to positive infinity. h

For the endemic equilibrium E�ðS�; I�Þ, we have the followingtheorem.

Theorem 3.2. If the endemic equilibrium E�ðS�; I�Þ of system (1.4)exists, then it is locally asymptotically stable.

Proof. According to Theorem 2.1, the endemic equilibriumE�ðS�; I�Þ exists if and only if R0 > 1 and u P u2. The Jacobian matrixof system (1.4) at E�ðS�; I�Þ is

J� ¼�b� b I�

1þaI� �b S�

ð1þaI�Þ2� bmþ pd

b I�

1þaI� b S�

ð1þaI�Þ2� pd� c� k

0@

1A: ð3:1Þ

Because S� and I� satisfy Eq. (2.1), by means of (2.1), the trace anddeterminant of J� are simplified into

trðJ�Þ ¼ � bþ ½bþ aðbþ pdþ cþ kÞ�I�

1þ aI�< 0;

detðJ�Þ ¼ baðpdþ cþ kÞ þ bðbmþ cþ kÞ1þ aI�

I� > 0:

Therefore, all eigenvalues of matrix J� have negative real parts whenR0 > 1; u P u2. It follows that E�ðS�; I�Þ is locally asymptoticallystable. h

Now we discuss the stability of endemic equilibria EiðSj; IjÞðj ¼1;2Þ. The Jacobian matrix of system (1.4) at EjðSj; IjÞ is

Jj ¼�b� b

Ij

1þaIj�b

Sj

ð1þaIjÞ2� bmþ pd

bIj

1þaIjb

Sj

ð1þaIjÞ2� pd� c

0B@

1CA; j ¼ 1;2: ð3:2Þ

Theorem 3.3. If the endemic equilibrium E1ðS1; I1Þ of system (1.4)exists, then it is unstable.

Proof. For the endemic equilibrium E1ðS1; I1Þ; S1 and I1 satisfy (2.2).By (2.2), (2.7) and (2.12), after some calculations, the determinantof the matrix J1 is

detðJ1Þ ¼�

ffiffiffiffiDp

1þ aI1< 0: ð3:3Þ

Therefore, E1ðS1; I1Þ is a saddle and unstable. h

For the endemic equilibrium E2ðS2; I2Þ, similarly, the determi-nant of J2 is

detðJ2Þ ¼ffiffiffiffiDp

1þ aI2> 0: ð3:4Þ

Thus, E2 may be a node, focus, or center.By (2.2), (2.7) and (2.12), the trace trðJ2Þ of matrix J2 is very

complicated, but the sign of trðJ2Þ is determined by

u ¼ �2bb0b2 þ b1u bðbþ baÞ � bðbmþ cÞ½ �� ½b2ðbþ aðbþ pdþ cÞÞ þ b0u�

ffiffiffiffiDp

: ð3:5Þ

(1) According to b1 < 0 and (3.5), if bðbþ baÞP bðbmþ cÞ, thenu < 0.

(2) If bðbþ baÞ < bðbmþ cÞ, because

� 2bb0b2 þ b1u bðbþ baÞ � bðbmþ cÞ½ �

¼ u �2b2 bðbmþ cÞ þ baðpdþ cÞ½ � þ bðpdþ c�mbÞ bðb½n

þbaÞ � bðbmþ cÞ� þ ðbaþ bÞ bðbþ baÞ � bðbmþ cÞ½ �uo;

ð3:6Þ

it is known from (3.6) that if

0 < bm� pd� c 62b½bðbmþ cÞ þ baðpdþ cÞ�

bðbmþ cÞ � bðbþ baÞ ¼ g1; ð3:7Þ

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6x 10−3

0.2

0.25

0.3

0.35

u

I

I2

I1

I*

u2u0

Fig. 3. The bifurcation diagram with multiple endemic equilibria versus u.

u P�2b2 bðbmþ cÞ þ baðpdþ cÞ½ � þ bðpdþ c�mbÞ bðbþ baÞ½ � bðbmþ cÞ�

ðbaþ bÞ bðbmþ cÞ � bðbþ baÞ½ � ¼ u3; ð3:8Þ



then u < 0.(3) It follows from (3.6) that if bðbþ baÞ < bðbmþ cÞ; bm�

pd� c > g1andthen u < 0.

