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

Abstract

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 [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 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 [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 𝑇(𝐼)=𝑟𝐼if0⩽𝐼⩽𝐼₀,𝑘if𝐼>𝐼₀, (1.1) 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 [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)𝛽₁, 𝛽₂ 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 𝜖 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: 𝑑𝑆𝑑𝑡=𝐴−𝑑𝑆−𝛽₁𝑆𝐼−𝛽₂𝑆𝑅,𝑑𝐼𝑑𝑡=𝛽₁𝑆𝐼+𝛽₂𝑆𝑅−𝑑+𝛾+𝜖𝐼−𝑇(𝐼),𝑑𝑅(𝐼)𝑑𝑡=𝛾𝐼+𝑇−𝑑𝑅, (1.2) 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: 𝑑𝑁𝑑𝑡=𝐴−𝑑𝑁−𝜖𝐼. (1.3) Then system (1.2) is equivalent to 𝑑𝑁𝑑𝑡=𝐴−𝑑𝑁−𝜖𝐼,𝑑𝐼=𝛽𝑑𝑡¹𝐼+𝛽₂𝑅(𝑁−𝐼−𝑅)−𝑑+𝛾+𝜖𝐼−𝑇(𝐼),𝑑𝑅(𝐼)𝑑𝑡=𝛾𝐼+𝑇−𝑑𝑅. (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⩽𝐼₀, system (1.4) becomes 𝑑𝑁𝑑𝑡=𝐴−𝑑𝑁−𝜖𝐼,𝑑𝐼=𝛽𝑑𝑡¹𝐼+𝛽₂𝑅(𝑁−𝐼−𝑅)−𝑑+𝛾+𝜖+𝑟𝐼,𝑑𝑅=𝑑𝑡𝛾+𝑟𝐼−𝑑𝑅. (1.5)

When 𝐼>𝐼₀, system (1.4) becomes 𝑑𝑁𝑑𝑡=𝐴−𝑑𝑁−𝜖𝐼,𝑑𝐼=𝛽𝑑𝑡¹𝐼+𝛽₂𝑅(𝑁−𝐼−𝑅)−𝑑+𝛾+𝜖𝐼−𝑘,𝑑𝑅𝑑𝑡=𝛾𝐼+𝑘−𝑑𝑅. (1.6)

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 𝛽𝐴−𝑑𝑁−𝜖𝐼=0,₁𝐼+𝛽₂𝑅()−(𝐼)(𝐼)𝑁−𝐼−𝑅𝑑+𝛾+𝜖𝐼−𝑇=0,𝛾𝐼+𝑇−𝑑𝑅=0. (2.1)

When 0⩽𝐼⩽𝐼₀, system (2.1) becomes 𝛽𝐴−𝑑𝑁−𝜖𝐼=0,₁𝐼+𝛽₂𝑅(𝑁−𝐼−𝑅)−𝑑+𝛾+𝜖+𝑟𝐼=0,𝛾+𝑟𝐼−𝑑𝑅=0. (2.2)

When 𝐼>𝐼₀, system (2.1) becomes 𝛽𝐴−𝑑𝑁−𝜖𝐼=0,₁𝐼+𝛽₂𝑅(𝑁−𝐼−𝑅)−𝑑+𝛾+𝜖𝐼−𝑘=0,𝛾𝐼+𝑘−𝑑𝑅=0. (2.3)

Form (2.2), 𝐼 satisfies the following equation: 𝛽₁+𝛽₂𝛾+𝑟𝑑𝐴−𝑑+𝜖+𝛾+𝑟𝐼𝑑=𝑑+𝜖+𝛾+𝑟. (2.4) Therefore, we obtain /𝛽𝐼=𝐴−𝑑𝑑+𝜖+𝛾+𝑟₁+𝛽₂𝛾+𝑟/𝑑.𝑑+𝜖+𝛾+𝑟 (2.5)

