# Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Authors: Yanju Xiao; Weipeng Zhang; Guifeng Deng; Zhehua Liu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.


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## Body

## 1. Introduction

In the mathematical modeling of epidemic transmission, there are several factors that substantially affect the dynamical behavior of the models. Incidence rate functions are seen as a major factor in producing the rich dynamics of epidemic models in many literatures (see [1–25]). In most classical models of epidemics, the incidence rate is taken to be mass action incidence with bilinear interaction, that is, βIS, where β is the probability of transmission per contact and S and I represent the number of susceptible and infected individuals, respectively. Epidemic models with such bilinear incidence rates usually have at most one endemic equilibrium, and then the diseases will be eradicated if the basic reproduction number is less than one and will persist otherwise. Besides, there are also many other types of incidence rate functions, such as nonlinear incidence rate, standard incidence rate, and saturated incidence rate. Recently, there are many studies that have demonstrated the nonlinear incidence rate which is one of the key factors that induce periodic oscillations in epidemic models (see [1, 2, 9, 10, 15]). Moreover, Liu et al. [12] introduced a nonlinear incidence rate of the form
(1)f(I)S=βIpS1+αIq,
where βIp means the infection force of the disease and 1/(1+αIq) measures the inhibition effect from the behavioral change of the susceptible individuals when the number of infective individuals increases. So we can see that the bilinear incidence rate βSI is a special case of (1) with p=1 and α=0 or q=0. Furthermore, Wang and Ruan in [16] studied the global dynamics of an SIRS model with the incidence function f(I)=βI2/(1+αI2); that is, p=q=2; they also showed that the SIRS epidemic model undergoes a Bogdanov-Takens bifurcation, that is, saddle-node, Hopf, and homoclinic bifurcations. To have a better understanding of the dynamics of the system, Tang et al. [17] calculated higher order Lyapunov values of the weak focus and reduced the system to a universal unfolding form for a cusp of codimension 3, and they gave the bifurcation surfaces and displayed all limit cycles and monoclinic loops of order up to 2. Wei and Cui explored an SIS epidemic model with standard incidence rate in [19]. They took the incidence rate form f(I)S=βIS/(I+S) and showed the dynamics and backward bifurcation of the SIS epidemic model.Recently, in order to prevent and control the spread of the infectious diseases such as measles, tuberculosis, and flu, many mathematicians (see [7, 11, 16, 20, 23, 25–29]) have begun to investigate the role of treatment functions in epidemiological models. In some classical epidemic models, the treatment function is an important method to decrease the spread of the epidemiological diseases. Generally speaking, the treatment function of the infective individuals is assumed to be proportional to the number of the infective individuals. But every community should have a maximal capacity for the treatment of a disease and the resources for treatment should be very large. Therefore, it is very important to adopt a suitable treatment function. In [16], Wang and Ruan introduced a constant treatment function of diseases in an SIR model; that is,
(2)T(I)=r,I>0,0,I=0.
This means that they use the maximal treatment capacity to cure infective individuals so that the disease can be eradicated. They also found that the model undergoes saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation. Further, a piecewise linear treatment function was considered in [20]; that is,
(3)T(I)=kI,0≤I≤I0,m,I>I0,
where m=kI0 and k and I0 are positive constants. This means that the treatment rate is proportional to the number of the infective individuals when the capacity of treatment has not been reached; otherwise it takes the maximal capacity of treatment kI0. By considering the above treatment function, Wang [20] found that a backward bifurcation takes place in an SIR epidemic model. In [30], J. C. Eckalbar and W. L. Eckalbar constructed an SIR epidemic model with a quadratic treatment function; that is, T(I)=max⁡{rI-gI2,0},   r,g>0. They found that the system has as many as four equilibria, with possible bistability, backward bifurcations, and limit cycles.Recently, saturated treatment function has been widely applied in many epidemic models. In particular, in paper [23], Zhang and Liu took a continuous and differentiable saturated treatment function T(I)=rI/(1+αI), where r>0, α≥0. r stands for the cure rate and α measures the extent of the effect of the infected being delayed for treatment. We can see that the treatment function T(I)~rI when I is small enough, whereas T(I)~r/α when I is large enough. It is more realistic and it has the convenience of being continuous and differential compared to the previous ones. Furthermore, the authors in [23] found that R0=1 is a critical threshold for disease eradication when this delayed effect for treatment is weak and a backward bifurcation will take place when this effect is strong. So, it is really important to adequately stress the interesting connection recently established between the choice of saturated treatment functions in epidemic models and the occurrence of backward bifurcation in the related system dynamics. In fact, recently, saturated-type treatment functions have been indicated as responsible for the occurrence of backward bifurcations for SIR [20, 23], for SIS [19, 21], and for SEIR [28, 29] models, supporting the general circumstance that saturated-type treatments can be one of the causes of backward bifurcations in epidemic models. In particular, in [29], such connection has also been shown and validated in a specific concrete disease-control setting.In the real world, some infectious diseases do not confer immunity. Such infections do not have a recovered state and individuals become susceptible again after infection. This type of disease can be modeled by the SIS type. So the SIS epidemic model has been adopted by many mathematicians (see [8, 11, 18, 19, 21, 25]). And SIS models are appropriate for some bacterial agent diseases such as meningitis, plague, and venereal diseases and for protozoan agent diseases such as malaria and sleeping sickness.Motivated by the above points, we will consider the following SIS model with bilinear incidence rate and saturated treatment function:(4)dSdt=A-dS-λSI+εI+rI1+αI,dIdt=λSI-(d+ε+μ)I-rI1+αI,
where S and I denote the numbers of susceptible and infective individuals, respectively. Positive constant A is the recruitment rate of the population. Positive constant d is the nature death rate of population. The bilinear incidence rate is λSI, where λ is positive. Positive constant ε is the natural recovery rate of infective individuals. Positive constant μ is the disease-related death rate. The saturated treatment function hI≜rI/(1+αI), where r is positive and α is nonnegative.This paper focuses on the detailed dynamics analysis of the model (4). The local stability of these equilibria is investigated, which enables us to classify the types of model equilibria (e.g., attractor, saddle, or repeller). We show that the system has backward bifurcation and Bogdanov-Takens bifurcation (i.e., Hopf bifurcation, saddle-node bifurcation, and homoclinic bifurcation) under some certain conditions. Finally, the three bifurcation curves and the complicated global bifurcation phase portraits are derived by applying the Bogdanov-Takens normal form and the corresponding parameters which satisfy the conditions that ensure Bogdanov-Takens bifurcation exists.The organization of this paper is as follows. In Section2, we study the existence and local stability of equilibria and backward bifurcation. In Section 3, we investigate the global stability of the model. In Section 4, we give the supercritical and subcritical bifurcation under two different conditions in system (4). In Section 5, we show that the system (4) undergoes Bogdanov-Takens bifurcation under some certain conditions. In Section 6, some numerical simulations are displayed in detail. We close with a discussion in Section 7 on our mathematical results and epidemiological implications.