(4) It is known from (3.5) and (3.6) that if bðbþ baÞ <bðbmþ cÞ; bm� pd� c > g1; u < u3 and

v ¼ D� �2bb0b2 þ b1u bðbþ baÞ � bðbmþ cÞ½ �f g2

½b2ðbþ aðbþ pdþ cÞÞ þ b0u�2> 0;

ð3:9Þ

then u < 0.

Summarizing the above discussions, we have the following re-sults on the stability of the equilibrium E2ðS2; I2Þ.

Theorem 3.4. Suppose that the endemic equilibrium E2ðS2; I2Þ exists,if one of following conditions is satisfied:

ðiÞ bðbþ baÞP bðbmþ cÞ;ðiiÞ bðbþ baÞ < bðbmþ cÞ and bm� pd� c 6 g1;ðiiiÞ bðbþ baÞ < bðbmþ cÞ; bm� pd� c > g1 and u P u3;ðivÞ bðbþ baÞ < bðbmþ cÞ; bm� pd� c > g1; u < u3 and v > 0,

then E2ðS2; I2Þ is locally asymptotically stable. It is unstable ifbðbþ baÞ < bðbmþ cÞ; bm� pd� c > g1; u < u3 and v < 0.

The existence of limit cycles plays an important role in deter-mining the dynamical behaviors of the system. For example, ifthere is no limit cycle in system (1.4) and its endemic equilibriumis unique and locally asymptotically stable, then it must be globallystable. Now, we consider the existence of limit cycles in system(1.4).

Theorem 3.5. Suppose R0 > R�0 and u < min u0;u2f g. If u > 0, thensystem (1.4) has at least a stable limit cycle which encircles E2.

Proof. As R0 > R�0 > 1 and u < u2, it is known from Theorem 2.1that the equilibrium E� of system (1.4) does not exist. Again,because R0 > R�0; u < u0 and u < u2, it follows from Theorem 2.2that the equilibrium E1 of system (1.4) does not exist, but the equi-librium E2 exists.

It is known from u > 0 that E2 is an unstable focus or node. It iseasy to check that the unstable manifold at E0ðm;0Þ which is asaddle point, is in the first quadrant. As the set D is positivelyinvariant for system (1.4), and system (1.4) does not have any

equilibrium in the interior of D n fE2g. It follows from Poincaré–Bendixson theorem that system (1.4) has at least a stable limitcycle which encircles E2. h

4. Numerical simulations

In this section, we present some numerical simulations of sys-tem (1.4) to illustrate our results.

Example 4.1 (Example 2.4 continued). Choose b ¼ 0:34;a ¼ 0:4;c ¼ 0:01; d ¼ 0:01; b ¼ 0:2; p ¼ 0:02; k ¼ 0:03 and m ¼ 0:3. We haveR0=2.53731, R�0 ¼ 2:12826;R0 > R�0;u0 ¼ 0:0101133 and u2 ¼0:00996346. A forward bifurcation diagram was given in Fig. 1.

If we select I0 ¼ 0:4, then u ¼ 0:012 > u2, the equilibriumE�ð0:133942;0:332115Þ exists, but E1 and E2 do not exist (seeTheorem 2.1(ii)). Its phase portrait is given in Fig. 4, which showsthat the unique equilibrium E� is globally asymptotically stable.

If we choose I0 ¼ 0:2, then u ¼ 0:006 < u2, the equilibriumE2ð0:0754805;0:55577Þ exists, but E1 and E� do not exist (seeTheorem 2.2(iv)). Its phase portrait is illustrated in Fig. 5. Theunique equilibrium E2 is globally asymptotically stable in D.

Example 4.2 (Example 2.5 continued). Choose b ¼ 0:2;a ¼ 0:4;c ¼ 0:01; d ¼ 0:01; b ¼ 0:2; p ¼ 0:02; k ¼ 0:03 and m ¼ 0:1. By cal-culations, R0 ¼ 0:497512 < 1;u0 ¼ 0:000590643;u1 ¼ 0:00266885and R0 < 1. A backward bifurcation diagram was given in Fig. 2.If we choose I0 ¼ 0:01, then u ¼ 0:0003 < u0, the equilibriaE1ð0:0929588;0:036941Þ and E2ð0:060531;0:238294Þ exist, butthe equilibrium E� does not exist. Its phase portrait is illustratedin Fig. 6. It shows that the equilibria E0 and E2 are asymptoticallystable.