Let 𝑅₀=𝐴𝛽₁+𝛽₂𝛾+𝑟/𝑑𝑑𝑑+𝜖+𝛾+𝑟. (2.6) 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 𝑁_∗=𝐴−𝜖𝐼_∗𝑑,𝐼_∗=𝐴−𝑑𝑑+𝜖+𝛾+𝑟/𝛽₁+𝛽₂𝛾+𝑟/𝑑,𝑅𝑑+𝜖+𝛾+𝑟_∗=𝛾+𝑟𝐼_∗𝑑. (2.7) Clearly, 𝐸_∗ is an endemic equilibrium of (1.4) if and only if 1<𝑅₀𝛽⩽1+¹+𝛽₂𝛾+𝑟/𝑑𝑑𝐼₀. (2.8)

According to (2.3), 𝐼 satisfies the following equation: 𝑎₀𝐼²+𝑎₁𝐼+𝑎₂=0, (2.9) where 𝑎₀=(𝛽₁+𝛽₂(𝛾/𝑑))(𝑑+𝜖+𝛾), 𝑎₁=𝑑𝑑+𝜖+𝛾+𝛽₂𝑘𝑑𝑑+𝜖+𝛾−𝛽₁+𝛽₂𝛾𝑑(),𝑎𝐴−𝑘₂=𝑑𝑘−𝛽₂𝑘𝑑(𝐴−𝑘). (2.10)

We only consider the case of 𝑎₂>0. If 𝑎₁⩾0, 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 𝑅₀𝛽⩾1+²(𝑘/𝑑)𝑑+𝜖+𝛾+𝛽₂(𝑟/𝑑)𝐴+𝛽₁+𝛽₂𝛾/𝑑𝑘−𝑑𝑟𝑑𝑑+𝜖+𝛾+𝑟=∶𝑝_∗. (2.11) It is easy that Δ=𝑎²₁−4𝑎₀𝑎₂=𝑅²₀𝑑²𝑑+𝜖+𝛾+𝑟²−2𝑅₀𝑑𝑑+𝜖+𝛾+𝑟𝑑+𝛽₂𝑘𝑑𝑑+𝜖+𝛾+𝛽₂𝑟𝑑𝐴+𝛽₁+𝛽₂𝛾𝑑𝑘+𝑑+𝛽₂𝑘𝑑𝑑+𝜖+𝛾+𝛽₂𝑟𝑑𝐴+𝛽₁+𝛽₂𝛾𝑑𝑘²−4𝛽₁+𝛽₂𝛾𝑑𝑑+𝜖+𝛾𝑑𝑘−𝛽₂𝑘𝑑(𝐴−𝑘). (2.12) It follows that Δ⩾0 is equivalent to 𝑅₀⩾𝑝_∗+2𝛽₁+𝛽₂𝛾/𝑑𝑑+𝜖+𝛾𝑑𝑘−𝛽₂(𝑘/𝑑)(𝐴−𝑘)𝑑𝑑+𝜖+𝛾+𝑟=∶𝑝₀, (2.13) or 𝑅₀⩽𝑝_∗−2𝛽₁+𝛽₂𝛾/𝑑𝑑+𝜖+𝛾𝑑𝑘−𝛽₂(𝑘/𝑑)(𝐴−𝑘)𝑑.𝑑+𝜖+𝛾+𝑟 (2.14) Thus 𝑎₁<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 𝐼₁ and 𝐼₂ where 𝐼₁=−𝑎₁−√Δ2𝑎₀,𝐼₂=−𝑎₁+√Δ2𝑎₀. (2.15) Set 𝑁_𝑖=(𝐴−𝜖𝐼_𝑖)/𝑑, 𝑅_𝑖=(𝛾𝐼_𝑖+𝑘)/𝑑 and 𝐸_𝑖(𝑁_𝑖,𝐼_𝑖,𝑅_𝑖)(𝑖=1,2). If 𝐼_𝑖>𝐼₀(𝑖=1,2), then 𝐸_𝑖 is an endemic equilibrium of (1.6).

