AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 647853 10.1155/2012/647853 647853 Research Article Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment Xue Yakui Wang Junfeng Zizovic Malisa R. Department of Mathematics North University of China Shanxi Taiyuan 030051 China nuc.edu.cn 2012 19 7 2012 2012 18 01 2012 27 05 2012 2012 Copyright © 2012 Yakui Xue and Junfeng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An epidemic model with infectious force in infected and immune period and treatment rate of infectious individuals is proposed to understand the effect of the capacity for treatment of infective on the disease spread. It is assumed that treatment rate is proportional to the number of infective below the capacity and is constant when the number of infective is greater than the capacity. It is proved that the existence and stability of equilibria for the model is not only related to the basic reproduction number but also the capacity for treatment of infective. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low.

1. Introduction

Recently, mathematical models describing the dynamics of human infectious diseases have played an important role in the disease control in epidemiology. Researchers have proposed many epidemic models to understand the mechanism of disease transmission. We assume that a susceptible individual first goes through a latent period after infection before becoming infectious. The resulting models are of SEI, SEIR, or SEIRS type, respectively. Zhang and Ma  studied the global stability of an SEI model with general contact rate. Yuan et al.  considered the local stability of the model having infectious force in both latent period and infected period. Li and Jin  studied the global stability of the epidemic model having infectious force in both latent period and infected period. Usually, these classical epidemic models have only one endemic equilibrium when the basic reproduction number R0>1, and the disease-free equilibrium is always stable when R0<1 and unstable when R0>1. So the bifurcation leading from a disease-free equilibrium to an endemic equilibrium is forward. But in recent years, the phenomenon of the backward bifurcations has arisen the interests in disease control (see ). In this case, the basic reproduction number cannot describe the necessary disease elimination effort any more. Thus, it is important to identify backward bifurcations and establish thresholds for the control of diseases.

In classical epidemic models, the treatment rate of the infective is assumed to be proportional to the number of the infective. Because the resources of treatment should be limited, every community should have a suitable capacity for treatment. This hypothesis is satisfactory when the number of the infective is small and the resources of treatment are enough and unsatisfactory when the number of the infective is large and the resources of treatment are limited. Thus, it is important to determine a suitable capacity for the treatment of a disease. A constant treatment rate of disease is adopted in . Note that a constant treatment rate is suitable when the number of infective is large. In , the treatment rate of the disease is modified into (1.1)T(I)={rIif0II0,kifI>I0, where k=rI0, r and I0 are positive constant. This means that the treatment rate of disease is proportional to the number of the infective when the capacity of treatment is not reached and, otherwise, takes the maximal capacity. This improves the classical proportional treatment and the constant treatment in .

In this paper, we study the backward bifurcation and global dynamics of an epidemic model with infectious force in infected and immune period and treatment function. To formulate our model, we will consider a population that is divided into three types: susceptible, infective, and recovered. Let S(t), I(t), and R(t) denote the numbers of susceptible, infective, recovered individuals at time t, respectively. The total population size at time t is denoted by N(t).

The basic assumptions in the paper are as the follows.

There is a positive constant recruitment rate of the population A.

Positive constant d is the nature death rate of population.

β1, β2 are the rate of the efficient contact in the infected and recovered period, respectively.

Positive constant γ is the natural recovery rate of infective individuals.

Positive constant ϵ is the disease-related death rate.

The treatment of a disease is T(I) in (1.1).

Under the assumptions above, an epidemic model to be studied takes the following form: (1.2)dSdt=A-dS-β1SI-β2SR,dIdt=β1SI+β2SR-(d+γ+ϵ)I-T(I),dRdt=γI+T(I)-dR, where S(t)+I(t)+R(t)=N(t). It is easy to verify that R+3 is positive invariant for system (1.2).