Example 4.3 (Example 2.6 continued). For system (1.4), wechoose parameter values as follows: b ¼ 0:34;a ¼ 0:4; c ¼ 0:01;d ¼ 0:01; b ¼ 0:2; p ¼ 0:02; k ¼ 0:03;m ¼ 0:2 and I0=0.1867. Thenwe have R0= 1.69154, R�0=2.274 05. bðbþ baÞ � bðbmþ cÞ ¼0:067 > 0;u0 ¼ 0:00582652; u1 ¼ 0:00719025; u2 ¼ 0:00548396and u ¼ 0:0056. The parameter values satisfy the conditions ofTheorem 3.2, the condition (ii) of Theorem 2.2 and the condition(i) of Theorem 3.4. Therefore, all four equilibriaE0ð0:2;0Þ; E�ð0:126881; 0:182799Þ; E1ð0:123039;0:195843Þ and

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

S

I

E*

E0

Fig. 4. The phase portrait of system (1.4) when E� is globally asymptotically stable.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

S

I

E2

E0

Fig. 5. The phase portrait of system (1.4) when E2 is globally asymptotically stable.



E2ð0:0917512;0:320995Þ exist. The phase portrait of system (1.4) isillustrated in Fig. 7, where the black thin lines are the separatricesof saddle points E0 and E1. It is known from Fig. 7 that the equilibriaE2 and E� are asymptotically stable, and E0 and E1 are unstable.Thus, system (1.4) has bistable endemic equilibria E� and E2.

We can see from Fig. 7 that the stable separatrices Cs of saddlepoint E1 separate the positive invariant set D into two regions, thebasin of attraction for the stable equilibrium E2 is the region aboveCs and the basin of attraction for the stable equilibrium E� is the re-gion below Cs.

Example 4.4. Choose b ¼ 0:2;a ¼ 0:8; c ¼ 0:01; d ¼ 0:03; b ¼ 0:01;p ¼ 0:2; k ¼ 0:04 and m ¼ 0:9. We have R0 ¼ 3:21429;R�0 ¼3:22723;u0 ¼ 0:00404965;u1 ¼ 0:00405539 and u2 ¼ 0:00404964.If we choose I0 ¼ 0:101225, then u ¼ 0:004049 < u0; trðJ2Þ ¼0:00407523 > 0, the equilibria E2ð0:29783;0:103826Þ exists but isunstable, and the equilibria E� and E1 do not exist. The parametervalues satisfy the conditions of Theorem 3.5. Its phase portrait isgiven in Fig. 8, which shows that system (1.4) has a stable limitcycle which encircles E2. Therefore, under some conditions, system(1.4) has a stable periodic orbit which encircles the equilibrium E2.

5. Discussion

In this paper, we have analyzed a SIR epidemic model to studythe effect of limited resources for the treatment of patients in thepublic-health system, which could occur when there is a very largenumber of patients but the medical facilities are insufficient, thenumber of beds is limited, or the number of health-care workersis short-handed. We also considered nonlinear incidence rate, ver-tical transmission and vaccination for the newborns of the suscep-tible and recovered individuals in the model. Theorem 2.2 andCorollary 2.3 imply that a backward bifurcation occurs whenR0 < 1, that is, the disease-free equilibrium coexists with an ende-mic equilibrium. Theorems 2.2 and 3.4 indicate that system (1.4)has multiple endemic equilibrium when R0 > 1 where a bifurcationdiagram displays forward bifurcations. When there are two ende-mic equilibria, one of them is always unstable and the other oneis stable under certain conditions. When there are three endemicequilibria, bistable endemic equilibria can occur. Numerical simu-lation confirmed that system (1.4) has a stable periodic orbit whichencircles an endemic equilibrium under some conditions. The exis-tence and stability of equilibria of system (1.4) can be summarizedin Table 1 (DNE = does not exist).