## 2. Equilibria and Backward Bifurcation

### 2.1. Disease-Free Equilibrium

Obviously, system (4) has a disease-free equilibrium E0=(A/d,0). The Jacobian matrix of (4) at E0 is
(5)M(E0)=-d-λAd+ε+r0λAd-(d+ε+μ)-r.
By using the next generation matrix of (4), we get the basic reproduction number R0=λA/d(d+ε+r+μ). M(E0) has negative eigenvalues if λA/d-(d+ε+μ)-r<0. Then we have the following result.Theorem 1.
The disease-free equilibriumE0 is locally asymptotically stable when R0<1 (see Figure 1) and is unstable when R0>1 (see Figure 2).Figure 1
The disease-free equilibriumE0 is locally asymptotically stable when R0<1, with the parameter values A=2, d=0.1, k=0.1, μ=0.01, λ=0.01, α=1, r=2, and ε=0.1.Figure 2
The disease-free equilibrium is unstable whenR0>1, with the parameter values A=15, d=0.1, k=0.1, μ=0.01, λ=0.01, α=1, r=0.8, and ε=0.1.


### 2.2. Endemic Equilibria

An endemic equilibrium always satisfies(6)A-dS-λSI+εI+rI1+αI=0,λSI-(d+ε+μ)I-rI1+αI=0.
In view of dS+(d+μ)I=A, we get S=(A-d+μI)/d and substitute it into the second equation of (6). When I≠0, we obtain
(7)λA-d+μId-(d+ε+μ)-r1+αI=0.
Then we have an equation of the form
(8)aI2+bI+c=0,
with
(9)a=αλ(d+μ),b=λ(d+μ)+αd(d+ε+μ)-αλA,c=d(d+ε+r+μ)-λA.
This equation may admit positive solution
(10)I1=-b-b2-4ac2a,I2=-b+b2-4ac2a.Obviously, ifR0=1, then c=0, if R0>1, then c<0, and if R0<1, then c>0. From (8), it is obvious that we have the following results.Theorem 2.
The following results hold.(H1)
Letα=0. Equation (8) is a linear equation with a unique solution I=-c/b. Then the system (4) has a unique endemic equilibrium when R0>1 and has no endemic equilibrium when R0≤1.(H2)
Letα>0. If b>0, system (4) has a unique endemic equilibrium when R0>1 and no endemic equilibrium when R0≤1.(H3)
Letα>0. If b<0, system (4) has a unique endemic equilibrium when R0≥1, no endemic equilibrium when R0<R0*, and two endemic equilibria E1 and E2 when R0*≤R0<1. When R0=R0* and E1=E2, one has b2-4ac=0 which is equivalent to
(11)R0=4aλAb2+4aλA=4λ2α(d+μ)Aλd+μ+αdd+ε+μ-αλA2+4λ2α(d+μ)A.We define the right-hand side of (11) as
(12)R0*≜4λ2α(d+μ)Aλd+μ+αdd+ε+μ-αλA2+4λ2α(d+μ)A.Therefore, according to the qualitative approach recently proposed by [31] which is based on the analysis of the equilibria curve in the neighboring of the critical threshold R0=1, we have the following theorem.Theorem 3.
Ifα>0, b<0, then system (4) has a backward bifurcation at R0=1 (see Figure 3).Figure 3
The figure of infective sizes at equilibria versusR0 when α=1, d=0.1, μ=0.01, λ=0.05, r=2, and ε=0.1.In order to verify the bifurcation curve (the graph ofI as a function of R0) in Figure 3, we think of r as a variable with the other parameters as constant. Through implicit differentiation of (8) with respect to r, we get
(13)(2aI+b)dIdr=-d<0.
From (13), we know that the sign of dI/dr is opposite to that of 2aI+b. And from the definition of R0 we know that R0 decreases when r increases. It implies that the bifurcation curve has positive slope at equilibrium values with 2aI+b>0 and negative slope at equilibrium values with 2aI+b<0. If there is no backward bifurcation at R0=1, then the unique endemic equilibrium for R0>1 satisfies
(14)2aI+b=b2-4ac>0,
and the bifurcation curve has positive slope at all points where I>0. If there is a backward bifurcation at R0=1, then there is an interval on which there are two endemic equilibria given by
(15)2aI+b=±b2-4ac.
The bifurcation curve has negative slope at the smaller one and positive slope at the larger one. Thus the bifurcation curve is as shown in Figure 3.Under the conditions of Theorem3, if a backward bifurcation takes place, we can see from Figure 3 that there is a critical value R0* at the turning point. In this case, the disease will not die out when R0<1. However, the disease will die out when R0<R0*. Therefore, the critical value R0* can be taken as a new threshold for the control of the disease.In the following, we give an explicit criterion of a backward bifurcation atR0=1.For convenience, we define(16)α0:=λ(d+μ)dr.Corollary 4.
Whenα>α0, system (4) has a backward bifurcation at R0=1.Proof.
WhenR0≤1⇔c≥0,