According to S(t)+I(t)+R(t)=N(t) and (1.1), N(t) satisfies the following equation: (1.3)dNdt=A-dN-ϵI. Then system (1.2) is equivalent to (1.4)dNdt=A-dN-ϵI,dIdt=(β1I+β2R)(N-I-R)-(d+γ+ϵ)I-T(I),dRdt=γI+T(I)-dR.

It is easy to verify that all solutions of system (1.4) initiating in set {(N,I,R)N>0,I0,R0,I+RN} eventually enter the set Ω={(N,I,R)0<NA/d,I0,R0,I+RN}. Therefore, Ω is positively invariant for system (1.4). We consider the solutions of system (1.4) in Ω below.

When 0II0, system (1.4) becomes (1.5)dNdt=A-dN-ϵI,dIdt=(β1I+β2R)(N-I-R)-(d+γ+ϵ+r)I,dRdt=(γ+r)I-dR.

When I>I0, system (1.4) becomes (1.6)dNdt=A-dN-ϵI,dIdt=(β1I+β2R)(N-I-R)-(d+γ+ϵ)I-k,dRdt=γI+k-dR.

The purpose of this paper is to show that system (1.4) has a backward bifurcation if the capacity for treatment is small. We obtain the sufficient conditions that the disease-free equilibrium and endemic equilibria of system (1.4) are stable. It is shown that (1.4) has bistable endemic equilibria if the capacity is small. The organization of this paper is as follows. In next section, we study the existence and bifurcations of equilibria for (1.4). We analyze the stability of equilibria for (1.4) and present the numerical simulations in Section 3.

2. The Existence of Equilibria

In this section, we consider the equilibria of system (1.4). Obviously, E0(A/d,0,0) is the disease-free equilibrium of (1.4). For the endemic equilibrium E(N,I,R) of (1.4), N, I and R satisfy (2.1)A-dN-ϵI=0,(β1I+β2R)(N-I-R)-(d+γ+ϵ)I-T(I)=0,γI+T(I)-dR=0.

When 0II0, system (2.1) becomes (2.2)A-dN-ϵI=0,(β1I+β2R)(N-I-R)-(d+γ+ϵ+r)I=0,(γ+r)I-dR=0.

When I>I0, system (2.1) becomes (2.3)A-dN-ϵI=0,(β1I+β2R)(N-I-R)-(d+γ+ϵ)I-k=0,γI+k-dR=0.

Form (2.2), I satisfies the following equation: (2.4)(β1+β2γ+rd)A-(d+ϵ+γ+r)Id=d+ϵ+γ+r. Therefore, we obtain (2.5)I=A-d(d+ϵ+γ+r)/(β1+β2((γ+r)/d))d+ϵ+γ+r.

Let (2.6)R0=A(β1+β2((γ+r)/d))d(d+ϵ+γ+r). Then R0 is a basic reproduction number of (1.4). If R0>1, then I>0; (2.2) admits a unique positive solution E*=(N*,I*,R*), where (2.7)N*=A-ϵI*d,I*=A-d(d+ϵ+γ+r)/(β1+β2((γ+r)/d))d+ϵ+γ+r,R*=(γ+r)I*d. Clearly, E* is an endemic equilibrium of (1.4) if and only if (2.8)1<R01+β1+β2((γ+r)/d)dI0.

According to (2.3), I satisfies the following equation: (2.9)a0I2+a1I+a2=0, where a0=(β1+β2(γ/d))(d+ϵ+γ), (2.10)a1=d(d+ϵ+γ)+β2kd(d+ϵ+γ)-(β1+β2γd)(A-k),a2=dk-β2kd(A-k).

