BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL WITH A NONLINEAR INCIDENCE

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 7, Number 1, Pages c 2011 Institute for Scientific Computing and Information BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL WITH A NONLINEAR INCIDENCE JUNFENG WANG AND YAKUI XUE Abstract. In this paper, a stage-structured epidemic model with a nonlinear incidence is investigated. By carrying out global qualitative and bifurcation analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation and the Bogdanov-Takens bifurcation. Moreover, several numerical simulations are given to support the theoretical analysis. Key Words. Nonlinear incidence, Bifurcation, Limit cycle. 1. Introduction In recent years, more and more researchers have taken into account oscillations in incidence rates and proposed many nonlinear incidence rates. With these nonlinear incidence rates, many interesting and complicated transmission namics of epidemics have been shown, such as multiple equilibria, periodic orbits, Hopf and Bogdanov-Takens bifurcations, which state clearer and more reasonable qualitative description of the disease namics and give better suggestions for the prediction or control of diseases. Liu et al. studied the codimension-1 bifurcation for SEIRS and SIRS models with the incidence rate βs p I q in [1]. Lizana and Rivero [2], Glendinning and Perry [3] and Derrick and van den Driessche [4] studied saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation of SIRS or SIR models with the incidence rate of βs p I q. Ruan and Wang [5] studied saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation and the existence of none, one and two limit cycles of an SIRS model with an incidence rate of ki 2 S/(1 + ai 2 ), which was also proposed by Liu et al. [1]. van den Driessche and Watmough [6] studied an incidence rate of the form βi(1 + vi k 1 )S, where β > 0,v 0, and k > 0.The incidence rate ki 2 S/(1+βI +ai 2 ) was proposed by Wang in [7], which was motivated by the work of Zhu et al. in [8] on a population model, and the stable disease-free equilibrium and endemic equilibrium were obtained. The hepatitis C virus (HCV) was identified in the year HCV is a singlestranded RNA virus with properties similar to those of flavivirus. The virus is mainly transmitted through transfusion of contaminated blood or blood products. Mathematical modelling has been proved to be valuable in understanding the namics of HCV infection and many excellent results have been obtained in [9,10]. Deuffic et al. [9] studied a mathematical model of hepatitis C, which is structured by age and sex. In [10], Neumann et al. analyzed a mathematical model in order to understand the namics of HCV and the antiviral effect of interferon-a; Received by the editors June 20, 2010 and, in revised form, August 22, Mathematics Subject Classification. 35R35, 49J40, 60G40. The work is supported by the National Sciences Foundation of China( ) and the National Sciences Foundation of Shanxi Province( ). 61

2 62 J. WANG AND Y. XUE Martcheva and Castillo-Chavez[11] considered a hepatitis C model with chronic infectious stage in varying population. We divide the population in three subclasses: susceptible (S(t)); infected with acute hepatitis C (stage-1 infections I(t) ); infected with chronic hepatitis C with or without cirrhosis (stage-2 infections J(t)); and the total number of people N(t) = S(t) +I(t)+J(t). And the following assumptions are used in the construction of the models. (1) Only the acute and chronic stages are differentiated. Patients with either acute or chronic infections are capable of transmitting the disease. (2) All infected individuals develop an acute hepatitis C first. Some individuals with an acute infection progress towards the chronic state and later on develop cirrhosis. (3) Since the disease-induced death rate is relatively low, it is ignored. The acute stage of infection is short and often asymptomatic and there is no possibility for treatment during this state. Liming Cai, Xuezhi Li[12] studied global stability of an SEI epidemic model with withacuteandchronicstages; In[13]L-M.C,MiniGhoshanalyzedamathematical model with nonlinear incidence rates of the form (β 1 I +β 2 J)S p,p > 0. In this paper, we will consider the nonlinear incidence rates of the form βs p I q and p = 1, q = 2. Thus, a stage-structured epidemic model with a nonlinear incidence can be written as: ds dt = B ds (β 1I 2 +β 2 J 2 )S +δj, di dt = (β 1I 2 +β 2 J 2 )S (d+γ)i, (1.1) dj dt = γi (d+δ)j, where B is the recruitment rate of the population, d is the death rate of the population, γ is the rate of progression to stage-2 from stage-1, δ is the recovery (treatment) rate from the individuals in stage-2 infection state; β i (i = 1,2) represent effective contact rates of infective individuals in stage-1 and stage-2 infection state respectively.all parameters are assumed to be positive. Before going into details,let us simplify this model. Summing up three equations in (1.1) and denoting the number of total population by N(t),we obtain dn dt = B dn. Since N(t) tends to a constant as t tends to infinity, we assume that the population is in equilibrium and investigate the behaviorofthe system on the plane S+I+J = N 0 > 0. Thus,we consider the reduced system di dt = (β 1I 2 +β 2 J 2 )(N 0 I J) (d+γ)i, dj dt = γi (d+δ)j, (1.2) β Rescaling(1.2)byX = 1 d+δ I, Y = β 1 d+δj, τ = (d+δ)t andstillwriting(x,y,τ) as (I,J,t). Then we obtain di dt = (I2 +mj 2 )(A I J) pi, dj dt = qi J, (1.3) where m = β 2 β 1, A = N 0 β1 d+δ, p = d+γ d+δ, q = γ d+δ.