(17)λA≤d(d+ε+μ+r).
The condition b<0 is equivalent to
(18)λ(d+μ)+αd(d+ε+μ)<αλA.
From (17) and (18), we get λ(d+μ)+αd(d+ε+μ)<αd(d+ε+r+μ), which reduces to
(19)α>λ(d+μ)dr≜α0.
It means that α is big enough to lead a backward bifurcation with two endemic equilibria when R0<1. Therefore, the proof is complete.Next, we consider the local stability of the unique endemic equilibrium whenR0>1.Theorem 5.
WhenR0>1 and 0≤α<λ/r, the unique endemic equilibrium E* is locally asymptotically stable.Proof.
Firstly, from Theorem2, we can know that system (4) has a unique endemic equilibrium E* when R0>1. Moreover, the Jacobian matrix of system (4) is
(20)M=-d-λI-λS+ε+r1+αI-rαI1+αI2λIλS-(d+ε+μ)-r1+αI+rαI1+αI2.
From the second equation of (6), we have
(21)-λS+ε+r1+αI=-d-μ.
From (21), the Jacobian matrix M reduces to
(22)M=-d-λI-d-μ-rαI1+αI2λIrαI1+αI2.
We obtain
(23)det⁡(M)=I1+αI2λd+μ1+αI2-rαd.
In fact, there is λ(d+μ)(1+αI)2>λd. Since R0>1 and 0≤α<λ/r, we have
(24)λ(d+μ)1+αI2-rαd>λd-rαd>0.
So we get det⁡(M)>0. The trace of M is given by
(25)tr⁡(M)=11+αI2-d+λI1+αI2+rαI.
In the same way as the above calculation of (24), we have
(26)-(d+λI)1+αI2+rαI<-λI+rαI<0.
So we get tr⁡(M)<0. The proof is complete.Now we consider the case that there are two endemic equilibriaE1 and E2; let Mi be the Jacobian matrix at Ei, i=1,2.Theorem 6.
The endemic equilibriumE1 is a saddle whenever it exists.Proof.
SinceI1=(-b-b2-4ac)/2a and Δ=b2-4ac, we have I1=(-b-Δ)/2a. Thus
(27)det⁡⁡M1=I11+αI12λd+μ1+αI12-rαd≜I11+αI12ψ(I1).
From (H3) of Theorem 2, we can know that if α>0, b<0, and R0*<R0<1, then E1 exists. Hence, we can get ψ(0)=λ(d+μ)-rαd<0 and ψ′(I1)=2λ(d+μ)(1+αI1)α>0. It follows that there exists a unique I*>0 such that
(28)ψ(I1)=0,whenI1=I*,ψ(I1)<0,when0<I1<I*,ψ(I1)>0,whenI1>I*,
where
(29)I*=rdαλ(d+μ)-1α.
On the other hand,
(30)I1=-b2a-Δ2a=αλA-λd+μ-αdd+ε+μ2αλd+μ-Δ2a=rdαλ(d+μ)-1α+1α-rdαλ(d+μ)+αλA-λ(d+μ)-αd(d+ε+μ)2αλ(d+μ)-Δ2a=I*+λd+μ+αλA-2rdαλd+μmmmmm-αdd+ε+μrdαλd+μ-Δ×2αλd+μ-1,(31)Δ=λd+μ-αλA-αdd+ε+μ2-4αλd+μdd+ε+r+μ-λA=rdαλ(d+μ)λ(d+μ)+αλA-αdd+ε+μmmm-2rdαλ(d+μ)2-4rdαλ(d+μ)-4λd+μαλA-αdd+ε+μ+4rdαλd+μλd+μ+αλA-αdd+ε+μ-4αλ(d+μ)[-λA+d(d+ε+r+μ)]≜rdαλd+μλd+μ+αλA-αdd+ε+μ-2rdαλd+μ2+P,
where
(32)P=-8rdαλd+μ+4rdαλd+μ×{λ(d+μ)+[αλA-αd(d+ε+μ)]}.
In the following we will show that P>0. Since R0*<R0, we have
(33)4αλd+μλd+μ+αdd+ε+μ-αλA2+4αλ2(d+μ)A<1dd+ε+r+μ;
that is,
(34)λd+μ+αdd+ε+μ-αλA2>4αλ(d+μ)d(d+ε+r+μ)-4αλ2(d+μ)A.
Therefore,
(35)λd+μ+αλA-αdd+ε+μ2=λd+μ-αλA-αdd+ε+μ2+4λd+μαλA-αdd+ε+μ>4αλ(d+μ)d(d+ε+r+μ)-4αλ2(d+μ)A+4λ(d+μ)[αλA-αd(d+ε+μ)]=4rdαλ(d+μ).
Obviously, we have P>0. From (30), one has I1<I*. So we get det⁡(M1)<0. Hence the endemic equilibrium E1 is a saddle. The proof is complete.In order to explore the stability of the endemic equilibriumE2, define
(36)m1≔2dαa2+λa2-εαa2-acλα2-b(2λαa+dα2a-bλα2),m2:=a2d-c(2λαa+dα2a-bλα2).Theorem 7.
Ifη>0, then endemic equilibrium E2 is locally asymptotically stable; if η<0, then endemic equilibrium E2 is unstable, where η:=2am2+m1(b2-4ac-b).Proof.
Consider(37)det⁡⁡M2=I21+αI22λd+μ1+αI22-εαd=I21+αI22×ψ(I2).
By carrying out arguments similar to that of Theorem 6, we have I2>I*. Therefore, det⁡(M2)>0. In addition, we have
(38)tr⁡M2=-(d+λI2)1+αI22-εαI21+αI22=-λα2I23+(2λα+dα2)I22+2dα+λ-εαI2+d1+αI22,
and then sgn⁡(tr⁡(M2))=-sgn⁡(G(I2)), where
(39)G(x)=λα2x3+(2λα+dα2)x2+(2dα+λ-εα)x+d.
Using the expression of m1=(2dαa2+λa2-εαa2-acλα2)-b(2λαa+dα2a-bλα2) and m2=a2d-c(2λαa+dα2a-bλα2), one has
(40)G(I2)=(aI22+bI2+c)φ0+m1I2+m2a2,
where φ0 is a first degree polynomial of I2. Since aI22+bI2+c=0, sgn⁡(tr⁡(M2))=-sgn⁡(G(I2))=-sgn⁡(m1I2+m2). From the expression of I2, we have
(41)sgn⁡m1I2+m2=sgn⁡(2am2+m1(b2-4ac-b))≜sgn⁡(η).
Thus, E2 is locally asymptotically stable if η>0 and E2 is unstable if η<0. The proof is complete.From the above discussion, we have the following conclusion. If two endemic equilibriaE1 and E2 exist, the stable manifolds of the saddle E1 split R+2 into two regions. The disease is persistent in the upper region and dies out in the lower region (see Figure 4).Figure 4
One region of disease persistence and one region of disease extinction whenA=10, α=1, d=0.1, μ=0.01, λ=0.01, r=0.1, and ε=0.1.


