Backward Bifurcation of an Epidemic Model with Infectious Force in Infected and Immune Period and Treatment
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.
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 [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 , and the disease-free equilibrium is always stable when and unstable when . 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 . 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 where , and 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 , , and denote the numbers of susceptible, infective, recovered individuals at time , respectively. The total population size at time is denoted by .
The basic assumptions in the paper are as the follows.(i)There is a positive constant recruitment rate of the population .(ii)Positive constant is the nature death rate of population.(iii), are the rate of the efficient contact in the infected and recovered period, respectively.(iv)Positive constant is the natural recovery rate of infective individuals.(v)Positive constant is the disease-related death rate.(vi)The treatment of a disease is in (1.1).
Under the assumptions above, an epidemic model to be studied takes the following form: where . It is easy to verify that is positive invariant for system (1.2).
It is easy to verify that all solutions of system (1.4) initiating in set eventually enter the set . Therefore, is positively invariant for system (1.4). We consider the solutions of system (1.4) in below.
When , system (1.4) becomes
When , 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 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
When , system (2.1) becomes
When , system (2.1) becomes
Form (2.2), satisfies the following equation: Therefore, we obtain
According to (2.3), satisfies the following equation: where ,
We only consider the case of . If , it is clear that (2.9) does not have positive real root. Let us suppose below. Note that is equivalent to It is easy that It follows that is equivalent to or Thus and if and only if (2.13) holds. Let us suppose that (2.13) holds. Then (2.9) has two positive solutions and where Set , and . If , then is an endemic equilibrium of (1.6).
By the definition of , we notice that is equivalent to This implies that . By immediate calculation, is equivalent to Further, demands that By immediate calculation, Therefore, holds if and only if and .
By similar discussions as previously mentioned, we have that holds if and only if either , or , .
Summarizing the discussions above, we have the following conclusion.
Theorem 2.1. is always the disease-free equilibrium of (1.5). is an endemic equilibrium of system (1.4) if and only if . Furthermore, is the unique equilibrium of system (1.4) if , and one of the following conditions is satisfied:(i), (ii).
By calculation, we have . Note that is equivalent to that .
Theorem 2.2. Endemic equilibria and do not exist if . Further, if , we have the following:(i)if , then both and exist when ,(ii)if , then does not exist but exists if ,(iii)letting , then does not exist. Further, exists when , and does not exist when .
We consider . If , a typical bifurcation diagram is illustrated in Figure 1, where the bifurcation from the disease-free equilibrium at is forward and there is a backward bifurcation from an endemic equilibrium at , which gives rise to the existence of multiple endemic equilibria. Further, if , a typical bifurcation diagram is illustrated in Figure 2, where the bifurcation at is forward, and (1.4) has one unique endemic equilibrium for all .
Note that a backward bifurcation with endemic equilibria when 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 if and .
Example 2.4. Fix , , , , , , , and . Then , , and . Thus, (1.4) has a backward bifurcation with endemic equilibria when in this case (see Figure 3).
As (the capacity of treatment resources) increases, by the definition we see that increases. When is so large that , it follows from Theorem 2.2 that there is no backward bifurcation with endemic equilibria when . If we increase to , (1.4) does not have a backward bifurcation because endemic equilibria and do not exist. This means that an insufficient capacity for treatment is a source of the backward bifurcation.
3. The Stability of Equilibria
We first determine the stability of the disease-free equilibrium . The Jacobian matrix of (1.4) at is Its characteristic equation is We obtain Therefore, we get the following theorem.
Theorem 3.1. The disease-free equilibrium is locally asymptotically stable if and unstable if .
Next, the stability of endemic equilibrium is analyzed. The Jacobian matrix of (1.4) at is where , .
Making use of (2.2), the characteristic equation of is simplified into where Therefore, the real part of the all eigenvalues of is negative when .
Theorem 3.2. If , then the endemic equilibrium of (1.4) is locally asymptotically stable.
Afterwards, we study the stability of endemic equilibrium . The characteristic equation of Jacobian matrix of (1.4) at is where , . After some calculations, we obtain Therefore, (3.7) has positive real part eigenvalues. Thus is unstable.
Theorem 3.3. If the endemic equilibrium of system (1.4) exists, then it is unstable.
Finally, we analyze the stability of endemic equilibrium . Its characteristic equation is where , . By some calculations, we obtain It follows that is equivalent to If then . Thus is locally asymptotically stable.
By complicated calculation, if , then (3.11) is equivalent to
Theorem 3.4. Suppose the endemic equilibrium of system (1.4) exists; if either or then it is locally asymptotically stable.
Theorem 3.5. The disease-free equilibrium of system (1.4) is globally asymptotically stable, if one of the following conditions is satisfied:(i) and ,(ii), and .
Proof . implies that does not exist. Suppose . It follows from the discussions for Theorem 2.2 that or exists only if , which is impossible since we have . Let us now suppose and . If , since , it follows from the discussions for (i), (ii) of Theorem 2.2 that or exists only if , which is impossible since we have . If , since , it follows from (iii) of Theorem 2.2 that and 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 and and , then the equilibrium does not exist, and there are three equilibria , , and . Its phase diagram is illustrated in Figure 4. Numerical calculations show that and are stable, but is unstable.
Example 4.2. For system (1.4), if and , there is the unique equilibrium which is stable. Its phase diagram is illustrated in Figure 5. Numerical calculations show that the unique equilibrium is globally stable.
Example 4.3. For system (1.4), if and and , the equilibria and are stable, and and are unstable; its phase diagram is illustrated in Figure 6. Numerical calculations show that the equilibria and are stable, and and unstable. Thus, we have bistable endemic equilibria.
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 and small treatment rate are not enough to eradicate the disease, but the basic reproduction number and large treatment rate may eradicate the disease. The disease cannot be eradicated for any treatment rate if the basic reproduction number . 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 , with respect to the existence and the local stability of the endemic equilibrium we only proved for the model (1.6) under the restriction . But the case of is an unsolved question.
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).
J. Zhang and Z. E. Ma, “Global analysis of the SEI epidemic model with constant inflows of different compartments,” Journal of Xi'an Jiaotong University, vol. 37, no. 6, pp. 653–656, 2003.View at: Google Scholar