Pine wilt disease (PWD) is a dramatic disease of pine caused by the pinewood nematode (Bursaphelenchus xylophilus), which constitutes a major threat to forest ecosystems worldwide, from both the economical point of view and the environmental perspective  - . PWD is vector-born disease which is transmitted by pine sawyer beetle. The healthy host pine trees are usually killed within a few months by nematodes . The first visible symptom is lack of resin exudation of bark wounds. The foliage becomes light grayish green, then becomes yellow, and finally it becomes reddish brown. The pine trees affected by pine wilt disease totally lack resin and wood becomes dry . During the 20th century, the greatest losses have occurred due to pine wilt disease and it had a devastating impact, spreading among highly susceptible Japanese Black Pine and Japanese red pine forest. In addition, it spread from Japan to neighboring East Asian countries such as China and Korea in 1982 and 1988, respectively, and it was detected in Portugal in 1999 . Pine wilt disease is a grave threat to forests worldwide and will potentially become a global epidemic.
Mathematical models play an important role in understanding the epidemiology of pine wilt diseases, control disease, and gain a deeper insight into the disease transmission dynamics in a population. There are many authors that establish mathematical models to describe the transmission of plant disease. In , Lee introduced a mathematical model on pine wilt disease and discussed the stability of pine wilt disease with optimal control. X. Y. Shi and G. H. Song  formulated a four-dimensional system involving only the pine andMonochamus alternatus and analyzed global dynamics of the model. The model is as follows:
Lee and Lashari introduced a mathematical model that incorporated the exposed class in the pine tree population and made a detailed discussion on the stability and optimal control. Ozair presented a mathematical model of the dynamics of pine wilt disease by dividing the host pine trees and vector beetles into susceptible and infected classes with nonlinear incidence and horizontal transmission . The classical epidemiological models are developed by the assumption of bilinear incidence rate and the standard incidence rate . Capasso and Serio  introduced a saturated incidence rate in an epidemic model. Liu   investigated nonlinear incidence rates of the form , and proposed a Holling-type nonlinear incidence function given by . A variety of nonlinear incidence rates has been utilized in epidemic models  - . In the real world, farmers’ experiences include replanting, and rouging diseased plants to control the disease spread. What’s more, the disease transmission is infected by season and temperature, et al. To address the time-varying impulse of dynamic model with general nonlinear, incidence rate has been analyzed.
The paper is organized as follows. In the next section, an impulsive epidemic model with general nonlinear incidence rate is formulated. In Sections 3 and 4, we proved the sufficient condition for the global attractivity of the disease-free equilibrium and the permanence of the model, respectively.
2. Model Formulation and Preliminary
In this section, we introduced a more advanced model that the pine trees population is subdivided into three groups: susceptible pine trees , asymptomatic carrier pine trees and infected pine trees . We assume that roguing asymptomatic carrier pine trees and infected pine trees at the same time. Additionally, we consider the disease mortality in the symptomatic pine trees population and assume that roguing asymptomatic and symptomatic group at different rate. Motivated by the above reasons, we build the epidemic model with general nonlinear incidence rate and time-varying impulse as follows:
The following assumptions are satisfied for the model.
(H1) , and are left continuous for , that is , , .
(H2) There is a maximum pine trees population size . Replant the pine trees population at a rate .
(H3) denotes the natural death rate of pine trees, is the disease induced death rate.
(H4) is symptomatic pine wilt disease at fixed time , and .
(H5) is pulse time. There exist a positive integer q and a positive such that for all .
(H6) and are both piecewise continuous and positive -periodic functions, which means incidence rates about group A and I, respectively. The form of it is as follows:
for all nonnegative integer n, and , for .
3. The Disease-Free Periodic Solution Is Global Attractive
From system (2), we can easily obtain that is the disease-free periodic solution. To analyze the global attractiveness of the disease-free periodic solution, we give the hypothesis as follows:
(B1) There exist two piecewise continuous and positive -periodic function , that is , for , such that , for all .
It’s easy to see that is the disease-free periodic solution of system (1).