3 BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL 63 The organization of this paper is as following. In the next section, we will qualitatively analyze the existence and stability of the disease-free equilibrium and endemic equilibria for this epidemic model. In Section 3, the periodic solutions will be discussed through the Hopf bifurcation theory, and the stability of the limit cycle can be obtained by computing the first Liapunov number. The BT bifurcation at the degenerate equilibrium will be shown in Section 4. The paper will finish with a brief discussion in Section Qualitative analysis In this section, we first consider the stability of the disease-free equilibrium. It is seen that system (1.3) has a unique disease equilibrium E 0 (0,0), and the Jacobian matrix of [(1.3) at (0,] 0) is p 0 M 0 =. q 1 It is easy to obtain the following result: Theorem 2.1 The disease free equilibrium E 0 (0,0) of system (1.3) is a stable node whatever the parameter values are. To find the positive equilibria, set which yields (I 2 +mj 2 )(A I J) pi = 0 qi J = 0. (1+mq 2 )()I 2 (1+mq 2 )AI +p = 0. (2.1) Letting = A 2 4p() 1+mq 2 and denote A 2 0 4p() 1+mq 2, then we can see that (i) there is no positive equilibrium if A 2 < A 2 0 ; (ii) there is one positive equilibrium if A 2 = A 2 0, where I = A 2(), J = qi. (iii) there are two positive equilibria if A 2 > A 2 0, where I 1 = A 2(), J 1 = qi 1 ; I 2 = A+ 2(), J 2 = qi 2. We first determine the stability of (I 1,J 1 ). The Jacobian matrix at (I 1,J 1 ) is [ 2I1 (A I M 1 = 1 J 1 ) (I1 2 +mj1) p 2 2mJ 1 (A I 1 J 1 ) (I1 2 +mj1) 2 q 1 After some algebra we can see that det(m 1 ) = (1+mq 2 )AI 1 2p < 0. Thus,the equilibrium (I 1,J 1 ) is a saddle point. ].

4 64 J. WANG AND Y. XUE Next we analyze the stability of the second positive equilibrium (I 2,J 2 ). The Jacobian matrix at (I 2,J 2 ) is [ 2I2 (A I M 2 = 2 J 2 ) (I2 2 +mj2 2 ) p 2mJ 2(A I 2 J 2 ) (I2 2 +mj2 2 ) ]. q 1 By a similar argument as above, we obtain that det(m 2 ) > 0. Thus, (I 2,J 2 ) is a node, or a focus, or a center. Note that the sign of tr(m 2 ) is determined by φ = [(3+2q +mq 2 )I 2 2 2AI 2 +(p+1)]. (2.2) Denote ξ = (1 + mq 2 )(1 + q)i 2 2 (1 + mq2 )AI 2 + p, then we can express φ as φ = (P 0 ξ +P 1 r 1 ), where P 0,P 1 is positive constant, and Clearly, r 1 = 0 is equivalent to it implies that And we have r 1 = (1+mq 2 )AI +pq + 2p() 1+mq 2. I = 2p(1+p) (1+mq2 )(pq +) (1+mq 2 ) 2, (2.3) A 2p(1+p) (1+mq 2 )(pq +) > 0. (2.4) 2p(1+p) (1+mq 2 )(pq +) (1+mq 2 ) 2 A = A+ A 2 4p() 1+mq 2, (2.5) 2() it is equivalent to 1 4p() (1+mq 2 )A 2 = 4p()2 2()(pq +)(1+mq 2 ) (1+mq 2 ) 2 A 2 1. (2.6) It may be proved that the right-hand side of (2.6) is more than zero,which implies A 2 < 2()[2p(1+p) (1+mq2 )(pq +)] (1+mq 2 ) 2 A 2 1. (2.7) Taking squares on both sides of (2.7) and by some algebra we have A 2 = it implies that ()[2p(1+p) (1+mq 2 )(pq +)] 2 (1+mq 2 ) 2 [2p(1+p) (1+mq 2 )(pq + +p)] A2 c. (2.8) 2p(1+p) (1+mq 2 )(pq + +p) > 0. (2.9) By compared, A 2 0 < A2 c < A 2 1 must satisfy 2p(1+p) (1+mq 2 )(pq + +2p) > 0. (2.10) It follows that (2.4) and (2.9) is valid if and only if (2.10). At the same time, we can see that r 1 + if A. Hence, tr(m 2 ) < 0 if both A 2 > A 2 c and (2.10) hold. Summarizing the above discussion, we can have the following results: Theorem 2.2 Suppose A 2 > A 2 0, i.e., there are two endemic equilibria E 1 (I 1,J 1 ) and E 2 (I 2,J 2 ). Then the equilibrium E 1 is a saddle point, and (i) (I 2,J 2 ) is stable if either of the following inequalities holds: A 2 > A 2 c,

