An SEIS epidemic model with a changing delitescence is studied. The disease-free equilibrium and the endemic equilibrium of the model are studied as well. It is shown that the disease-free equilibrium is globally stable under suitable conditions. Moreover, we also show that the unique endemic equilibrium of the system is globally asymptotically stable under certain conditions.

Infectious diseases have tremendous influence on human life. Every year millions of human beings suffer from or die of various infectious diseases. It has been an increasingly complex issue to control infectious diseases. In order to predict the spreading of infectious diseases among regions, many epidemic models have been proposed and analyzed in recent years (see [

In recent years, the spread of infectious diseases is diversiform, such as H1N1 disease. The diversity of the delitescence period in each infected body who is infected with H1N1 virus is mainly due to the variation of the virus and the distinct constitution of different people. The study of the models with a changing delitescence plays an important role in controlling infectious diseases. In this paper, we consider an SEIS epidemic model with a changing delitescence and a nonlinear incidence rate, and we study the existence and stability of the equilibriums of the SEIS epidemic model.

By a standard nonlinear incidence rate

The model is given as follows:

Now, we study equilibria of model (

If

If

From the first and second equations of (

By substituting (

It is easy to see that (

If

If

The linearization of model (

If

Consider a Liapunov function as

Hence, according to Theorem

Now, we study the local stability of the endemic equilibrium

Substituting

Then, the Jacobi matrix of (

Denote

According to [

If

If

From the above discussion, we get the following conclusion.

If

From the third equation of (

From the above discussion, we get the following conclusion.

There is no limit cycle, and the endemic equilibrium

When

Figures

Dynamical behavior of system (

Dynamical behavior of system (

Dynamical behavior of system (

Dynamical behavior of system (

Dynamical behavior of system (

With

For the initial values (20,30,500), (30,20,700), (20,20,500), (30,40,800), (20,50,900), (50,40,900), (20,40,900), Figures

From Figures

From Figures

With

Dynamical behavior of system (

Because of distinct constitutions of individuals, some infected individuals who are not yet infectious become exposed individuals, while other infected individuals immediately become infectious. So, we establish an epidemic model with a changing delitescence between SEIS and SIS model by using the proportional number

This paper mainly considers the existence and stability of equilibriums. We use Jacobi matrix to discuss the local asymptotical stability of the endemic equilibrium, and we obtain sufficient conditions for this. When we discuss its global asymptotical stability, we use the Dulac function in its limit system (

After the discussion of the stability of the equilibriums, we have made numerical simulations of the epidemic model with different values of