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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 219173, 11 pages
Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate
1Department of Mathematics, Pusan National University, 30 Geumjeong-Gu, Busan 609-735, Republic of Korea
2School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan
Received 29 June 2013; Revised 6 November 2013; Accepted 20 November 2013; Published 6 January 2014
Academic Editor: Elena Braverman
Copyright © 2014 Kwang Sung Lee and Abid Ali Lashari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction number . Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that if , the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. If , a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists.
Pine wilt disease (PWD) is caused by the pinewood nematode Bursaphelenchus xylophilus Nickle, which is vectored by the Japanese pine sawyer beetle Monochamus alternatus. The first epidemic of PWD was recorded in 1905 in Japan . Since PWD was found in Japan, the pinewood nematode has spread to Korea, Taiwan, and China and has devastated pine forests in East Asia. Furthermore, it was also found in Portugal in 1999 . The greatest losses to pine wilt have occurred in Japan. During the 20th century, the disease spread through highly susceptible Japanese black (P. thunbergiana) and Japanese red (P. densiflora) pine forests with devastating impact. Iowa, Illinois, Missouri, Kentucky, eastern Kansas, and southeastern Nebraska have experienced heavy losses of Scots pine. Thus, PWD has become the most serious threat to forest worldwide .
Mathematical modeling is useful in understanding the process of transmission of a disease, and determining the different factors that influence the spread of the disease. In this way, different control strategies can be developed to limit the spread of infection. Lately, some mathematical models have been formulated on pest-tree dynamics, such as PWD transmission model which was investigated by Lee and Kim  and Shi and Song .
The incidence rate of the transmission of the disease plays an important role in the study of mathematical epidemiology. In classical epidemiological models, the incidence rate is assumed to be bilinear given by , where is the probability of transmission per contact rate, is susceptible, and is infective populations, respectively. However, actual data and evidence observed for many diseases show that dynamics of disease transmission are not always as simple as it is shown in these rates. In 1978, Capasso and Serio  introduced a saturated incidence rate in epidemic models where , , . This incidence rate is important because the number of effective contacts between infected and susceptible individuals may be saturated at high infective levels in order to avoid the overcrowding effect of infective individuals.
There are many papers for mathematical models with nonlinear incidence rates [7–15]. Lee and Kim  introduced a model of a pine wilt disease with nonlinear incidence rate. Their model does not include an exposed class for the host population and falls within the susceptible-infected (SI) category of models. When the pine tree has been infected by the nematode, the pine tree stopped the cessation of oleoresin exudation in 2-3 weeks. We consider the role of incubation period during disease transmission, that is, exposed pine trees , the tree has been infected by the nematode but still sustains the ability for oleoresin exudation.
In this paper, we propose a mathematical model with nonlinear incidence rates to describe the host-vector interaction between pines and pine sawyers carrying nematode by means of ordinary differential equation. The vector (beetles) population is described by a system for the susceptible and infected vector and the dynamics of the host (pine trees) are described by SEI model. The ODE model shows that the dynamics are completely determined by the basic reproduction number . If , the disease-free equilibrium is globally stable and the disease dies out. If , a unique endemic equilibrium exists and is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.
The paper is organized as follows. In Section 2, the host-vector model for pine wilt disease with nonlinear incidence rates is presented, where the dynamics of hosts and vectors are described by SEI and SI models, respectively. The stability of disease free equilibrium and the stability of endemic equilibrium are investigated in Sections 3 and 4, respectively. In Section 5, the global stability of endemic equilibrium is proved using the geometric approach method for global stability, due to Li and Muldowney . Some numerical results and conclusions are presented in Section 6.
2. Model Frame Work
This model regards Monochamus alternatus as vector and pine tree as host, and establishes the host-vector epidemic model.
The total host population at time , is divided into three subclasses of susceptible pine trees at time , ; that is, the susceptible pine trees have a potential to be infected by the nematode and can exude oleoresin which acts as a physical barrier to beetle oviposition, and beetles cannot oviposit on them. Exposed pine trees have been infected by the nematode but still sustain the ability for oleoresin exudation, and infected pine trees have been infected by the nematode and the oleoresin exudation ability have been lost and also beetles can oviposit on it. Furthermore, we assume that the class of recovered is negligible because every infectious pine tree dies within the year of infection or in the next year. The number of total host population is denoted by . And then, we assume that the total vector population at time , is split into two subclasses the number of susceptible adult beetles which does not carry pinewood nematode at time and the number of infective adult beetles which does carry pinewood nematode at time , so that total vector population is denoted by . Our model excludes the immature beetles which are in the egg stage, a pupal stage, because they do not participate in the infection cycle. The parameters in the system are as follows: the parameter is the constant increase rate of pine tree at time and is the constant emergence rate of adult beetles at time (the period of emergence). And is the natural death rate of pine tree host and is the natural death rate of beetles as vectors. The parameter is denoted by the probability that infectious beetles transmit nematode by means of contact and is the probability of having pinewood nematode when the beetle emerges out in the . And the parameter is the average number of contact per day of the vectors adult beetles during maturation feeding period. The parameter denotes the transfer rates between the exposed and the infectious.