5 BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL 65 2p(1+p) (1+mq 2 )(pq + +2p) > 0. (2.11) (ii) (I 2,J 2 ) is unstable if A 2 < A 2 c, 2p(1+p) (1+mq 2 )(pq + +2p) > 0. (2.12) For example, assuming A = 9.60,n = 0.3,m = 0.02,p = 5, and q = 5.9, one has the parameters are satisfied with condition (i) of Theorem 2.2,then from above analysis, bistable equilibria E 0 (0,0) (stable node), E 2 (1.8056,6.3195)(stable focus) and unstable saddle point E 1 (0.4166,1.4583) are obtained in system (1.3), which is shown in Fig R I Fig. 1. Bistable equilibria. A = 9.60, m = 0.02, p = 5, q = 4.2. Equilibria of (1.3) are: (0,0) (stable node), (I 1,J 1 ) = (0.4166,1.4583) (saddle) and (I 2,J 2 ) = (1.8056, ) (stable focus). 3. Hopf bifurcation In this section, we stu the Hopf bifurcation of (1.3). Since R axis is invariant and (I 1,J 1 ) must be a saddle, there is no closed orbit surrounding (0,0) or (I 1,J 1 ). ThusHopfbifurcationsonly emergeat(i 2,J 2 ).For(1.3), makex = I I 2,y = J J 2 to translate (I 2,J 2 ) to the origin. Then (1.3) becomes dt = a 11x+a 12 y +f(x,y), dt = a 21x+a 22 y, where a 11 = 2I 2 (A I 2 J 2 ) (I 2 2 +mj2 2 ) p, a 12 = 2mJ 2 (A I 2 J 2 ) (I 2 2 +mj 2 2), (3.1)

6 66 J. WANG AND Y. XUE a 21 = q, a 22 = 1, f(x,y) = x 2 (A 3I 2 J 2 ) 2xy(I 2 + mj 2 ) + my 2 (A I 2 3J 2 ) (x 3 + x 2 y + mxy 2 +my 3 ). To obtain the Hopf bifurcation, we fix parameters such that tr(m 2 ) = 0, which is equivalent to a 11 = 1. Let X = x, Y = x+a 12 y. Then (3.1) is reduced to dx dt = Y +f 1(X, Y X a 12 ), dt = kx+f 1(X, Y X a 12 ), (3.2) where k = det(m 2 ) = (a 12 ). Let u = X, v = Y/ k. We obtain the normal form for the Hopf bifurcation du dt = kv +F 1 (u,v), dv dt = kv +F 2 (u,v), (3.3) where F 1 (u,v) = f 1 ( u,(v k +u)/a 12 ), F 2 (u,v) = F 1 (u,v)/ k. Set the Liapunov number by σ = 1 16 [ 3 F 1 u F 1 u v F 2 u 2 v + 3 F 2 v 3 ] [ 2 F 1 u v ( 2 F 1 u F 1 v 2 ) 2 F 2 u v ( 2 F 2 u F 2 v 2 ) 2 F 1 u 2 2 F 2 u F 1 v 2 2 F 2 v 2 ] We have the following Hopf bifurcation results: Theorem 3.1 For (1.3),assume tr(m 2 ) = 0,i.e.,either of the following inequalities holds: A 2 = A 2 c, 2p(1+p) (1+mq 2 )(pq + +2p) > 0. If σ < 0, then there is a family of stable periodic orbits in (1.3); If σ > 0, there is a family of unstable periodic orbits in (1.3); If σ = 0, there are at least two limit cycles in (1.3) under suitable perturbations. For example, fixed the parameters as A = ,p = 4,q = 3.2,m = 0.01, the conditions of Theorem 3.1 hold. So we can determine σ = , and then there is an stable periodic orbit, which is shown in Fig Bogdanov-Takens bifurcations The purpose of this section is to stu the Bogdanov-Takens bifurcations of (1.3) when there is a unique degenerate positive equilibrium. Assume that (H1) A 2 = A 2 0. Then (1.3) admits a unique positive equilibrium (I,J ); where I = A 2(), J = qi.