By the definition of 𝐼₁, we notice that 𝐼₁>𝐼₀ is equivalent to −√Δ>2𝑎₀𝐼₀+𝑎₁. (2.16) This implies that 2𝑎₀𝐼₀+𝑎₁<0. By immediate calculation, 2𝑎₀𝐼₀+𝑎₁<0 is equivalent to 𝑅₀>𝑝_∗+2𝛽₁+𝛽₂𝛾/𝑑𝑑+𝜖+𝛾𝐼₀𝑑𝑑+𝜖+𝛾+𝑟=∶𝑝₁. (2.17) Further, 𝐼₁>𝐼₀ demands that 2𝑎₀𝐼₀+𝑎₁²>Δ. (2.18) By immediate calculation, 𝑅₀𝛽<1+¹+𝛽₂𝛾+𝑟/𝑑𝑑𝐼₀=∶𝑝₂. (2.19) 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. 𝐸₀(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<𝑅₀⩽𝑝₂. Furthermore, 𝐸_∗ is the unique equilibrium of system (1.4) if 1<𝑅₀⩽𝑝₂, and one of the following conditions is satisfied:(i)𝑅₀<𝑝₀, (ii)𝑝₀⩽𝑅₀<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 𝑅₀⩾𝑝₀, we have the following:(i)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,(ii)if [𝑑−𝛽₂(𝐴−𝑘)/𝑑]𝑟>(𝛽₁+𝛽₂(𝛾/𝑑))(𝑑+𝜖+𝛾)𝐼₀, then 𝐸₁ does not exist but 𝐸₂ exists if 𝑅₀⩾𝑝₂,(iii)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 𝑅₀⩽𝑝₂.

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.

Figure 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.

Figure 2 

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 𝑝₁≈0.6779, 𝑝₀≈0.8878, 𝑝₂≈1.225 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.

Figure 3 

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 −𝑑−𝜖00𝛽₁𝐴𝑑−𝑑+𝜖+𝛾+𝑟𝛽₂𝐴𝑑0𝛾+𝑟−𝑑. (3.1) Its characteristic equation is (𝜆+𝑑)𝜆²+𝑑−𝛽₁𝐴𝑑+𝑑+𝜖+𝛾+𝑟𝜆+𝑑𝑑+𝜖+𝛾+𝑟𝛽₁𝐴𝑑−𝛽₂𝐴𝑑𝛾+𝑟=0. (3.2) We obtain 𝜆₁𝑑=−𝑑<0,𝑑+𝜖+𝛾+𝑟−𝛽₁𝐴𝑑−𝛽₂𝐴𝑑𝛾+𝑟=𝑑𝑑+𝜖+𝛾+𝑟1−𝑅₀. (3.3) 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 𝐽_∗=𝑏−𝑑−𝜖0₁𝛽₁𝑏₂−𝑎₁−𝑑+𝜖+𝛾+𝑟𝛽₂𝑏₂−𝑏₁0𝛾+𝑟−𝑑, (3.4) where 𝑐₀=𝛽₁𝐼_∗+𝛽₂𝑅_∗, 𝑏₀=𝑁_∗−𝐼_∗−𝑅_∗.

Making use of (2.2), the characteristic equation of 𝐽_∗ is simplified into (𝜆+𝑑)𝜆²+𝑑+𝑐₀+𝑑+𝜖+𝛾+𝑟−𝛽₁𝑏₀𝜆+𝑑𝑐₀+𝑑+𝜖+𝛾+𝑟−𝛽₁𝑏₀+𝑐₀−𝛽₂𝑏₀𝛾+𝑟+𝑐₀𝜖=0, (3.5) where 𝑑+𝑐₀+𝑑+𝜖+𝛾+𝑟−𝛽₁𝑏₀=𝑑+𝛽₁+𝛽₂𝛾+𝑟𝑑𝐼_∗+𝛽𝑑+𝜖+𝛾+𝑟1−¹𝛽₁+𝛽₂𝑑𝑐𝛾+𝑟/𝑑>0,₀+𝑑+𝜖+𝛾+𝑟−𝛽₁𝑏₀+𝑐₀−𝛽₂𝑏₀𝛾+𝑟+𝑐₀𝜖=𝛽₁+𝛽₂𝛾+𝑟𝑑𝐼_∗𝑑+𝜖+𝛾+𝑟>0. (3.6) Therefore, the real part of the all eigenvalues of 𝐽_∗ is negative when 1<𝑅₀⩽𝑝₂.