We only consider the case of a2>0. If a10, it is clear that (2.9) does not have positive real root. Let us suppose a1<0 below. Note that a1<0 is equivalent to (2.11)R01+β2(k/d)(d+ϵ+γ)+β2(r/d)A+(β1+β2(γ/d))k-drd(d+ϵ+γ+r)=:p*. It is easy that (2.12)Δ=a12-4a0a2=R02d2(d+ϵ+γ+r)2-2R0d(d+ϵ+γ+r)[(d+β2kd)(d+ϵ+γ)+β2rdA+(β1+β2γd)k]+[(d+β2kd)(d+ϵ+γ)+β2rdA+(β1+β2γd)k]2-4(β1+β2γd)(d+ϵ+γ)(dk-β2kd(A-k)). It follows that Δ0 is equivalent to (2.13)R0p*+2(β1+β2(γ/d))(d+ϵ+γ)[dk-β2(k/d)(A-k)]d(d+ϵ+γ+r)=:p0, or (2.14)R0p*-2(β1+β2(γ/d))(d+ϵ+γ)[dk-β2(k/d)(A-k)]d(d+ϵ+γ+r). Thus a1<0 and Δ0 if and only if (2.13) holds. Let us suppose that (2.13) holds. Then (2.9) has two positive solutions I1 and I2 where (2.15)I1=-a1-Δ2a0,I2=-a1+Δ2a0. Set Ni=(A-ϵIi)/d, Ri=(γIi+k)/d and Ei(Ni,Ii,Ri)  (i=1,2). If Ii>I0  (i=1,2), then Ei is an endemic equilibrium of (1.6).

By the definition of I1, we notice that I1>I0 is equivalent to (2.16)-Δ>2a0I0+a1. This implies that 2a0I0+a1<0. By immediate calculation, 2a0I0+a1<0 is equivalent to (2.17)R0>p*+2(β1+β2(γ/d))(d+ϵ+γ)I0d(d+ϵ+γ+r)=:p1. Further, I1>I0 demands that (2.18)(2a0I0+a1)2>Δ. By immediate calculation, (2.19)R0<1+β1+β2((γ+r)/d)dI0=:p2. Therefore, I1>I0 holds if and only if R0>p1 and R0<p2.

By similar discussions as previously mentioned, we have that I2>I0 holds if and only if either R0>p1, or R0>p2, R0<p1.

Summarizing the discussions above, we have the following conclusion.

Theorem 2.1.

E 0 ( A / d , 0,0 ) is always the disease-free equilibrium of (1.5). E*(N*,I*,R*) is an endemic equilibrium of system (1.4) if and only if 1<R0p2. Furthermore, E* is the unique equilibrium of system (1.4) if 1<R0p2, and one of the following conditions is satisfied:

R0<p0,

p0R0<p1.

By calculation, we have p2-p1=[d-β2(A-k)/d]r-(β1+β2((γ+r)/d))I0(d+ϵ+γ). Note that [d-β2(A-k)/d]r>(β1+β2(γ/d))(d+ϵ+γ)I0 is equivalent to that p1<p2.

Theorem 2.2.

Endemic equilibria E1 and E2 do not exist if R0<p0. Further, if R0p0, we have the following:

if [d-β2(A-k)/d]r>(β1+β2(γ/d))(d+ϵ+γ)I0, then both E1 and E2 exist when p1<R0<p2,

if [d-β2(A-k)/d]r>(β1+β2(γ/d))(d+ϵ+γ)I0, then E1 does not exist but E2 exists if R0p2,

letting [d-β2(A-k)/d]r(β1+β2(γ/d))(d+ϵ+γ)I0, then E1 does not exist. Further, E2 exists when R0>p2, and E2 does not exist when R0p2.

We consider p0>1. If [d-β2(A-k)/d]r>(β1+β2(γ/d))(d+ϵ+γ)I0, a typical bifurcation diagram is illustrated in Figure 1, where the bifurcation from the disease-free equilibrium at R0=1 is forward and there is a backward bifurcation from an endemic equilibrium at R0=1.71, which gives rise to the existence of multiple endemic equilibria. Further, if [d-β2(A-k)/d]r(β1+β2(γ/d))(d+ϵ+γ)I0, a typical bifurcation diagram is illustrated in Figure 2, where the bifurcation at R0=1 is forward, and (1.4) has one unique endemic equilibrium for all R0>1.

