Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment

and Applied Analysis 3 where S t I t R t N t . It is easy to verify that R3 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: dN dt A − dN − I. 1.3 Then system 1.2 is equivalent to


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 1 studied the global stability of an SEI model with general contact rate.Yuan et al. 2 considered the local stability of the model having infectious force in both latent period and infected period.Li and Jin 3-5 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 R 0 > 1, and the disease-free equilibrium is always stable when R 0 < 1 and unstable when R 0 > 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 6-15 .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 16 .Note that a constant treatment rate is suitable when the number of infective is large.In 17 , the treatment rate of the disease is modified into where k rI 0 , r and I 0 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 16 .
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.
i There is a positive constant recruitment rate of the population A.
ii Positive constant d is the nature death rate of population.
iii β 1 , β 2 are the rate of the efficient contact in the infected and recovered period, respectively.
iv Positive constant γ is the natural recovery rate of infective individuals.
v Positive constant is the disease-related death rate.
vi 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: 1.4 It is easy to verify that all solutions of system 1.4 initiating in set { N, I, R | N > 0, I 0, R 0, I R N} eventually enter the set Ω { N, I, R | 0 < N A/d, I 0, R 0, I R N}.Therefore, Ω is positively invariant for system 1.4 .We consider the solutions of system 1.4 in Ω below.
When 0 I I 0 , system 1.4 becomes

1.5
When I > I 0 , system 1.4 becomes 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 diseasefree 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.

The Existence of Equilibria
In this section, we consider the equilibria of system 1.4 .Obviously, E 0 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 A − dN − I 0,

2.1
When 0 I I 0 , system 2.1 becomes

2.2
When I > I 0 , system 2.1 becomes

2.3
Form 2.2 , I satisfies the following equation: Therefore, we obtain Then R 0 is a basic reproduction number of 1.4 .If R 0 > 1, then I > 0; 2.2 admits a unique positive solution E * N * , I * , R * , where

2.7
Clearly, E * is an endemic equilibrium of 1.4 if and only if According to 2.3 , I satisfies the following equation: where

2.10
We only consider the case of a 2 > 0. If a 1 0, it is clear that 2.9 does not have positive real root.Let us suppose a 1 < 0 below.Note that a 1 < 0 is equivalent to It is easy that

2.12
It follows that Δ 0 is equivalent to Thus a 1 < 0 and Δ 0 if and only if 2.13 holds.Let us suppose that 2.13 holds.Then 2.9 has two positive solutions I 1 and I 2 where By the definition of I 1 , we notice that I 1 > I 0 is equivalent to

2.18
By immediate calculation,

2.19
Therefore, I 1 > I 0 holds if and only if R 0 > p 1 and R 0 < p 2 .By similar discussions as previously mentioned, we have that Summarizing the discussions above, we have the following conclusion.The bifurcation from the disease-free equilibrium at R 0 1 is forward, and there is a backward bifurcation from an endemic equilibrium at R 0 1, which leads to the existence of multiple endemic equilibria.
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 < R 0 p 2 .Furthermore, E * is the unique equilibrium of system 1.4 if 1 < R 0 p 2 , and one of the following conditions is satisfied: By calculation, we have Theorem 2.2.Endemic equilibria E 1 and E 2 do not exist if R 0 < p 0 .Further, if R 0 p 0 , we have the following: Further, E 2 exists when R 0 > p 2 , and E 2 does not exist when R 0 p 2 .1, where the bifurcation from the disease-free equilibrium at R 0 1 is forward and there is a backward bifurcation from an endemic equilibrium at R 0 1.71, which gives rise to the existence of multiple endemic equilibria.Further, 2, where the bifurcation at R 0 1 is forward, and 1.4 has one unique endemic equilibrium for all R 0 > 1.Note that a backward bifurcation with endemic equilibria when R 0 < 1 is very interesting in applications.We present the following corollary to give conditions for such a backward bifurcation to occur.424.Thus, 1.4 has a backward bifurcation with endemic equilibria when R 0 < 1 in this case see Figure 3 .