## 2.1. Disease-Free Equilibrium

Obviously, system (4) has a disease-free equilibrium E0=(A/d,0). The Jacobian matrix of (4) at E0 is
(5)M(E0)=-d-λAd+ε+r0λAd-(d+ε+μ)-r.
By using the next generation matrix of (4), we get the basic reproduction number R0=λA/d(d+ε+r+μ). M(E0) has negative eigenvalues if λA/d-(d+ε+μ)-r<0. Then we have the following result.Theorem 1.
The disease-free equilibriumE0 is locally asymptotically stable when R0<1 (see Figure 1) and is unstable when R0>1 (see Figure 2).Figure 1
The disease-free equilibriumE0 is locally asymptotically stable when R0<1, with the parameter values A=2, d=0.1, k=0.1, μ=0.01, λ=0.01, α=1, r=2, and ε=0.1.Figure 2
The disease-free equilibrium is unstable whenR0>1, with the parameter values A=15, d=0.1, k=0.1, μ=0.01, λ=0.01, α=1, r=0.8, and ε=0.1.


## 2.2. Endemic Equilibria

An endemic equilibrium always satisfies(6)A-dS-λSI+εI+rI1+αI=0,λSI-(d+ε+μ)I-rI1+αI=0.
In view of dS+(d+μ)I=A, we get S=(A-d+μI)/d and substitute it into the second equation of (6). When I≠0, we obtain
(7)λA-d+μId-(d+ε+μ)-r1+αI=0.
Then we have an equation of the form
(8)aI2+bI+c=0,
with
(9)a=αλ(d+μ),b=λ(d+μ)+αd(d+ε+μ)-αλA,c=d(d+ε+r+μ)-λA.
This equation may admit positive solution
(10)I1=-b-b2-4ac2a,I2=-b+b2-4ac2a.Obviously, ifR0=1, then c=0, if R0>1, then c<0, and if R0<1, then c>0. From (8), it is obvious that we have the following results.Theorem 2.
The following results hold.(H1)
Letα=0. Equation (8) is a linear equation with a unique solution I=-c/b. Then the system (4) has a unique endemic equilibrium when R0>1 and has no endemic equilibrium when R0≤1.(H2)
Letα>0. If b>0, system (4) has a unique endemic equilibrium when R0>1 and no endemic equilibrium when R0≤1.(H3)
Letα>0. If b<0, system (4) has a unique endemic equilibrium when R0≥1, no endemic equilibrium when R0<R0*, and two endemic equilibria E1 and E2 when R0*≤R0<1. When R0=R0* and E1=E2, one has b2-4ac=0 which is equivalent to
(11)R0=4aλAb2+4aλA=4λ2α(d+μ)Aλd+μ+αdd+ε+μ-αλA2+4λ2α(d+μ)A.We define the right-hand side of (11) as
(12)R0*≜4λ2α(d+μ)Aλd+μ+αdd+ε+μ-αλA2+4λ2α(d+μ)A.Therefore, according to the qualitative approach recently proposed by [31] which is based on the analysis of the equilibria curve in the neighboring of the critical threshold R0=1, we have the following theorem.Theorem 3.
Ifα>0, b<0, then system (4) has a backward bifurcation at R0=1 (see Figure 3).Figure 3
The figure of infective sizes at equilibria versusR0 when α=1, d=0.1, μ=0.01, λ=0.05, r=2, and ε=0.1.In order to verify the bifurcation curve (the graph ofI as a function of R0) in Figure 3, we think of r as a variable with the other parameters as constant. Through implicit differentiation of (8) with respect to r, we get
(13)(2aI+b)dIdr=-d<0.
From (13), we know that the sign of dI/dr is opposite to that of 2aI+b. And from the definition of R0 we know that R0 decreases when r increases. It implies that the bifurcation curve has positive slope at equilibrium values with 2aI+b>0 and negative slope at equilibrium values with 2aI+b<0. If there is no backward bifurcation at R0=1, then the unique endemic equilibrium for R0>1 satisfies
(14)2aI+b=b2-4ac>0,
and the bifurcation curve has positive slope at all points where I>0. If there is a backward bifurcation at R0=1, then there is an interval on which there are two endemic equilibria given by
(15)2aI+b=±b2-4ac.
The bifurcation curve has negative slope at the smaller one and positive slope at the larger one. Thus the bifurcation curve is as shown in Figure 3.Under the conditions of Theorem3, if a backward bifurcation takes place, we can see from Figure 3 that there is a critical value R0* at the turning point. In this case, the disease will not die out when R0<1. However, the disease will die out when R0<R0*. Therefore, the critical value R0* can be taken as a new threshold for the control of the disease.In the following, we give an explicit criterion of a backward bifurcation atR0=1.For convenience, we define(16)α0:=λ(d+μ)dr.Corollary 4.
Whenα>α0, system (4) has a backward bifurcation at R0=1.Proof.
WhenR0≤1⇔c≥0,
(17)λA≤d(d+ε+μ+r).
The condition b<0 is equivalent to
(18)λ(d+μ)+αd(d+ε+μ)<αλA.
From (17) and (18), we get λ(d+μ)+αd(d+ε+μ)<αd(d+ε+r+μ), which reduces to
(19)α>λ(d+μ)dr≜α0.
It means that α is big enough to lead a backward bifurcation with two endemic equilibria when R0<1. Therefore, the proof is complete.Next, we consider the local stability of the unique endemic equilibrium whenR0>1.Theorem 5.
WhenR0>1 and 0≤α<λ/r, the unique endemic equilibrium E* is locally asymptotically stable.Proof.
Firstly, from Theorem2, we can know that system (4) has a unique endemic equilibrium E* when R0>1. Moreover, the Jacobian matrix of system (4) is
(20)M=-d-λI-λS+ε+r1+αI-rαI1+αI2λIλS-(d+ε+μ)-r1+αI+rαI1+αI2.
From the second equation of (6), we have
(21)-λS+ε+r1+αI=-d-μ.
From (21), the Jacobian matrix M reduces to
(22)M=-d-λI-d-μ-rαI1+αI2λIrαI1+αI2.
We obtain
(23)det⁡(M)=I1+αI2λd+μ1+αI2-rαd.
In fact, there is λ(d+μ)(1+αI)2>λd. Since R0>1 and 0≤α<λ/r, we have
(24)λ(d+μ)1+αI2-rαd>λd-rαd>0.
So we get det⁡(M)>0. The trace of M is given by
(25)tr⁡(M)=11+αI2-d+λI1+αI2+rαI.
In the same way as the above calculation of (24), we have
(26)-(d+λI)1+αI2+rαI<-λI+rαI<0.
So we get tr⁡(M)<0. The proof is complete.Now we consider the case that there are two endemic equilibriaE1 and E2; let Mi be the Jacobian matrix at Ei, i=1,2.Theorem 6.
The endemic equilibriumE1 is a saddle whenever it exists.Proof.
SinceI1=(-b-b2-4ac)/2a and Δ=b2-4ac, we have I1=(-b-Δ)/2a. Thus
(27)det⁡⁡M1=I11+αI12λd+μ1+αI12-rαd≜I11+αI12ψ(I1).
From (H3) of Theorem 2, we can know that if α>0, b<0, and R0*<R0<1, then E1 exists. Hence, we can get ψ(0)=λ(d+μ)-rαd<0 and ψ′(I1)=2λ(d+μ)(1+αI1)α>0. It follows that there exists a unique I*>0 such that
(28)ψ(I1)=0,whenI1=I*,ψ(I1)<0,when0<I1<I*,ψ(I1)>0,whenI1>I*,
where
(29)I*=rdαλ(d+μ)-1α.
On the other hand,
(30)I1=-b2a-Δ2a=αλA-λd+μ-αdd+ε+μ2αλd+μ-Δ2a=rdαλ(d+μ)-1α+1α-rdαλ(d+μ)+αλA-λ(d+μ)-αd(d+ε+μ)2αλ(d+μ)-Δ2a=I*+λd+μ+αλA-2rdαλd+μmmmmm-αdd+ε+μrdαλd+μ-Δ×2αλd+μ-1,(31)Δ=λd+μ-αλA-αdd+ε+μ2-4αλd+μdd+ε+r+μ-λA=rdαλ(d+μ)λ(d+μ)+αλA-αdd+ε+μmmm-2rdαλ(d+μ)2-4rdαλ(d+μ)-4λd+μαλA-αdd+ε+μ+4rdαλd+μλd+μ+αλA-αdd+ε+μ-4αλ(d+μ)[-λA+d(d+ε+r+μ)]≜rdαλd+μλd+μ+αλA-αdd+ε+μ-2rdαλd+μ2+P,
where
(32)P=-8rdαλd+μ+4rdαλd+μ×{λ(d+μ)+[αλA-αd(d+ε+μ)]}.
In the following we will show that P>0. Since R0*<R0, we have
(33)4αλd+μλd+μ+αdd+ε+μ-αλA2+4αλ2(d+μ)A<1dd+ε+r+μ;
that is,
(34)λd+μ+αdd+ε+μ-αλA2>4αλ(d+μ)d(d+ε+r+μ)-4αλ2(d+μ)A.
Therefore,
(35)λd+μ+αλA-αdd+ε+μ2=λd+μ-αλA-αdd+ε+μ2+4λd+μαλA-αdd+ε+μ>4αλ(d+μ)d(d+ε+r+μ)-4αλ2(d+μ)A+4λ(d+μ)[αλA-αd(d+ε+μ)]=4rdαλ(d+μ).
Obviously, we have P>0. From (30), one has I1<I*. So we get det⁡(M1)<0. Hence the endemic equilibrium E1 is a saddle. The proof is complete.In order to explore the stability of the endemic equilibriumE2, define
(36)m1≔2dαa2+λa2-εαa2-acλα2-b(2λαa+dα2a-bλα2),m2:=a2d-c(2λαa+dα2a-bλα2).Theorem 7.
Ifη>0, then endemic equilibrium E2 is locally asymptotically stable; if η<0, then endemic equilibrium E2 is unstable, where η:=2am2+m1(b2-4ac-b).Proof.
Consider(37)det⁡⁡M2=I21+αI22λd+μ1+αI22-εαd=I21+αI22×ψ(I2).
By carrying out arguments similar to that of Theorem 6, we have I2>I*. Therefore, det⁡(M2)>0. In addition, we have
(38)tr⁡M2=-(d+λI2)1+αI22-εαI21+αI22=-λα2I23+(2λα+dα2)I22+2dα+λ-εαI2+d1+αI22,
and then sgn⁡(tr⁡(M2))=-sgn⁡(G(I2)), where
(39)G(x)=λα2x3+(2λα+dα2)x2+(2dα+λ-εα)x+d.
Using the expression of m1=(2dαa2+λa2-εαa2-acλα2)-b(2λαa+dα2a-bλα2) and m2=a2d-c(2λαa+dα2a-bλα2), one has
(40)G(I2)=(aI22+bI2+c)φ0+m1I2+m2a2,
where φ0 is a first degree polynomial of I2. Since aI22+bI2+c=0, sgn⁡(tr⁡(M2))=-sgn⁡(G(I2))=-sgn⁡(m1I2+m2). From the expression of I2, we have
(41)sgn⁡m1I2+m2=sgn⁡(2am2+m1(b2-4ac-b))≜sgn⁡(η).
Thus, E2 is locally asymptotically stable if η>0 and E2 is unstable if η<0. The proof is complete.From the above discussion, we have the following conclusion. If two endemic equilibriaE1 and E2 exist, the stable manifolds of the saddle E1 split R+2 into two regions. The disease is persistent in the upper region and dies out in the lower region (see Figure 4).Figure 4
One region of disease persistence and one region of disease extinction whenA=10, α=1, d=0.1, μ=0.01, λ=0.01, r=0.1, and ε=0.1.