The figure of infective sizes at equilibria versus R0 when I0=30, A=80, β1=0.01, β2=0.01, γ=0.01, d=0.9, ϵ=0.01, r=1, p0=1.6782, p1=1.541, p2=1.7074, where (i) of Theorem 2.2 holds. The bifurcation from the disease-free equilibrium at R0=1 is forward, and there is a backward bifurcation from an endemic equilibrium at R0=1, which leads to the existence of multiple endemic equilibria.

The diagram of I*, I2 versus R0 when I0=30, A=100, β1=0.01, β2=0.01, γ=0.01, d=0.9, ϵ=0.01, r=1, p0=1.6889, p1=1.7882, p2=1.7074, where (iii) of Theorem 2.2 holds. The bifurcation at R0=1 is forward, and (1.4) has a unique endemic equilibrium for R0>1.

Note that a backward bifurcation with endemic equilibria when R0<1 is very interesting in applications. We present the following corollary to give conditions for such a backward bifurcation to occur.

Corollary 2.3.

Equation (1.4) has a backward bifurcation with endemic equilibria when R0<1 if [d-β2(A-k)/d]r>(β1+β2(γ/d))(d+ϵ+γ)I0 and p0<1.

Example 2.4.

Fix I0=10, A=60, β1=0.01, β2=0.005, γ=0.1, d=1, ϵ=0.1, and r=3. Then p10.6779, p00.8878, p21.225 and [d-β2(A-k)/d]r-(β1+β2γ/d)I0(d+ϵ+γ)=2.424. Thus, (1.4) has a backward bifurcation with endemic equilibria when R0<1 in this case (see Figure 3).

As I0 (the capacity of treatment resources) increases, by the definition we see that p0 increases. When I0 is so large that p0>1, it follows from Theorem 2.2 that there is no backward bifurcation with endemic equilibria when R0<1. If we increase I0 to R0<p0, (1.4) does not have a backward bifurcation because endemic equilibria E1 and E2 do not exist. This means that an insufficient capacity for treatment is a source of the backward bifurcation.

The figure of I*, I1 and I2 versus R0 that shows a backward bifurcation with endemic equilibrium when R0<1, where Corollary 2.3 holds.

3. The Stability of Equilibria

We first determine the stability of the disease-free equilibrium E0(A/d,0,0). The Jacobian matrix of (1.4) at E0(A/d,0,0) is (3.1)(-d-ϵ00β1Ad-(d+ϵ+γ+r)β2Ad0γ+r-d). Its characteristic equation is (3.2)(λ+d)[λ2+(d-β1Ad+(d+ϵ+γ+r))λ+d(d+ϵ+γ+r)β1Ad-β2Ad(γ+r)]=0. We obtain (3.3)λ1=-d<0,d[(d+ϵ+γ+r)-β1Ad]-β2Ad(γ+r)=d(d+ϵ+γ+r)(1-R0). Therefore, we get the following theorem.

Theorem 3.1.

The disease-free equilibrium E0(A/d,0,0) is locally asymptotically stable if R0<1 and unstable if R0>1.

Next, the stability of endemic equilibrium E*(N*,I*,R*) is analyzed. The Jacobian matrix of (1.4) at E*(N*,I*,R*) is (3.4)J*=(-d-ϵ0b1β1b2-a1-(d+ϵ+γ+r)β2b2-b10γ+r-d), where c0=β1I*+β2R*, b0=N*-I*-R*.

