Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate

and Applied Analysis 3 Theorem 1. Let (S h , E h , I h , SV, IV) be the solution of the system (1) with initial conditions S h (0) = S 0 h , E h (0) = E 0 h , I h (0) = I 0 h , SV(0) = S 0 V , and IV(0) = I 0 V and the compact set Ω = { (S h , E h , I h , SV, IV) ∈ R 5 + | 0 ≤ S h + E h + I h ≤ a h μ 1 , 0 ≤ SV + IV ≤ bV μ 2 } . (2) Then, Ω is positively invariant and attracting under the flow described by (1). Proof. Consider the following Lyapunov function: V (t) = (V 1 (t) , V 2 (t)) = (S h + E h + I h , SV + IV) . (3) Its time derivative is


Introduction
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 [1].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 [2].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 [3].
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 [4] and Shi and Song [5].
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 [6] introduced a saturated incidence rate () in epidemic models where () = /(1 + ),  > 0,  > 0. 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][8][9][10][11][12][13][14][15].Lee and Kim [4] 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 2 Abstract and Applied Analysis 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  ℎ , 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  0 .If  0 ≤ 1, the disease-free equilibrium is globally stable and the disease dies out.If  0 > 1, 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 hostvector 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 [16].Some numerical results and conclusions are presented in Section 6.

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 ,  ℎ () is divided into three subclasses of susceptible pine trees at time ,  ℎ (); 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  ℎ () have been infected by the nematode but still sustain the ability for oleoresin exudation, and infected pine trees  ℎ () 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  ℎ () 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  ℎ () =  ℎ () +  ℎ () +  ℎ ().And then, we assume that the total vector population at time ,  V () is split into two subclasses the number of susceptible adult beetles  V () which does not carry pinewood nematode at time  and the number of infective adult beetles  V () which does carry pinewood nematode at time , so that total vector population is denoted by  V () =  V () +  V ().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  ℎ is the constant increase rate of pine tree at time  and  V is the constant emergence rate of adult beetles at time  (the period of emergence).And  1 is the natural death rate of pine tree host and  2 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  V ().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  ℎ  V /(1 +  V ) denotes the rate at which the pine trees host  ℎ gets infected by infectious adult beetles  V () which do carry pinewood nematode at time , and  ℎ  V /(1+ ℎ ) refers to the rate at which the susceptible pine sawyers  V have pinewood nematode when it emerges in the infected pine trees  ℎ 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,  > 0,  > 0,  > 0,  > 0,  ℎ > 0,  V > 0,  1 > 0,  2 > 0, in Ω.The above systems for the host population and the vector are also equipped with initial conditions as follows: V , and  V (0) =  0 V .The total host population dynamics are given by  ℎ / =  ℎ −  1  ℎ .
The given initial conditions make sure that  ℎ (0) ≥ 0. The total dynamics of vector population is given by  V / =  V −  2  V .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 lim  → ∞  ℎ () =  ℎ / 1 and lim  → ∞  V () =  V / 2 .This implies that in our model, we assume without loss of generality that V , and  V (0) =  0 V and the compact set ( Then, Ω is positively invariant and attracting under the flow described by (1).
Proof.Consider the following Lyapunov function: Its time derivative is With this in mind, we can get that Then, it follows from (5) that / ≤ 0 which implies that Ω is a positively invariant set.On the other hand, a standard comparison theorem [17] can be used to show that where  1 (0) and  2 (0) are in the initial conditions of  1 () and  2 (), respectively.Thus, as  → ∞, 0 ≤ ( 1 ,  2 ) ≤ ( ℎ / 1 ,  V / 2 ) and one can conclude that Ω is an attractive set.
The values of  ℎ and  V can be determined correspondingly by by the results of theorem [18].Also, we can reduce system (1) to a 3-dimensional system by eliminating  ℎ and  V , respectively, in the feasible region Therefore, from now on, we will investigate the following 3-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  3 + denotes the nonnegative cone of  3 including its lower-dimensional faces.
With respect to system (7), we have the following result.