Theorem 3.2. If 1<𝑅₀⩽𝑝₂, 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 (𝜆+𝑑)𝜆²+𝑑+𝑐₁+𝑑+𝜖+𝛾−𝛽₁𝑏₁𝜆+𝑑𝑐₁+𝑑+𝜖+𝛾−𝛽₁𝑏₁+𝛾𝑐₁−𝛽₂𝑏₁+𝑐₁𝜖=0, (3.7) where 𝑐₁=𝛽₁𝐼₁+𝛽₂𝑅₁, 𝑏₁=𝑁₁−𝐼₁−𝑅₁. After some calculations, we obtain 𝑑𝑐₁+𝑑+𝜖+𝛾−𝛽₁𝑏₁+𝛾𝑐₁−𝛽₂𝑏₁+𝑐₁𝜖=2𝑎₀𝐼₁+𝑎₁=−√Δ<0. (3.8) 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 (𝜆+𝑑)𝜆²+𝑑+𝑐₂+𝑑+𝜖+𝛾−𝛽₁𝑏₂𝜆+𝑑𝑐₂+𝑑+𝜖+𝛾−𝛽₁𝑏₂+𝛾𝑐₂−𝛽₂𝑏₂+𝑐₂𝜖=0, (3.9) where 𝑐₂=𝛽₁𝐼₂+𝛽₂𝑅₂, 𝑏₂=𝑁₂−𝐼₂−𝑅₂. By some calculations, we obtain 𝑑𝑐₂+𝑑+𝜖+𝛾−𝛽₁𝑏₂+𝛾𝑐₂−𝛽₂𝑏₂+𝑐₂𝜖=2𝑎₀𝐼₂+𝑎₁=√Δ>0,𝑑+𝑐₂+𝑑+𝜖+𝛾−𝛽₁𝑏₂=𝑑+𝛽₁+𝛽₂𝛾𝑑𝐼₂++𝛽𝑑+𝜖+𝛾²𝑘𝑑+𝛽₁𝑑+𝜖+𝛾𝐼₂𝑑+𝛽₁(𝑘−𝐴)𝑑. (3.10) It follows that 𝑑+𝑐₂+(𝑑+𝜖+𝛾)−𝛽₁𝑏₂>0 is equivalent to √Δ>𝑎₁+−2𝑎₀𝑑+𝛽₂𝑘/𝑑+𝑑+𝛾+𝜖−𝛽₁((𝐴−𝑘)/𝑑)𝛽₁+𝛽₂𝛾/𝑑+𝛽₁𝑑+𝛾+𝜖/𝑑. (3.11) If 𝑎₁𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑−2𝑎₀𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑<0, (3.12) 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 𝑎₂𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑²<𝑎₁𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑−𝑎₀𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑². (3.13)

Theorem 3.4. Suppose the endemic equilibrium 𝐸₂(𝑁₂,𝐼₂,𝑅₂) of system (1.4) exists; if either 𝑎₁𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑−2𝑎₀𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑<0, (3.14) or 𝑎₁𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑−2𝑎₀𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑𝑎>0,₂𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑²<𝑎₁𝛽₁+𝛽₂𝛾𝑑+𝛽₁𝑑+𝛾+𝜖𝑑𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑−𝑎₀𝛽𝑑+²𝑘𝑑+𝑑+𝛾+𝜖−𝛽₁𝐴−𝑘𝑑², (3.15) 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:(i)𝑅₀<1 and 𝑝₀>1,(ii)𝑅₀<1, 𝑝₀<1 and 𝑝₁⩾1.

Proof . 𝑅₀<1 implies that 𝐸_∗ does not exist. Suppose 𝑝₀⩾1. 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 𝑝₁⩾1. 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.

Figure 4 

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.

Figure 5 

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.

Figure 6 

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).