Making use of (2.2), the characteristic equation of J* is simplified into (3.5)(λ+d)[λ2+(d+c0+(d+ϵ+γ+r)-β1b0)λ+d(c0+d+ϵ+γ+r-β1b0)+(c0-β2b0)(γ+r)+c0ϵλ2]=0, where (3.6)d+c0+(d+ϵ+γ+r)-β1b0=d+(β1+β2γ+rd)I*+(d+ϵ+γ+r)(1-β1β1+β2((γ+r)/d))>0,d(c0+d+ϵ+γ+r-β1b0)+(c0-β2b0)(γ+r)+c0ϵ=(β1+β2γ+rd)I*(d+ϵ+γ+r)>0. Therefore, the real part of the all eigenvalues of J* is negative when 1<R0p2.

Theorem 3.2.

If 1<R0p2, then the endemic equilibrium E* of (1.4) is locally asymptotically stable.

Afterwards, we study the stability of endemic equilibrium E1(N1,I1,R1). The characteristic equation of Jacobian matrix of (1.4) at E1(N1,I1,R1) is (3.7)(λ+d)[λ2+(d+c1+(d+ϵ+γ)-β1b1)λ+d(c1+d+ϵ+γ-β1b1)+γ(c1-β2b1)+c1ϵ]=0, where c1=β1I1+β2R1, b1=N1-I1-R1. After some calculations, we obtain (3.8)d(c1+d+ϵ+γ-β1b1)+γ(c1-β2b1)+c1ϵ=2a0I1+a1=-Δ<0. Therefore, (3.7) has positive real part eigenvalues. Thus E1(N1,I1,R1) is unstable.

Theorem 3.3.

If the endemic equilibrium E1(N1,I1,R1) of system (1.4) exists, then it is unstable.

Finally, we analyze the stability of endemic equilibrium E2(N2,I2,R2). Its characteristic equation is (3.9)(λ+d)[λ2+(d+c2+(d+ϵ+γ)-β1b2)λ+d(c2+d+ϵ+γ-β1b2)+γ(c2-β2b2)+c2ϵ]=0, where c2=β1I2+β2R2, b2=N2-I2-R2. By some calculations, we obtain (3.10)d(c2+d+ϵ+γ-β1b2)+γ(c2-β2b2)+c2ϵ=2a0I2+a1=Δ>0,d+c2+(d+ϵ+γ)-β1b2=d+(β1+β2γd)I2+(d+ϵ+γ)+β2kd+β1(d+ϵ+γ)I2d+β1(k-A)d. It follows that d+c2+(d+ϵ+γ)-β1b2>0 is equivalent to (3.11)Δ>a1+-2a0[d+β2k/d+(d+γ+ϵ)-β1((A-k)/d)]β1+β2(γ/d)+β1((d+γ+ϵ)/d). If (3.12)a1[β1+β2γd+β1d+γ+ϵd]-2a0[d+β2kd+(d+γ+ϵ)-β1A-kd]<0, then d+c2+(d+ϵ+γ)-β1b2>0. Thus E2(N2,I2,R2) is locally asymptotically stable.

By complicated calculation, if a1[β1+β2(γ/d)+β1((d+γ+ϵ)/d)]-2a0[d+β2k/d+(d+γ+ϵ)-β1((A-k)/d)]>0, then (3.11) is equivalent to (3.13)a2[β1+β2γd+β1d+γ+ϵd]2<a1[β1+β2γd+β1d+γ+ϵd][d+β2kd+(d+γ+ϵ)-β1A-kd]-a0[d+β2kd+(d+γ+ϵ)-β1A-kd]2.

Theorem 3.4.

Suppose the endemic equilibrium E2(N2,I2,R2) of system (1.4) exists; if either (3.14)a1[β1+β2γd+β1d+γ+ϵd]-2a0[d+β2kd+(d+γ+ϵ)-β1A-kd]<0, or (3.15)a1[β1+β2γd+β1d+γ+ϵd]-2a0[d+β2kd+(d+γ+ϵ)-β1A-kd]>0,a2[β1+β2γd+β1d+γ+ϵd]2<a1[β1+β2γd+β1d+γ+ϵd][d+β2kd+(d+γ+ϵ)-β1A-kd]-a0[d+β2kd+(d+γ+ϵ)-β1A-kd]2, then it is locally asymptotically stable.