We consider p
As I 0 the capacity of treatment resources increases, by the definition we see that p 0 increases.When I 0 is so large that p 0 > 1, it follows from Theorem 2.2 that there is no backward bifurcation with endemic equilibria when R 0 < 1.If we increase I 0 to R 0 < p 0 , 1.4 does not have a backward bifurcation because endemic equilibria E 1 and E 2 do not exist.This means that an insufficient capacity for treatment is a source of the backward bifurcation.

The Stability of Equilibria
We first determine the stability of the disease-free equilibrium E 0 A/d, 0, 0 .The Jacobian matrix of 1.4 at

Its characteristic equation is
We obtain

3.3
Therefore, we get the following theorem.

Theorem 3.1. The disease-free equilibrium
Next, the stability of endemic equilibrium Making use of 2.2 , the characteristic equation of J * is simplified into where

3.6
Therefore, the real part of the all eigenvalues of J * is negative when 1 < R 0 p 2 .
Afterwards, we study the stability of endemic equilibrium E 1 N 1 , I 1 , R 1 .The characteristic equation of Jacobian matrix of 1.4 at where After some calculations, we obtain Therefore, 3.7 has positive real part eigenvalues.Thus Finally, we analyze the stability of endemic equilibrium By some calculations, we obtain

3.10
It If

3.15
then it is locally asymptotically stable.Theorem 3.5.The disease-free equilibrium E 0 of system 1.4 is globally asymptotically stable, if one of the following conditions is satisfied: it follows from iii of Theorem 2.2 that E 1 and E 2 do not exist.In summary, endemic equilibria do not exist under the assumptions.

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 R 0 < 1 and R 0 > p 0 and p 1 < R 0 < p 2 , then the equilibrium E * does not exist, and there are three equilibria E 0 , E 1 , and E 2 .Its phase diagram is illustrated in Figure 4. Numerical calculations show that E 0 and E 2 are stable, but E 1 is unstable.
Example 4.2.For system 1.4 , if R 0 > 1 and R 0 < p 0 , 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.
Example 4.3.For system 1.4 , if R 0 > 1 and R 0 > p 0 and p 1 < R 0 < p 2 , the equilibria E 2 and E * are stable, and E 0 and E 1 are unstable; its phase diagram is illustrated in Figure 6.Numerical calculations show that the equilibria E 2 and E * are stable, and E 0 and E 1 unstable.Thus, we have bistable endemic equilibria.

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 R 0 < 1 and small treatment rate are not enough to eradicate the disease, but the basic reproduction number R 0 < 1 and large treatment rate may eradicate the disease.The disease cannot be eradicated for any treatment rate if the basic reproduction number R 0 > 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 > I 0 , with respect to the existence and the local stability of the endemic equilibrium we only proved for the model 1.6 under the restriction a 2 > 0. But the case of a 2 < 0 is an unsolved question.

IFigure 3 :
Figure 3: The figure of I * , I 1 and I 2 versus R 0 that shows a backward bifurcation with endemic equilibrium when R 0 < 1, where Corollary 2.3 holds.

:
The figure of infective sizes at equilibria versus R 0 when I 0 30, A 80, β 1 0.01, β 2 0.01, IFigure 1 and p 1 1.Proof .R 0 < 1 implies that E * does not exist.Suppose p 0 1.It follows from the discussions for Theorem 2.2 that E 1 or E 2 exists only if R 0 > p 0 , which is impossible since we have R 0 < 1.Let us now suppose p 0 < 1 andp 1 1.If d−β 2 A−k /d r > β 1 β 2 γ/d I 0 d γ , since p 1 < p 2 ,it follows from the discussions for i , ii of Theorem 2.2 that E 1 or E 2 exists only if R 0 > p 1 , which is impossible since we