7 BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL R I Fig. 2. A stable equilibria and a stable limit cycle. A = ,p = 4,q = 3.2,m = 0.01.Equilibriaof (1.3) are: (0,0) (stable node), (I 1,J 1 ) = (0.6038,1.9321)(saddle) and (I 2,J 2 ) = (1.4308,4.5786) (unstable focus). Let x = I I,y = R R to translate (I,J ) to the origin. Then (1.3) becomes where dt = b 11x+b 12 y +g 1 (x,y), dt = b 21x+b 22 y, b 11 = 2I (A I J ) (I 2 +mj2 ) p, b 12 = 2mJ (A I J ) (I 2 +mj 2 ), b 21 = q, b 22 = 1, (4.1) g(x,y) = x 2 (A 3I J ) 2xy(I + mj ) + my 2 (A I 3J ) (x 3 + x 2 y + mxy 2 +my 3 ). Then by (H1) and I = A 2(),J = qi,the linearized matrix of (1.3) at (I,J ) is [ ] b11 b M = 12. q 1 and det(m ) = b 11 b 12 q = 0. Since we are interested in codimension 2 bifurcations, we assume further (H2) b 11 = 1.

8 68 J. WANG AND Y. XUE Then b 12 = 1/q. By (H1) and (H2), (4.1) becomes dt = x (1/q)y+c 21x 2 +c 22 xy +P(x,y), dt = qx y, (4.2) where P is a smooth function in (x,y) at least of order three and c 12 = A 3I J = A 2 A, c 22 = 2(I +mj ) = (1+mq)A ) < 0. Set X = x, Y = x y/q. Then (4.2) is transformed into dx dt = Y +(c 21 +qc 22 )X 2 c 22 qxy +P 1 (X,Y), dy dt = (c 21 +qc 22 )X 2 c 22 qxy +P 1 (X,Y), (4.3) where P 1 is a smooth function in (X,Y) at least of order three. In order to obtain the canonical normal forms, we perform the transformation of variables by Then, we obtain u = X +c 22 q/2, v = Y +(c 21 +qc 22 )X 2. du dt = v +R 1(u,v), dv dt = (c 21 +qc 22 )u 2 +(2c 21 +qc 22 )uv +R 2 (u,v), where R i are smooth functions in (u,v) at least of the third order. Then we have c 21 +qc 22 = A 2 A (1+mq)qA < 0, 2c 21 +qc 22 = A 2A (1+mq)qA < 0. This implies that (I,J ) is a cusp of codimension 2. (4.4) Theorem 4.1 Suppose that (H1) and (H2) hold. Then the equilibrium (I,J ) of (1.3) is a cusp of codimension 2, i.e., it is a Bogdanov-Takens singularity. In the following, we will find the versal unfolding depending on the original parameters in(1.3). In this way, we will know the approximating bifurcation curves. We choose A and p as bifurcation parameters. Suppose A 0,m,p 0,q satisfy (H1) and (H2). Let A = A 0 +λ 1, p = p 0 +λ 2, I = A 0 2(), J = qi. Clearly if λ 1 = 0,λ 2 = 0, (I,J ) is a degenerate equilibrium of (1.3). Substituting x = I I,y = J J into (1.3) and using the Taylor expansion, we obtain that dt = a 0 +a 1 x+a 2 y +a 3 x 2 +a 4 xy +W 1 (x,y,λ), dt = qx y, (4.5) where W 1 is a smooth function of x,y,and λ = (λ 1,λ 2 ) at least of order three in x and y,and