Theorem 3.5.

The disease-free equilibrium E0 of system (1.4) is globally asymptotically stable, if one of the following conditions is satisfied:

R0<1 and p0>1,

R0<1, p0<1 and p11.

Proof .

R 0 < 1 implies that E* does not exist. Suppose p01. It follows from the discussions for Theorem 2.2 that E1 or E2 exists only if R0>p0, which is impossible since we have R0<1. Let us now suppose p0<1 and p11. If [d-β2(A-k)/d]r>(β1+β2(γ/d)I0(d+ϵ+γ), since p1<p2, it follows from the discussions for (i), (ii) of Theorem 2.2 that E1 or E2 exists only if R0>p1, which is impossible since we have R0<1. If [d-β2(A-k)/d]r>(β1+β2γ/d)I0(d+ϵ+γ), since 1<p2, it follows from (iii) of Theorem 2.2 that E1 and E2 do not exist. In summary, endemic equilibria do not exist under the assumptions.

4. The Simulation of Model

In this section, we give the numerical simulations of system (1.4) for the conclusions gained previously.

Example 4.1.

For system (1.4), if R0<1 and R0>p0 and p1<R0<p2, then the equilibrium E* does not exist, and there are three equilibria E0, E1, and E2. Its phase diagram is illustrated in Figure 4. Numerical calculations show that E0 and E2 are stable, but E1 is unstable.

The phase diagram of system (1.4) when I0=5, A=80, β1=0.015, β2=0.001, γ=0.01, d=0.8, ϵ=0.01, r=1, p0=0.8607, p1=0.6589, p2=1.1016, R0=0.8935.

Example 4.2.

For system (1.4), if R0>1 and R0<p0, there is the unique equilibrium E* which is stable. Its phase diagram is illustrated in Figure 5. Numerical calculations show that the unique equilibrium E* is globally stable.

The phase diagram of system (1.4) when I0=40, A=100, β1=0.02, β2=0.01, γ=0.01, d=0.9, ϵ=0.01, r=1.5, p0=2.6340, p1=2.6606, p2=2.6346, R0=1.6886.

Example 4.3.

For system (1.4), if R0>1 and R0>p0 and p1<R0<p2, the equilibria E2 and E* are stable, and E0 and E1 are unstable; its phase diagram is illustrated in Figure 6. Numerical calculations show that the equilibria E2 and E* are stable, and E0 and E1 unstable. Thus, we have bistable endemic equilibria.

The phase diagram of system (1.4) when I0=10, A=100, β1=0.015, β2=0.001, γ=0.01, d=0.8, ϵ=0.01, r=1, p0=1.0462, p1=0.8156, p2=1.2033, R0=1.1169.

5. Discussion

In this paper, we have proposed an epidemic model with infectious force in infected and immune period and treatment rate of infectious individuals to understand the effect of the capacity for treatment of infective on the disease transmission, which can occur when patients have to be hospitalized but there are limited beds or medical establishments in hospitals, or there is not enough medicine for treatment. We have shown in Theorem 2.2 and Corollary 2.3 that backward bifurcations occur because of the insufficient capacity for treatment. We have also shown that system (1.4) has bistable endemic equilibria because of the limited resources. This means that the basic reproduction number R0<1 and small treatment rate are not enough to eradicate the disease, but the basic reproduction number R0<1 and large treatment rate may eradicate the disease. The disease cannot be eradicated for any treatment rate if the basic reproduction number R0>1. Therefore, the level of initial infectious invasion must be lowered to a threshold so that the disease dies out or approaches a lower endemic steady state for a range of parameters.

In Sections 2 and 3, when I>I0, with respect to the existence and the local stability of the endemic equilibrium we only proved for the model (1.6) under the restriction a2>0. But the case of a2<0 is an unsolved question.

