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We study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. A numerical analysis is given to explain the analytical findings.

There have been lots of researches about SEIRS, SIRS models, in which the infectious diseases spread in a single population [

There has been a great deal of work about using mathematical models to study malaria [

Motivated by the work of [

In [

Motivated by system (

The quantities

This paper is organized as follows. In Section

In this section, we first introduce the following assumptions for system (

The initial conditions of (

If the functions

Firstly we discuss the permanence of the system (

For a continuous and bounded function

The system (

Set

We will give the following Propositions

The solution

Since the functions

From the fifth equation of system (

Last, from the third equation of system (

The solution

By Proposition

Therefore, from the first equation of system (

Set

Since

From that condition (

Similarly, since

Firstly, from the second and the fourth equation of (

We claim that it is impossible that

For

Similarly, from the third equation of system (

Let us take

Suppose that it is not true; then there exists

Thus

Finally, we will show that

Let

If

Using the second equation of system (

The solution

From the third equation of system (

Assume that

From the fourth equation of system (

Thus, the system (

In this paper, we only find the inferior limit of

Next, we will use the following lemma to discuss the extinction of the epidemic.

Consider an autonomous delay differential equation

Set

Note that

From Proposition

Next we will discuss the extinction in two cases.

If

Using the Lemma

If

In this section, we derive sufficient conditions for the global asymptotic stability of system (

System (

Assume that

If there exist

Assume that

The right-upper derivatives of

Define

Calculating the right-upper derivatives of

The right-upper derivative of

Let

Integrating (

By assumptions about

It follows from (

If there exist

Specially, if system (

If system (

To demonstrate the theoretical results obtained in this paper, we will give some numerical simulations.

Firstly, for system (

(a)–(e) show that system (

Now, we consider the general case; that is, we take account of malaria model with periodic environment, we chose the parameter values as

(a)–(e) show that system (

Next, we also consider the special case firstly; we chose the parameter values as

(a)–(e) show that the disease in system (

Now, we consider the general case; we chose the parameter values as

(a)–(e) show that the disease in system (

In this paper, we study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay, that is, (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referees for their careful reading of the original paper and their many valuable comments and suggestions that greatly improve the presentation of this work. This work is supported by Natural Science Foundation of Shanxi province (2013011002-2).