## 3. Global Analysis

Firstly, we consider the global stability of the disease-free equilibriumE0. Let N=S+I be the total population size. Now we note that the equation for total population is given by dN/dt=A-dS-(d+μ)I≤A-dN. It follows that lim⁡t→+∞⁡N(t)≤A/d. Let
(42)R=S,I∈R+2:S+I≤Ad
which is positive invariant with respect to system (4).Theorem 8.
IfR0<R0*, the disease-free equilibrium E0(A/d,0) is globally asymptotically stable; that is, the disease dies out.Proof.
SupposeR0<R0*. From the (H3) of Theorem 2, we know that the model has no endemic equilibrium. From the corollary of Poincaré-Bendixson theorem [32], we know that there is no periodic orbit in R as there is a disease-free equilibrium in R. Since R is a bounded positively invariant region and E0 is the only equilibrium in R, the local stability of E0 implies that every solution initiating in R approaches E0. Thus, the disease-free equilibrium E0 is globally asymptotically stable. The proof is complete.Now we analyze the global dynamics of the endemic equilibrium whenR0>1.Theorem 9.
IfR0>1 and 0≤α<λ/r, the system (4) has no limit cycles.Proof.
We use Dulac theorem to exclude the limit cycle. Let(43)P(S,I)=A-dS-λSI+εI+rI1+αI,Q(S,I)=λSI-(d+ε+μ)I-rI1+αI,
and take the Dulac function
(44)D=1I.
According to 0≤α<λ/r, we can get
(45)∂PD∂S+∂QD∂I=-dI-λ+αr1+αI2<-dI-λ+λ1+αI2=1I1+αI2-d1+αI2-λI1+αI2-1<0.
Hence, the system (4) has no limit cycles. The proof is complete.Therefore, we obtain the global result of the unique endemic equilibrium.Theorem 10.
IfR0>1 and 0≤α<λ/r, the unique endemic equilibrium E* is globally asymptotically stable (see Figure 5).Figure 5
The unique endemic equilibriumE* is globally asymptotically stable when A=15, α=0.01, d=0.1, μ=0.01, λ=0.01, r=0.8, and ε=0.1.