9 BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL 69 a 0 = (1+mq 2 )I 2 (A 0 +λ 1 ()I ) (p 0 +λ 2 )I, a 1 = 2I (A 0 +λ 1 ()I ) (1+mq 2 )I 2 (p 0 +λ 2 ), a 2 = 1 q, a 3 = 2I +A 0 +λ 1 ()I, a 4 = 2(1+mq 2 )I. Let X = x,y = a 0 +a 1 x+a 2 y +a 3 x 2 +a 4 xy +W 1 (x,y,λ). Then (4.5) becomes dx dt = Y, dy dt = a 0 +(a 2 q +a 1 )X +a 5 Y +(a 3 +a 4 q)x 2 +a 6 XY + a4 a 2 Y 2 +W 2 (X,Y,λ), where a 5 = a 1 1 a0a4 a 2, a 6 = 2a 3 + a2 4 a0 a1a4 a 2 2 a 2. (4.6) If λ i 0, it is easyto see that a 6 A0(1+mq2 ) < 0. By setting x = X+ a5 a 6,y = Y. (4.6) becomes dt = y, dt = h 0 +h 1 x+h 2 x 2 +h 3 xy +h 4 y 2 +W 3 (x,y,λ), (4.7) where W 3 (x,y,λ) is a smooth function of x,y and λ at least of order three and h 0 = a 0 (a 1 1)a 5 /a 6 +(a 3 +a 4 q)a 2 5/a 2 6, h 1 = a 1 1 2(a 3 +a 4 q)a 5 /a 6, h 2 = a 3 +a 4 q = A0 + A0 2 +λ 1 (1+mq)A0q, h 3 = a 6 = 2a 3 +q 2 a 2 4 a 0 +qa 1 a 4, h 4 = a 4 /a 2 = qa 4 = (1+mq 2 )qa 0 /() > 0. It is not difficult to obtain that when λ 1 0,λ 2 0, have h 0 0, h 1 0, h 2 A0(+2mq2 ) 2() < 0, h 3 A0(1+mq2 ) < 0, h 4 (1+mq 2 )qa 0 /() > 0. Now, introduce the new time τ by dt = (1 h 4 x)dτ, and rewrite τ as t, we obtain that dt = y(1 h 4x), dt = (1 h 4x)(h 0 +h 1 x+h 2 x 2 +h 3 xy +h 4 y 2 )+W 4 (x,y,λ), (4.8) where W 4 (x,y,λ) is a smooth function of x,y and λ at least of order three. Let X = x,y = y(1 h 4 x), still write (X,Y) as (x,y).then we have dt = y, dt = c 1 +c 2 x+c 3 x 2 +c 4 xy +W 5 (x,y,λ), (4.9)

