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 [

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 [

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

The basic assumptions in the paper are as the follows.

There is a positive constant recruitment rate of the population

Positive constant

Positive constant

Positive constant

The treatment of a disease is

Under the assumptions above, an epidemic model to be studied takes the following form:

According to

It is easy to verify that all solutions of system (

When

When

The purpose of this paper is to show that system (

In this section, we consider the equilibria of system (

When

When

Form (

Let

According to (

We only consider the case of

By the definition of

By similar discussions as previously mentioned, we have that

Summarizing the discussions above, we have the following conclusion.

By calculation, we have

Endemic equilibria

if

if

letting

We consider

The figure of infective sizes at equilibria versus

The diagram of

Note that a backward bifurcation with endemic equilibria when

Equation (

Fix

As

The figure of

We first determine the stability of the disease-free equilibrium

The disease-free equilibrium

Next, the stability of endemic equilibrium

Making use of (

If

Afterwards, we study the stability of endemic equilibrium

If the endemic equilibrium

Finally, we analyze the stability of endemic equilibrium

By complicated calculation, if

Suppose the endemic equilibrium

The disease-free equilibrium

In this section, we give the numerical simulations of system (

For system (

The phase diagram of system (

For system (

The phase diagram of system (

For system (

The phase diagram of system (

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

In Sections

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).