## 4. Hopf Bifurcation

In this section, we study the Hopf bifurcation of system (4). From the above discussion, we know that there is no closed orbit surrounding E0 or E1 because the S-axis is invariant with respect to system (4) and E1 is always a saddle. Therefore, Hopf bifurcation can only occur at E2. Set
(46)σ=-λα-c1c2+2c4a12D4+a112c1a12-λαa112-2λαa11a12+c2-2a11c1a12mmm×a11c2-a112c1a12mmm-2a112c1a12-a11c2+2c4a11-c3a12mmm×c4-a11c1a12a112c1a12D2+a11a112c1a12-a11c2+c4a11-c3a12×a12c3-2a11c4-a11c2,
where aij(i,j=1,2), ck(k=1,2,3,4), and D are defined by (50) and (52).Theorem 11.
System (4) undergoes a Hopf bifurcation if η=0. Moreover, if σ<0, there is a family of stable period orbits of system (4) as η decreases from 0; that is, a supercritical Hopf bifurcation occurs; if σ>0, there is a family of unstable period orbits of system (4) as η increases from 0; that is, a subcritical Hopf bifurcation occurs.Proof.
The proof of Theorem7 shows that tr⁡(M2)=0 if and only if η=0, and det⁡(M2)>0 when E2 exists. Therefore, the eigenvalues of M2 are a pair of pure imaginary roots if and only if η=0. The direct calculations show that
(47)dtr⁡M2dηη=0=-12a31+αI22<0.
By Theorem  3.4.2 in [24], η=0 is the Hopf bifurcation point for system (4).
To be concise in notations, rescale (4) by τ=t/(1+αI). For simplicity, we still use t instead of τ. Then we obtain
(48)dSdt=(A-dS-λSI+εI)(1+αI)+rI,dIdt=λSI(1+αI)-(d+ε+μ)I(1+αI)-rI.
Let x=S-S2 and y=I-I2; then (48) becomes
(49)dxdt=a11x+a12y+c1y2+c2xy-λαxy2,dydt=a21x+a22y+c3xy+c4y2+λαxy2,
where
(50)a11=-(d+λI2)(1+αI2),a12=-λS2+r1+αI2+α(A-dS2-λS2I2+rI2)+ε,a21=λI2(1+αI2),a22=λS2-d+r+μ1+αI2+α[λS2I2-(d+r+μ)I2]-ε,c1=α(-λS2+r),c2=α(-d-λI2)-λ(1+αI2),c3=λ(1+αI2)+λI2α,c4=[λS2-(d+r+μ)]α.
Let E* denote the origin of x-y plane. Since E2=(S2,I2) satisfies (6), we obtain
(51)det⁡⁡ME*=-(d+λI2)αεI2+(d+μ)1+αI222λI2+αλεI22=I2×ψ(I2).
From the proof of Theorem 7, it follows that ψ(I2) is always positive. It is easy to verify that a11+a22=0 if and only if η=0. Set
(52)D=det⁡(M(E*))
and let u=-x and v=(a11/D)x+(a12/D)y; then the normal form of (48) for Hopf bifurcation reads
(53)dudt=-Dv+f(u,v),dvdt=Du+g(u,v),
where
(54)fu,v=a11a12c2-a112a122c1u2-D2c1a122v2+Dc2a12-2Da11c1a122uv-λαa112a122u3-2Da11λαa122u2v-D2λαa122uv2,gu,v=a11Da112a122c1-a11a12c2+a11a12c4-c3u2+Da11c1a122+Dc4a12v2+2a112a122c1-a11a12c2+2a11a12c4-c3uv+λαa112a12D1a12-1u3+2λαa11a12a11a12-1u2v+Dλαa12a11a12-1uv2.
Now, we evaluate the first Lyapunov coefficient Γ of system (4) as follows:
(55)Γ=116fuuu+fuvv+guuv+gvvv+116D×[fuv(fuu+fvv)-guv(guu+gvv)-fuuguu+fvvgvv],
where fuv denotes (∂2f/∂u∂v)(0,0) and so forth. Then
(56)Γ=18a122D2-λα-c1c2+2c4a12D4mmmmmm+2a112c1a12-a11c2+2c4a11-c3a12-λαa112-2λαa11a12+c2-2a11c1a12mmmimmmmm×a11c2-a112c1a12mmmmimmmm-2a112c1a12-a11c2+2c4a11-c3a12mmmimmmmm×c4-a11c1a122a112c1a12-a11c2+2c4a11-c3a12D2mmmmmm+a11a112c1a12-a11c2+c4a11-c3a12mmmmmm×a12c3-2a11c4-a11c2-λα-c1c2+2c4a12=σ8a122D2.
Obviously, the sign of  Γ is determined by σ. By Theorem 3.4.2 and (3.4.11) in [24], the rest of the claims in Theorem 11 are valid. The proof is complete.