Theorem 1. If system (2) satisfies and the (B1), then the disease-free periodic solution is globally attractive.
Proof: Let be any solution of system (2), since , we have
where is a sufficiently small number.
From the first equation of (2), we have . By the comparison theorem, for arbitrarily small number , there exists a positive constant , such that
Therefore, let , we can obtain that .
Next, from (5) and the second equation of system (2), for and , we have
where is same as (3). Thus
where is same as (3).
By the similar method, we get that for .
Especially, we obtain that
Then, for an any positive integer s, we obtain
It follows from (4) and (8), we have that
From (7) and (8), we obtain that
that is, for any arbitrarily small number , there exists , such that for . Then, from the first equation of system (2), we know that
for , where . Solving the differential inequality, we can obtain
Combining (5) with (12), we get that
Because is an arbitrary positive small number, (12) means that
By (10) and (13), we know that the disease-free periodic solution is global attractive.
In this section, the sufficient conditions of system (2) for the permanence are obtained. At firstly, we give the hypothesis (B1) as following:
(B2) There exist two piecewise continuous and positive -periodic functions , such that , , for all .
Theorem 2. The system (1) is permanence, when and the assumptions (B1) and (B2) hold.
Proof: We have , there exists an arbitrarily small number such that
For proving the conclusion, we prove the disease is uniformly weakly persistent, firstly. There exists a constant , such that . By contradiction, for , there exists such that for .
In view of the hypothesis (B1) and the first equation of model (2), we have
for all , where is defined in (10). Consider the following auxiliary system:
is the solution of system (16), it is easy to know that
as . By the comparison theorem, we get . Therefore, for any arbitrarily small , there have an integer , such that
For above and a positive integer , we have . Therefore, for all , according to the second equation of system (2) and (17), we get
where is same as (14). Furthermore, from the system (2), we obtain
where can be seen in (14).
Then, we consider the following auxiliary impulsive system:
By solving the impulsive differential equation, we infer that for ,
Thus, from (15) and (19), we get
that is to say, as , we have . By the comparison theorem, we have , which is a contradiction to . Thus the claim is proved.
By the claim, we need to consider the following two possible cases. The proven method is same as to .
Case 1. for all large t;
Case 2. oscillates about for all large t.
Obviously, the conclusion is evident in Case 1. For Case 2, set and be large enough such that
There are two possible subcases for .
Subcase (I). If (n is a nonnegative integer and ), then and , where is same as (18). It’s easy to know that there have a positive constant m, such that , for all . Then, we consider two possibilities in the size of and .
1) If , where is same as (17), then from system (2), we have
From (20), we get
for all .
2) If , in view of the discussion in (i), we have for all . Next, we show that for all . Otherwise, there exists a constant such that , for all , and , for .
Next, we consider two possibilities separately:
a) For any , .
Obviously, system (18) holds on . Then, we can choose a proper , such that . Using the comparison theorem, we have
In addition, (15) implies that
Then, we have
Then , for , which is a contradiction. Therefore, for all .
b) We have a such that . The proof of (b) is same as to (a), we omit it.
Subcase (II). If for any , then . Using the same methods of subcase (I), it is easy to get a positive constant m, such that , for all .
Thus, we get that for all . Since the interval is chosen in an arbitrary way, we have that for all large t.
From the above analysis, m is independent of the positive solution of model (2), and we have proved that the solution of system (2) satisfies for sufficiently large t, that is to say, . We can easily see that, there exists a positive constant , such that . Therefore, the permanence of system (2) is proved.
In this paper, we studied a pine wilt disease epidemic model with general nonlinear incidence rate and pulse rouging strategy. According to the above analysis of the systems, we know that the disease-free equilibrium is globally attractive determined by the basic reproduction number , and in such a case, the endemic equilibrium does not exist. In Section 4, we proved that the disease uniformly persists only if .
The research has been supported by the Science and Technology Plan Projects of Jiangxi Provincial Education Department (GJJ191443, GJJ181359), Guidance Project of Ji’an Science and Technology Bureau.
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