The Disease-Free Equilibrium and Its Stability
Direct calculations show that the system (7) always has the disease-free equilibrium point given by  0 = (0, 0, 0).The dynamics of the disease are described by the quantity .  0 is the critical threshold of model ( 7) that is called the basic reproduction number in the epidemic model.Using Theorem 2 in [19], at first, the following results are established.Theorem 3. If  0 < 1, the disease-free equilibrium  0 of the model (7) is locally asymptotically stable, and is unstable if  0 > 1.
Proof.We linearize the system (7) around the disease-free equilibrium  0 .The matrix of the linearization at  0 is given by The characteristic equation of this matrix is given by det( − ( 0 )) = 0, where  is the 3 × 3 unit matrix.Expanding the determinant into a characteristic equation, we obtain the following equation, which is equivalent to where These three eigenvalues have negative real part if they satisfy the Routh-Hurwitz Criteria [20], such that   > 0 for  = 1, 2, 3, with  1 > 0,  3 > 0, and According to the Routh-Hurwitz Criteria, the disease-free equilibrium  0 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  0 ≤ 1, the disease-free equilibrium  0 of the model (7) is globally asymptotically stable in Γ.
Proof.We construct the following Lyapunov function: where Its derivative along the solutions to the system (7) is Thus,   () is negative if  0 ≤ 1.Furthermore,  > 0 along the solution of the system and is zero if and only if  ℎ ,  V , and  V are zero.Also,   ≤ 0. If  0 ≤ 1, then   = 0 if and only if  V = 0, and in the case . By Lasalle's Invariance Principle [21], then it implies that  0 is globally asymptotically stable in Γ.

The Endemic Equilibrium and Its Stability
Here, we study the existence and stability of the endemic equilibrium points.By straightforward computation, if  0 > 1, then the host-vector model system (7) has a unique endemic equilibrium given by  * = ( * ℎ ,  * ℎ ,  * V ) in Γ, 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 J [2] (E * ) = ( ( ( ) .
The following lemma stated and proved in McCluskey and van den Driessche [24] is used to demonstrate the local stability of endemic equilibrium point  * .
Lemma 5. Let M be a 3 × 3 real matrix.If tr(), det(), and det( [2] ) 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  0 > 1, the endemic equilibrium  * of the model (7) is locally asymptotically stable in Γ.
Proof.From the Jacobian matrix ( * ), we have tr Because From (21), it is easy to see that Thus, Computing directly the determinant of  [2] ( * ), we can get det ( [2] ( * )) Hence, by lemma, the endemic equilibrium  * of the model ( 7) is locally asymptotically stable in Γ.

Global Stability of the Endemic Equilibrium
We now prove the global stability of the endemic equilibrium  * , when the reproduction number  0 is greater than the unity.For this, first we will prove the following result.
Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibrium  * in the feasible region Ω.We have omitted the detailed introduction of this approach and we refer the interested readers to see [16].For the applications of the Li and Muldowney approach to host-vector models (see [26,27]).We summarize this approach as follows.
Consider a  1 map  :   → () from an open set  ⊂   to   such that each solution (,  0 ) to the differential equation is uniquely determined by the initial value (0,  0 ).We have the following assumptions: ( 1 )  is simply connected; ( 2 ) there exists a compact absorbing set  ⊂ ; ( 3 ) (29) has unique equilibrium  in .
Let  be the Lozinskiȋ measure with respect to the | ⋅ |.Define a quantity  2 as where  =    −1 +  [2]  −1 , the matrix   is obtained by replacing each entry  of  by its derivative in the direction of , (  )  , and  [2] is the second additive compound matrix of the Jacobian matrix  of (19).The following result has been established in Li and Muldowney [16].
We choose a suitable vector norm | ⋅ | in  3 and a 3 × 3 matrix valued function Obviously,  is  1 and nonsingular in the interior of Ω. Linearizing system (2) about an endemic equilibrium  * gives the following Jacobian matrix: The second additive compound matrix of ( * ) is given by  [2]   = ( ( ( ( ) .

Discussion
We know that the basic reproduction number of the model  0 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  0 does not depend on ,  definitely; numerical simulations indicate that when the disease is endemic, the steady state value of the exposed host  * ℎ , infected host  * ℎ decreases as m increases (see Figures 1 and 2), and the steady state value of the infective vector  * V decreases as  increases (see Figure 3).The numerical simulations are carried out using  ℎ (0) = 300,  ℎ (0) = 30,  ℎ (0) = 20,  V (0) = 65,  V (0) = 20,  ℎ = 0.009041,  V = 0.002691,  = 0.00166,  = 0.2,  = 0.057142,  = 0.00305,  1 = 0.0000301,  2 = 0.011764,  1 = 0.01,  2 = 0.03,  3 = 0.07,  4 = 0.09,  = 0.01,  1 = 0.02, and  2 = 0.03.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  0 .More specifically, by constructing suitable Lyapunov function, we proved that if  0 ≤ 1, then disease-free equilibrium  0 is globally asymptotically stable in Γ, and thus the disease always dies out.If  0 > 1, the unique endemic equilibrium  * exists and is globally asymptotically stable, so that the disease persists at the endemic equilibrium if it is initially present.

Figure 1 :Figure 2 :
Figure 1: Plot of the exposed host population.