In this model, the nonlinear incidence term denotes the rate at which the pine trees host gets infected by infectious adult beetles which do carry pinewood nematode at time , and refers to the rate at which the susceptible pine sawyers have pinewood nematode when it emerges in the infected pine trees and determine the level at which the force of infection saturates. The incidence function forms reflect a saturating effect of diseases transmission. All parameters are assumed to be positive based on some biological reasons. Thus, a host-vector epidemic model with nonlinear incidence can be described by the following system of differential equations: Considering ecological signification, we restrict our attention to the dynamics of the model in . We make some reasonable technical assumptions on the parameters of the model, namely, , , , , , , , , in . The above systems for the host population and the vector are also equipped with initial conditions as follows: , , , , and .
The total host population dynamics are given by .
The given initial conditions make sure that .
The total dynamics of vector population is given by . It is easily seen that both for the host population and for the vector population, the corresponding total population sizes are asymptotically constant such as and . This implies that in our model, we assume without loss of generality that , for all provided that , .
Proof. Consider the following Lyapunov function:
Its time derivative is
With this in mind, we can get that
Then, it follows from (5) that which implies that is a positively invariant set. On the other hand, a standard comparison theorem  can be used to show that
where and are in the initial conditions of and , respectively.
Thus, as , and one can conclude that is an attractive set.
The values of and can be determined correspondingly by , by the results of theorem . Also, we can reduce system (1) to a -dimensional system by eliminating and , respectively, in the feasible region
Therefore, from now on, we will investigate the following -dimensional nonlinear system so that the dynamics of system (1) and (7) are qualitively equivalent to the dynamics of system. It is easy to verify that all of the solutions of system (7) exist and are nonnegative. The feasible region for the system (2) is where denotes the nonnegative cone of including its lower-dimensional faces.
With respect to system (7), we have the following result.
Theorem 2. Let be the solution of the system (7) with initial conditions , , , and the closed set . Then, is positively invariant with respect to system (7) and attracting under the flow described by (7).
3. The Disease-Free Equilibrium and Its Stability
Direct calculations show that the system (7) always has the disease-free equilibrium point given by . The dynamics of the disease are described by the quantity . is the critical threshold of model (7) that is called the basic reproduction number in the epidemic model. Using Theorem 2 in , at first, the following results are established.
Theorem 3. If , the disease-free equilibrium of the model (7) is locally asymptotically stable, and is unstable if .
Proof. We linearize the system (7) around the disease-free equilibrium . The matrix of the linearization at is given by
The characteristic equation of this matrix is given by , where is the unit matrix. Expanding the determinant into a characteristic equation, we obtain the following equation, which is equivalent to
These three eigenvalues have negative real part if they satisfy the Routh-Hurwitz Criteria , such that for , with , , and . If , then According to the Routh-Hurwitz Criteria, the disease-free equilibrium of the model (7) is locally asymptotically stable.
Now, we study the global behavior of the disease-free equilibrium for system (7).
Theorem 4. If , the disease-free equilibrium of the model (7) is globally asymptotically stable in .
Proof. We construct the following Lyapunov function: where Its derivative along the solutions to the system (7) is Thus, is negative if . Furthermore, along the solution of the system and is zero if and only if , , and are zero. Also, . If , then if and only if , and in the case , if and only if or . Hence, the largest compact invariant set in when , is the singleton . By Lasalle’s Invariance Principle , then it implies that is globally asymptotically stable in .
4. The Endemic Equilibrium and Its Stability
Here, we study the existence and stability of the endemic equilibrium points. By straightforward computation, if , then the host-vector model system (7) has a unique endemic equilibrium given by in , with where
In order to investigate the stability of the endemic equilibrium, the additive compound matrices approach as in [22, 23] is used. We will linearize system (7) about an endemic equilibrium and get the following Jacobian matrix
From the Jacobian matrix , the second additive compound matrix is given by
The following lemma stated and proved in McCluskey and van den Driessche  is used to demonstrate the local stability of endemic equilibrium point .
