We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

Let

Incidence rate plays a very important role in the research of epidemiological models; it should generally be written as

Motivated by the work of Capasso and Serio [

For continuity of initial conditions, we require

It is easy to show that all solutions of system (

The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (

In this section, we discuss the local stability of each of nonnegative equilibria of system (

System (

The characteristic equation of system (

The characteristic equation of system (

Let

The characteristic equation of system (

Based on the discussions above, we have the following result.

For system (

if

if

In this section, we discuss the global stability of the disease-free equilibrium and the endemic equilibrium of system (

We first consider the subsystem of (

To study the global dynamics of system (

Consider the following equation:

if

if

Let

Let

From Lemma

For

For

For

For

For

For

For

Continuing this process, we derive four sequences

If

From (

In the following, we show the existence of

By Lemma

Therefore, we derive from the third equation of system (

Again, for

If

Choose

On the other hand, we derive from the first equation of system (

In this section, we give two examples to illustrate the main theoretical results above.

In system (

The numerical solution of system (

In system (

The numerical solution of system (

In this paper, we have discussed the effect of stage structure and saturation incidence rate on an SIR epidemic model with time delay. The basic reproduction number

The authors wish to thank the reviewers for their valuable comments and suggestions that greatly improved the presentation of this work. This work was supported by the National Natural Science Foundation of China (No. 10671209) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.