## 5. Bogdanov-Takens Bifurcations

The purpose of this section is to study the Bogdanov-Takens bifurcation of (4). Now, we assume the following:(A1)
b2-4ac=0,(A2)
2ad2+bλμ=0.Then (6) admits a unique positive equilibrium E*=(S*,I*), where
(57)I*=-b2a,S*=A-(d+μ)I*d.
The Jacobian matrix of system (4) at E* is
(58)M*=-d-λI*-λS*+ε+r1+αI*2λI*λS*-(d+ε+μ)-r1+αI*2.
By (58), we have
(59)det⁡M*=I*1+αI*2[(d+μ)λ1+αI*2-rdα]=0,
because of
(60)1+αI*2=4a2-4aαb+α2b24a2=4a2-4aαb+α24ac4a2=a-αb+α2ca=α2rda.
Furthermore, (A2) implies that
(61)tr⁡(M*)=0.
Thus, (A1) and (A2) imply that the Jacobian matrix has a zero eigenvalue with multiplicity 2. This suggests that (4) may admit a Bogdanov-Takens singularity.Theorem 12.
Suppose that(A1), (A2), and 2b1+b4≠0 hold. Then the endemic equilibrium E*=(S*,I*) of (4) is a cusp of codimension 2; that is, it is a Bogdanov-Takens singularity. Here, b1 and b4 are defined by (65).Proof.
Using the transformation ofx=I-I* and y=S-S*, system (4) becomes
(62)dxdt=a1x+a2y+λxy+a^11x2-P1(x),dydt=-a12a2x-a1y-λxy-a^11x2+P1(x),
where P1(x) is a smooth function of x at least of the third order and
(63)a1=λS*-(d+ε+μ)-r1+αI*2>0,a2=λI*>0,a^11=rα1+αI*3>0.
Set X=x,   Y=a1x+a2y. Then (62) is transformed into
(64)dXdt=Y+b1X2+b2XY+Q1(X),dYdt=b3X2+b4XY+Q2(X),
where Qi(X) are smooth functions of X at least of the third order and
(65)b1=a^11-λa1a2,b2=λa2,b3=a1a^11-λa12a2+a1λ-a2a^11,b4=λa1a2-λ.
In order to obtain the canonical normal form, we perform the transformation of variables by
(66)u=X-b22X2,v=Y+b1X2.
Then, we obtain
(67)dudt=v+R1(u),dvdt=b3u2+(2b1+b4)uv+R2(u),
where Ri(u) are smooth functions of u at least of the third order. Note that b3<0 and 2b1+b4≠0. It follows from [33–35] that (4) admits a Bogdanov-Takens bifurcation.In the following, we will find the versal unfolding in terms of the original parameters in (4). In this way, we will know the approximate saddle-node, Hopf, and homoclinic bifurcation curves. We choose A and d as bifurcation parameters. Fix λ=λ0, ε=ε0, r=r0, α=α0, and μ=μ0. Let A=A0+θ1 and d=d0+θ2, where θ1 and θ2 are parameters which vary in a small neighborhood of the origin.Suppose thatA=A0, d=d0, λ=λ0, ε=ε0, r=r0, α=α0, and μ=μ0 satisfy (A1) and (A2). Consider the following system:
(68)dIdt=λ0SI-d0+θ2+ε0+μ0I-r0I1+α0I,dSdt=A0+θ1-(d0+θ2)S-λ0SI+ε0I+r0I1+α0I.
By the transformations of x=I-I* and y=S-S*, system (68) becomes
(69)dxdt=-θ2I*+c^1x+c^2y+c11x2+λ0xy-w1(x),dydt=(θ1-θ2S*)+c^3x+c^4y-c11x2-λ0xy+w1(x),
where w1(x) is a smooth function of x at least of the third order and
(70)c^1=λ0S*-(d0+θ2+ε0+μ0)-r01+α0I*2,c^2=λ0I*,c^3=-λ0S*+ε0+r01+α0I*2,c^4=-(d0+θ2)-λ0I*,c11=r0α01+α0I*3.
Making the change of variables X=x, Y=-θ2I*+c^1x+c^2y+c11x2+λ0xy-w1(x) and rewriting X, Y as x and y, respectively, we have
(71)dxdt=y,dydt=e0+e1x+e2y+e11x2+e12xy+e22y2+w2(x,y,θ),
where θ=(θ1,θ2), w2(x,y,θ) is a smooth function of x, y, θ1, and θ2 at least of the third order, and
(72)e0=c^2(θ1-θ2S*)+c^4θ2I*,e1=λ0(θ1-θ2S*)+c^2c^3-c^1c^4-λ0θ2I*,e2=c^1+c^4+λ0θ2I*c^2,e11=λ0c^3-c^4c11-c11c^2+c^1λ0,e12=-λ0+2c11+λ0-c^1c^2-λ0θ2I*c^22,e22=λ0c^2.
Next, introduce a new time variable τ by dt=(1-(λ0/c^2)x)dτ. Rewriting τ as t, we obtain
(73)dxdt=y1-λ0c^2x,dydt=1-λ0c^2x×e0+e1x+e2y+e11x2mmm+e12xy+e22y2+w2(x,y,θ).
Let X=x,  Y=y(1-(λ0/c^2)x) and rename X and Y as x and y; we have
(74)dxdt=y,dydt=e0+g1x+e2y+g11x2+g12xy+w3(x,y,θ),
where θ=(θ1,θ2), w3(x,y,θ) is a smooth function of x, y, θ1, and θ2 at least of the third order, and
(75)g1=-2e0λ0c^2+e1,g11=e11-2e1λ0c^2+e0λ02c^22,g12=e12-e2λ0c^2.
Now, we assume that g11≠0 and g12≠0 when λi are small. Set x=X-e2/g12 and rewrite X as x; we can get
(76)dxdt=y,dydt=f0+f1x+g11x2+g12xy+w4(x,y,θ),
where θ=(θ1,θ2), w4(x,y,θ) is a smooth function of x, y, θ1, and θ2 at least of the third order, and
(77)f0=e0-g1e2g12+g11e22g122,f1=g1-2g11e2g12.
Making the final change of variables by X=g122x/g11, Y=g123y/g112, and τ=g11t/g12 and then denoting them again by x, y, and t, respectively, we obtain
(78)dxdt=y,dydt=τ1(θ1,θ2)+τ2(θ1,θ2)x+x2+xy+w5(x,y,θ),
where θ=(θ1,θ2) and w5(x,y,θ) is a smooth function of x, y, θ1, and θ2 at least of the third order.We substitute values ofα0=1, λ0=1/4, d0=1/10, ε0=1/10, μ0=1/100, A0=537/500, and r0=55/8 for the above system and these values satisfy conditions (A1) and (A2). And we obtain the following equations:
(79)τ1θ1,θ2=f0g124g113=-23729160585184θ1+25485118377851840θ2+Oθ1,θ22,τ2θ1,θ2=f1g122g112=-1722251936θ1+39832533872θ2+Oθ1,θ22,
where g11=-11/500-(1/16)θ1+341/800θ2+(1/4)θ22<0 and g12=-83/200+(1/4)θ2<0 in a small neighborhood of (θ1,θ2)=(0,0). Let
(80)J=∂τ1∂θ1∂τ1∂θ2∂τ2∂θ1∂τ2∂θ2.
And after simple calculation we obtain that
(81)det⁡(J)|λ=0=-1683939874882582458112≠0.
Thus, τ1 and τ2 are regular maps in a small neighborhood of (θ1,θ2)=(0,0). By the Bogdanov and Takens bifurcation theorems [36], we obtain the following conclusion.Theorem 13.
Suppose thatA0, d0, λ0, ε0, r0, α0, and μ0 satisfy (A1), (A2), g11≠0, and g12≠0 when θi are small. Then (4) admits the following bifurcation behavior. (1)
There is a saddle-node bifurcation curveSN={(θ1,θ2):4f0g11=f12+o(|(θ1,θ2)|2), f1≠0}={(θ1,θ2):(5907/6250)θ2-(11/125)θ1+(2313/400)θ1θ2-(5/16)θ12−(1028637/40000)θ22+o(|(θ1,θ2)|2)=0, (47991/16600)θ2-(1/4)θ1+(25/83)θ1θ2-(13645/13778)θ22+o(|(θ1,θ2)|2)≠0}.(2)
There is a Hopf bifurcation curveH={(θ1,θ2):f0+o(|(θ1,θ2)|2)=0,f1<0}={(θ1,θ2):-(537/50)θ2+θ1+(50/83)θ1θ2-(74667/6889)θ22+o(|(θ1,θ2)|2)=0, (47991/16600)θ2-(1/4)θ1+(25/83)θ1θ2-(13645/13778)θ22+o(|(θ1,θ2)|2)<0}.(3)
There is a homoclinic bifurcation curveHL={(θ1,θ2):25g11f0+6f12=o(|(θ1,θ2)|2), f1<0}={(θ1,θ2):(5907/1000)θ2-(11/20)θ1+(611979/33200)θ1θ2-(19/16)θ12−(16075834839/275560000)θ22+o(|(θ1,θ2)|2)=0,(47991/16600)θ2−(1/4)θ1+(25/83)θ1θ2−(13645/13778)θ22+o(|(θ1,θ2)|2)<0}.