Acknowledgment

This work is supported by the National Science Foundation of China (10471040), Science Foundation of China (2009011005-1) as well as Science and Technology Research Developmental item of Shan xi Province Education Department (20061025).

Zhang J. Ma Z. E. Global analysis of the SEI epidemic model with constant inflows of different compartments Journal of Xi'an Jiaotong University 2003 37 6 653 656 1996196 Yuan S. L. Han L. T. Ma Z. E. A kind of epidemic model having infectious force in both latent period and infected period Journal of Biomathematics 2001 16 4 392 398 1881859 ZBL1059.92509 Li G. H. Jin Z. Global stability of an SEI epidemic model Chaos, Solitons and Fractals 2004 21 4 925 931 10.1016/j.chaos.2003.12.031 2042810 ZBL1045.34025 Li G. H. Jin Z. Global stability of an SEI epidemic model with general contact rate Chaos, Solitons and Fractals 2005 23 3 997 1004 10.1016/j.chaos.2004.06.012 2093784 ZBL1062.92062 Li G. H. Jin Z. Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period Chaos, Solitons and Fractals 2005 25 5 1177 1184 10.1016/j.chaos.2004.11.062 2144662 ZBL1065.92046 Arino J. McCluskey C. C. van den Driessche P. Global results for an epidemic model with vaccination that exhibits backward bifurcation SIAM Journal on Applied Mathematics 2003 64 1 260 276 10.1137/S0036139902413829 2029134 ZBL1034.92025 Dushoff J. Huang W. Castillo-Chavez C. Backwards bifurcations and catastrophe in simple models of fatal diseases Journal of Mathematical Biology 1998 36 3 227 248 10.1007/s002850050099 1608613 ZBL0917.92022 van den Driessche P. Watmough J. A simple SIS epidemic model with a backward bifurcation Journal of Mathematical Biology 2000 40 6 525 540 10.1007/s002850000032 1770939 ZBL0961.92029 Hadeler K. P. van den Driessche P. Backward bifurcation in epidemic control Mathematical Biosciences 1997 146 1 15 35 10.1016/S0025-5564(97)00027-8 1604288 ZBL0904.92031 Kuznetsov Y. A. Piccardi C. Bifurcation analysis of periodic SEIR and SIR epidemic models Journal of Mathematical Biology 1994 32 2 109 121 10.1007/BF00163027 1258722 ZBL0786.92022 Ruan S. Wang W. Dynamical behavior of an epidemic model with a nonlinear incidence rate Journal of Differential Equations 2003 188 1 135 163 10.1016/S0022-0396(02)00089-X 1954511 ZBL1028.34046 Zhang X. Liu X. Backward bifurcation of an epidemic model with saturated treatment function Journal of Mathematical Analysis and Applications 2008 348 1 433 443 10.1016/j.jmaa.2008.07.042 2449361 ZBL1144.92038 Hu Z. Liu S. Wang H. Backward bifurcation of an epidemic model with standard incidence rate and treatment rate Nonlinear Analysis. Real World Applications 2008 9 5 2302 2312 10.1016/j.nonrwa.2007.08.009 2442344 ZBL1156.34320 Li X.-Z. Li W.-S. Ghosh M. Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment Applied Mathematics and Computation 2009 210 1 141 150 10.1016/j.amc.2008.12.085 2504129 ZBL1159.92036 Zhang X. Liu X. Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment Nonlinear Analysis. Real World Applications 2009 10 2 565 575 10.1016/j.nonrwa.2007.10.011 2474246 ZBL1167.34338 Wang W. Ruan S. Bifurcation in an epidemic model with constant removal rate of the infectives Journal of Mathematical Analysis and Applications 2004 291 2 775 793 10.1016/j.jmaa.2003.11.043 2039086 Wang W. Backward bifurcation of an epidemic model with treatment Mathematical Biosciences 2006 201 1-2 58 71 10.1016/j.mbs.2005.12.022 2252078 ZBL1093.92054