Lemma 5. Let M be a real matrix. If , , and are all negative, then all eigenvalues of M have negative real part.
Using the above Lemma, we will study the stability of the endemic equilibrium.
Theorem 6. If , the endemic equilibrium of the model (7) is locally asymptotically stable in .
Proof. From the Jacobian matrix , we have
From (21), it is easy to see that
Computing directly the determinant of , we can get
Hence, by lemma, the endemic equilibrium of the model (7) is locally asymptotically stable in .
5. Global Stability of the Endemic Equilibrium
We now prove the global stability of the endemic equilibrium , when the reproduction number is greater than the unity. For this, first we will prove the following result.
Theorem 7. If , then system (7) is uniformly persistent; that is, there exists (independent of initial conditions), such that , , .
Proof. Let be a semidynamical system (7) in , Let be a locally compact metric space, and . The set is a compact subset of and is positively invariant set of system (7). Let be defined by and set , where is sufficiently small so that . Assume that there is a solution such that for each , we have . Let us consider the following: where is a sufficiently small constant so that . By a direct calculation, we have Let Thus, we have The above inequality (28) implies that as . However, is bounded on the set . According to Theorem 1 in , we complete the proof of Theorem 7.
Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibrium in the feasible region . We have omitted the detailed introduction of this approach and we refer the interested readers to see . For the applications of the Li and Muldowney approach to host-vector models (see [26, 27]). We summarize this approach as follows.
Consider a map from an open set to such that each solution to the differential equation is uniquely determined by the initial value . We have the following assumptions: is simply connected;there exists a compact absorbing set ;(29) has unique equilibrium in .
Let be a nonsingular matrix-valued function which is in and a vector norm on , where .
Let be the Lozinskiĭ measure with respect to the . Define a quantity as where , the matrix is obtained by replacing each entry of by its derivative in the direction of , , and is the second additive compound matrix of the Jacobian matrix of (19). The following result has been established in Li and Muldowney .
Theorem 8. Suppose that , , and hold; then the unique endemic equilibrium is globally stable in if .
We choose a suitable vector norm in and a matrix valued function
Obviously, is and nonsingular in the interior of . Linearizing system (2) about an endemic equilibrium gives the following Jacobian matrix:
The second additive compound matrix of is given by
The matrix can be written in block form as where
Consider the norm in as where denotes the vector in . The Lozinskiĭ measure with respect to this norm is defined as , where
From system (2), we can write Since is a scalar, its Lozinskiĭ measure with respect to any vector norm in will be equal to . Thus, and will become Also, . and are the operator norms of and which are mapping from to and from to , respectively, and is endowed with the norm. is the Lozinskiĭ measure of matrix with respect to norm in .Consider the following:
So, if and .
Thus, if .
Since (2) is uniformly persistent when , so for such that implies that , , , and for all . Thus, for all . The condition is satisfied provided that . Therefore, all the conditions of Theorem 7 are satisfied. Hence, unique endemic equilibrium is globally stable in .
We know that the basic reproduction number of the model is proportional to the total number of the host tree population available as oviposition sites for the vector beetles and the number of vector population and host infectious rates and vector infectious rate , respectively. The basic reproduction number does not depend on , definitely; numerical simulations indicate that when the disease is endemic, the steady state value of the exposed host , infected host decreases as m increases (see Figures 1 and 2), and the steady state value of the infective vector decreases as increases (see Figure 3). The numerical simulations are carried out using , , , , , , , , , , , , , , , , , , , and . Furthermore, from the expression of the basic reproduction number, we can observe that more effective control strategy seems to reduce the total number of infection and the rates of transmission and decrease the carrying capacity of the environment for vector beetles using conventional controls, such as aerial spraying of pesticide to kill pine sawyer adults, injection procedures and physical treatment (chipping and burning), or chemical treatment of wilt pines to kill their larvae.
This paper presents a host-vector model for pine wilt disease with nonlinear incidence rate. The mathematical analysis is carried out for a model for forest insect pests with pine wilt disease. The global dynamics of the model are shown to be determined by the basic reproduction number . More specifically, by constructing suitable Lyapunov function, we proved that if , then disease-free equilibrium is globally asymptotically stable in , and thus the disease always dies out. If , the unique endemic equilibrium exists and is globally asymptotically stable, so that the disease persists at the endemic equilibrium if it is initially present.
Conflict of Interests
The authors declared that there is no conflict of interests.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2013R1A6A3A01060805). A. A. Lashari thanks Daniah Tahir for her help in removing grammatical errors.
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