## 6. Numerical Simulations

Whenα=1, λ=1/4, d=1/10, ε=1/10, μ=1/100, A=537/500, and r=55/8, σ>0. By applying PPLANE8 and Photoshop software, the (θ1, θ2)-plane near the origin is divided into 4 regions by these bifurcation curves as shown in Figure 6. Fix θ1<0 and decrease θ2 from 0; our conclusions are summarized as follows.(a)
When (θ1, θ2) lies in region I, there is no positive equilibrium, which implies that the positive orbits of (4) meet the positive S-axis in finite time, and therefore the disease disappears.(b)
There is a saddle-node point when (θ1, θ2) lies on curve SN.(c)
When (θ1, θ2) crosses SN into region II, two positive equilibria which are an unstable focus and a saddle appear.(d)
When the parameters lie on the curve H, there is also a saddle and an unstable focus. An unstable limit cycle appears with the parameters crossing H into III.(e)
A homoclinic cycle appears as the parameters passing III into HL through the homoclinic bifurcation.(f)
The homoclinic loop breaks with the parameter crossing HL into IV, and a saddle and a stable focus appear.Figure 6
The bifurcation set and the corresponding phase portraits for system (68).Furthermore, using PPLANE8, some numerical simulations of system (68) are depicted in Figures 7–10. When (θ1,θ2)=(0,0), that is (A0,d0)=(537/500,1/10), there is a unique positive equilibrium (S*,I*)=(317/50,4), which is a cusp of codimension 2 (Figure 7). When (θ1,θ2)=(-0.1,-0.0093) lies in region I, there is no positive equilibrium (Figure 8). When (θ1,θ2)=(-0.02,-0.001863798), there is a homoclinic loop (Figure 9). When (θ1,θ2)=(-0.02,-0.0020) lies in region IV, the homoclinic loop is broken, and there is a stable focus and a saddle (Figure 10).Figure 7
When(θ1,θ2)=(0,0), the unique positive equilibrium is a cusp of codimension 2.Figure 8
When(θ1,θ2)=(-0.1,-0.0093) lies in region I, there is no positive equilibrium.Figure 9
When(θ1,θ2)=(-0.02,-0.001863798), there is a homoclinic loop.Figure 10
When(θ1,θ2)=(-0.02,-0.0020) lies in region IV, there is a stable focus and a saddle.


## 7. Discussion

In this paper, we focus on the bifurcation analysis of an SIS epidemic model with bilinear incidence rate and saturated treatment. Generally speaking, in many epidemic models, the basic reproduction number, which is the key concept in epidemiology, can be decreased below unity to eradicate the disease. However, in our model, the basic reproduction number below 1 is not enough to eradicate the disease. According to our analysis in this paper, we demonstrate not only the global stability of the disease-free equilibrium whenR0<R0* and the local asymptotic stability of the endemic equilibrium E2 when R0*<R0<1, but also the global stability of the unique endemic equilibrium E* when R0>1. Moreover, it has been shown in Corollary 4 that backward bifurcations occur if the effect of the infected being delayed for treatment is strong. That is to say, we should get prompt treatment for patients. Through Figure 4, we can see that there is a region such that the disease will persist if the initial position lies in the region and disappear if the initial position lies outside this region. So, in order to restrain the spread of the disease, governments should take timely measures to control the initial infected individuals in a lower level. In addition, from Theorem 8, we can see that the disease will die out if R0 is small enough. Therefore, from the definition of R0, one knows that we can decrease the incidence rate λ and increase the cure rate r so as to eradicate the diseases or to make them controlled in a lower endemic steady state.The stability analysis of the model equilibria enables us to completely analyze their local bifurcation behavior, such as Hopf, saddle-node, and Bogdanov-Takens bifurcation. By computing the first Lyapunov coefficient, we can determine that the Hopf bifurcation is a supercritical Hopf bifurcation or a subcritical Hopf bifurcation. We also show that, under assumptions(A1) and (A2), the model undertakes Bogdanov-Takens bifurcation; that is, there are saddle-node bifurcation, Hopf bifurcation, and homoclinic bifurcation in the system. The normal norm of the Bogdanov-Takens bifurcation is derived in Section 5 which is very helpful to obtain the three kinds of bifurcation curves. By analytical techniques, A and d are chosen as bifurcation parameters and other parameters are fixed, and we easily get a clear picture about the rich dynamics behaviors of our model. Through studying the bifurcations of the SIS epidemic model, we are suggesting that, in order to eradicate the disease, more medical facilities as well as medical professionals are needed and the medical standard needs to be improved as well.



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*Source: 745732-2015-01-19.xml*