10 70 J. WANG AND Y. XUE where c 1 = h 0,c 2 = h 1 2h 0 h 4,c 3 = h 0 h 2 4 2h 1h 4 +h 2,c 4 = h 3. and W 5 (x,y,λ) is a smooth function of x,y and λ at least of order three. Then when λ 1 0,λ 2 0, have c 1 0, c 2 0, c 3 A0(+2mq2 ) 2() < 0, h 3 A0(1+mq2 ) < 0. Make the final change of variables by X = c 2 4x/c 2,Y = c 3 4y/c 2 3,τ = c 3 t/c 4. Write (X,Y,τ) as (x,y,t),then (4.9) becomes dt = y, dt = τ 1 +τ 2 x+x 2 +xy +W 6 (x,y,λ), where W 6 (x,y,λ) is a smooth function of x,y and λ at least of order three and τ 1 = c 1c 4 4 c 3 3, τ 2 = c 2c 2 4 c 2. 3 (4.10) By the theorems in Bogdanov[14] and Kuznetsov [15], we obtain the following local representations of the bifurcation curves in a small neighborhood of the origin. Theorem 4.2 Let (H1) and (H2) hold. Then (1.3) admits the following bifurcation behavior: (i) there is a saddle-node bifurcation curve SN = {λ 1,λ 2 ) : 4c 1 c 3 = c 2 2 +o( λ )2 }; (ii) there is a Hopf bifurcation curve H = {λ 1,λ 2 ) : c 1 = 0+o( λ ) 2,c 2 < 0}; (iii) there is a homoclinic bifurcation curve HL = {λ 1,λ 2 ) : 25c 1 c 3 +6c 2 2 = 0+o( λ ) 2,c 2 < 0}. By using numerical simulations, an interesting phenomenon is that qualitative structures of (1.3) is sensitive to parameter m for some combination of parameters, which is shown in the following example.let us fix A = ,p = 4,q = 3.2,m = in system (1.3), then there exist two limit cycles, which is shown in Fig. 3. The general pattern to produce two limit cycles is illustrated in the following.when fix A = ,p = 4,q = 3.2 and take m as a bifurcation parameter in system (1.3), an unstable limit cycle bifurcates from a homoclinic orbit at m = and contracts as m decreases. A stable limit cycle emerges from the Hopf bifurcation of an endemic equilibrium m = 0.01 and enlarges as m decreases. There exist two limit cycles for < m and two limit cycles collide at m = (see Fig. 4). 5. Discussion In this paper, we considered a hepatitis C model with chronic infectious stage in varying population.we divide the population in three subclasses: susceptible S(t); infected with acute hepatitis C (stage-1 infections I(t)); infected with chronic hepatitis C with or without cirrhosis (stage-2 infections J(t)); and consider the nonlinear incidence rates of the form βs p I q and p = 1,q = 2. The existence and stability of the disease-free equilibrium and endemic equilibria are investigated. There exists a region such that the disease becomes extinct in a finite time when the initial position lies outside the region and persist if the initial position lies in this region. The various types of Hopf bifurcation, i.e. the subcritical and supcritical Hopf bifurcation are determined by computing the first Lyapunov coefficient. Codimension-two Bogdanov-Takens bifurcation, i.e. the saddle-node bifurcation, Hopf bifurcations

11 BIFURCATION ANALYSIS OF A STAGE-STRUCTURED EPIDEMIC MODEL J I Fig. 3. Fix the parameter A = ,p = 4,q = 3.2,m = , there exist two limit cycles in (1.3) Period m x 10 3 Fig. 4. Bifurcation curve of period of cycles versus m where A = ,p = 4,q = 3.2. and homoclinic bifurcation has investigated in this epidemic model. By carrying out this global analysis, the rich namical behavior of system (1.3) has been shown near the degenerate equilibrium.

12 72 J. WANG AND Y. XUE References [1] Liu WM, Hethcote HW, Levin SA. Dynamical behavior of epidemiological models with nonlinear incidence rates. J Math Biol, 1987 ; 25: [2] Lizana M, Rivero H. Multiparameteric bifurcations for a model in epidemiology. J Math Biol, 1996 ; 35: [3] Glendinning P, Perry LP. Melnikov analysis of chaos in a simple epidemiological model. J Math Biol, 1997 ; 35: [4] Derrick WR, van den Driessche P. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discret Contin Dynam Systems Ser B, 2003; 3: [5] Ruan S, Wang W. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J Math BiolJ Differn Equat 2003;188:135C63. [6] van den Driessche P, Watmough J. A simple SIS epidemic model with a backward bifurcation. J Math Biol, 2000; 40: [7] W.D. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 2006 ; 3: [8] H. Zhu, S.A. Campbell, G.S.K. Wolkowicz, Bifurcation analysis of a predatorcprey system with nonmonotonic functional response, SIAM J. Appl. Math., 2002 ; 63: [9] S. Deuffic, L. Buffat, et al., Modelling hepatitis C virus epidemic in France, Hepatology, 1999; 29(5): [10] A. Neumann, N. Lam, H. Dahari, D. Gretch, T. Wiley, T. Layden, A. Perelson, Hepatitis C viral namics in vivo and the antiviral efficacy of Interferon-a therapy, Science, 1998; 282: [11] M. Martcheva, C. Castillo-Chavez, Disease with chronic stage in a population with varying size, Math. Biosci., 2003; 182: [12] Liming Cai, Xuezhi Li,A note on global stability of an SEI epidemic model with acute and chronic stages, Applied Mathematics and Computation, 2008 ; 196: [13] Li-Ming Cai, Xue-Zhi Li, Mini Ghosh,Global stability of a stage-structured epidemic model with a nonlinear incidence, Applied Mathematics and Computation, 2009; 214: [14] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Math. Soviet., 1981; 1: [15] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, Department of Mathematics, North University of China, Taiyuan, Shanxi, , China wangjunfeng44413@163.com and xyk5152@163.com, ykxue@nuc.edu.cn