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This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction number

Pine wilt is a dramatic disease of pine caused by the pinewood nematode (

In [

Schematic representation of the interrelationships between the pinewood nematode,

Pine wilt disease causes significant economic losses in natural coniferous forests in Eastern Asia (especially Japan, China, and South Korea) and Western Europe (especially Portugal). As such, the pine wilt disease is among the most important pests included in the quarantine lists of many countries around the world [^{3}/year in the 1980s. Since then, it has spread to other Asian countries and regions such as China, Taiwan, and Korea, causing serious losses and economic damage. The pinewood nematode was first detected in Portugal’s Setúbal region in 1999. Immediately, several governments of the European Union prompted actions to assess the extent of the nematode’s presence and to contain

Experience from control actions in Japan included aerial spraying of insecticides to control the insect vector (the cerambycid beetle

In recent years, many attempts have been made to develop realistic mathematical models for investigating the transmission dynamics of the pine wilt disease, see [

The population of pine is divided into two classes: pine which is normal and susceptible and pine which has been infected by Bursaphelenchus xylophilus.

The pine is infected by the

The dynamic of the whole model is indeed determined by the four-dimensional system involving only the pine and

Let

The initial conditions of the system (

This paper is organized as follows. In Section

It is important to prove that the solutions of the system (

Let

All solutions

The first statement is trivial. It easily follows from the argument for reduction in the last equation that

For all infectious diseases, the basic reproduction number, sometimes called the basic reproductive rate or the basic reproductive ratio, is one of the most useful threshold parameters which characterize mathematical problems concerning infectious diseases. This metric is useful because it helps determine whether or not an infectious disease will spread through a population. In this section, we will calculate the basic reproduction number of the system (

It is easy to see that the system (

In the following, we will discuss the local and global stability of the disease-free equilibrium. From above and [

The disease-free equilibrium

The disease-free equilibrium

Define a Lyapunov function

This above result is of outmost importance because it shows that if at any time, through appropriate interventions, we are able to lower

(a) Time series diagram of the system (

In this section, we will discuss the local and global stability of the endemic equilibrium. The endemic equilibrium

Clearly, the system (

In the following, we will consider the locally asymptotical stability of the positive equilibrium when

If

Jacobian matrix of the system (

The eigenvalue problem for the Jacobian matrix (

Note that

Expressing

We have shown that

Let us consider the system of differential equations

The system (

Hirsch [

We recall additional definitions that we will use later. We first recall the basic definitions in [

We say that the system (

The following lemma is the main tool to prove the global stability of the endemic equilibrium with disease.

Assume that

In order to apply this lemma to prove the globally asymptotically stability of the endemic equilibrium, we will prove the persistence of the system (

On the boundary of

The vector field is transversal to the boundary of

To prove the second part of the position, we consider the function

By looking at the Jacobian matrix of the system (

If

Since the system (

Let

To prove that (

Function (

Therefore,

Define

Thus, we obtain

In this paper, we investigate the dynamical behavior of a pine wilt disease model that incorporates a standard incidence rate. Qualitative analysis of the model is presented. The model has two equilibria, the disease-free equilibrium and endemic equilibrium. The behavior of the system (

We obtain the basic reproduction number of the system (

By simple calculation, we can find

But it is difficult to make

From Theorem

We can also get

(a) Simulation results showing the effect of decreasing infected pine trees to parameter

This work is supported by the National Natural Science Foundation of China (11171284 and 11071011), program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN006), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (104200510011), Natural Science Foundation of the Education Department of Henan Province (2011B110028), and Key Project for the Education Department of Henan Province (13A110771).