We can see that when the basic reproduction number R0 < 1and the treatment term u < minfu0;u1g, a backward bifurcationoccurs with a disease-free equilibrium and two endemic equilibria.Note that the disease-free equilibrium is always stable sinceR0 < 1, the stability of the endemic equilibria depends on otherconditions. Recall that u ¼ kI0, where k is the treatment parameterand I0 represents the infective level at which the health-care sys-tem reaches capacity,

R0 ¼bm

pdþ cþ k;

u0 ¼b ðbm� pd� cÞðabþ bÞ þ 2 bðbmþ cÞ þ abðpdþ cÞ½ � � 2gf g

ðabþ bÞ2;

u1 ¼bkðbm� pd� cÞ

2½bðbmþ cÞ þ baðpdþ cÞ� þ kðbaþ bÞ ;

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

S

I

E2

E0

E*E1

Γs

Γs

Fig. 7. The phase portrait of system (1.4) with bistable endemic equilibria.

0.265 0.27 0.275 0.28 0.285 0.29 0.295 0.3 0.305 0.310.095

0.1

0.105

0.11

0.115

0.12

0.125

S

I

E2

Fig. 8. A stable limit cycle of system (1.4) encircling the unstable equilibrium E2.

0 0.05 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

S

I

E2

E1

E0

Fig. 6. The phase portrait of system (1.4) when E2 and E0 are stable and E1 isunstable.



and

u2 ¼kbðpdþ cþ kÞðR0 � 1Þ

bðbmþ cþ kÞ þ abðkþ cþ dpÞ :

Therefore, in order to eradicate the disease, the basic reproductionnumber R0 must be lowered than a threshold or the treatmentu > maxfu0;u1g by other control measures, possibly a combinationof different types of measures. Epidemic models with two differenttypes of interventions have been studied by some other researchers,see Brauer [6] and Sun and Yang [32]. For example, reduce the rateof vertical transmission rate qð¼ 1� pÞ, increase the vaccinationrate m0, or increase the treatment rate k, so that the disease ap-proaches a lower endemic steady state for a range of parameters.If the capacity for treatment increases further to a certain level, thenthe disease may be eradicated.

We would like to mention that related models have been stud-ied and similar results have been obtained by other researchers, forexample, Cui et al. [10], Li et al. [25], and Hu et al. [22]. However,we have considered more factors and components in our modeland obtained different and new results. For example, we assumedthat the disease can be transmitted vertically and vaccination ap-plies to the newborns of susceptible and recovered individuals.Moreover, we established the existence and stability of a limit cy-cle when there is a unique unstable endemic equilibrium.

As pointed out by Greenhalgh and Griffiths [15], the study ofbackward bifurcation in epidemic modeling is relatively new andthere are many issues deserving further investigation for thisnew and interesting phenomenon. So far most studies focus onthe theoretical aspects of backward bifurcation and very few relateto real communicable diseases (see Greenhalgh and Griffiths [15]on a Bovine Respiratory Syncytial virus epidemic model and Blay-neh et al. [4] and Wan and Zhu [35] on West Nile virus epidemicmodels). As mentioned in the Introduction, we started the projectwith modeling the SARS outbreaks in 2003 [38]. The second SARSoutbreak in Toronto, Canada in 2003 (see Gumel et al. [16]) maybe explained as a result of backward bifurcation since the basicreproduction number was certainly less than unity after the firstoutbreak when restrictive and suitable infection-control proce-dures were being taken [19]. We expect that our results in this pa-per might be helpful to study the endemic of hepatitis B virus inChina [43] and the outbreaks of cholera in Haiti or Zimbabwe[28]. We propose to study these in the future.

Acknowledgments

We are very grateful to the referees and the handling editor fortheir helpful comments and suggestions. We also would like tothank Professor Wendi Wang for helpful discussions.

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Table 1Existence and stability of equilibria for system (1.4).

Conditions E0 E� E1 E2 Figure

R0 < 1 Stable DNE DNE DNEu < minfu0; u1g Stable DNE Unstable Stable Fig. 6

R0 > 1 u2 6 u Unstable Stable DNE DNE Fig. 4R0 < R�0 u2 6 u 6 minfu0; u1g Unstable Stable Unstable Stable Fig. 7R�0 < R0 u 6minfu0;u2g Unstable DNE DNE Stable Fig. 5

Unstable Fig. 8



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