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

Authors: Kwang Sung Lee; Abid Ali Lashari
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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 numberR0. 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
R0≤1, the disease free equilibrium is globally
asymptotically stable, and the disease will be eliminated.
If R0>1, 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.


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## Body

## 1. Introduction

Pine wilt disease (PWD) is caused by the pinewood nematodeBursaphelenchus 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 byβSI, where β is the probability of transmission per contact rate, S is susceptible, and I 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 g(I)S in epidemic models where g(I)=βI/(1+αI), β>0, α>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 [7–15]. 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 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 Eh, 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 numberR0. If R0≤1, the disease-free equilibrium is globally stable and the disease dies out. If R0>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 Section2, 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 [16]. Some numerical results and conclusions are presented in Section 6.


## 2. Model Frame Work

This model regardsMonochamus alternatus as vector and pine tree as host, and establishes the host-vector epidemic model.The total host population at timet, Nh(t) is divided into three subclasses of susceptible pine trees at time t, Sh(t); 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 Eh(t) have been infected by the nematode but still sustain the ability for oleoresin exudation, and infected pine trees Ih(t) 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 Rh(t) 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 Nh(t)=Sh(t)+Eh(t)+Ih(t). And then, we assume that the total vector population at time t, Nv(t) is split into two subclasses the number of susceptible adult beetles Sv(t) which does not carry pinewood nematode at time t and the number of infective adult beetles Iv(t) which does carry pinewood nematode at time t, so that total vector population is denoted by Nv(t)=Sv(t)+Iv(t). 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 ah is the constant increase rate of pine tree at time t and bv is the constant emergence rate of adult beetles at time t (the period of emergence). And μ1 is the natural death rate of pine tree host and μ2 is the natural death rate of beetles as vectors. The parameter α is denoted by the probability that infectious beetles transmit nematode by means of contact and γ is the probability of having pinewood nematode when the beetle emerges out in the Iv(t). And the parameter ϕ is the average number of contact per day of the vectors adult beetles during maturation feeding period. The parameter β denotes the transfer rates between the exposed and the infectious.In this model, the nonlinear incidence termαϕShIv/(1+mIv) denotes the rate at which the pine trees host Sh gets infected by infectious adult beetles Iv(t) which do carry pinewood nematode at time t, and γIhSv/(1+nIh) refers to the rate at which the susceptible pine sawyers Sv have pinewood nematode when it emerges in the infected pine trees Ih and m,n 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:
(1)dShdt=ah-αϕShIv1+mIv-μ1Sh,dEhdt=αϕShIv1+mIv-(β+μ1)Eh,dIhdt=βEh-μ1Ih,dSvdt=bv-γIhSv1+nIh-μ2Sv,dIvdt=γIhSv1+nIh-μ2Iv.
Considering ecological signification, we restrict our attention to the dynamics of the model in Ω={(Sh,Eh,Ih,Sv,Iv)∈R+5∣Sh≥0,Eh≥0,Ih≥0,Sv≥0,Iv≥0}. We make some reasonable technical assumptions on the parameters of the model, namely, α>0, ϕ>0, β>0, γ>0, ah>0, bv>0, μ1>0, μ2>0, in Ω. The above systems for the host population and the vector are also equipped with initial conditions as follows: Sh(0)=Sh0, Eh(0)=Eh0, Ih(0)=Ih0, Sv(0)=Sv0, and Iv(0)=Iv0.The total host population dynamics are given bydNh/dt=ah-μ1Nh.The given initial conditions make sure thatNh(0)≥0.The total dynamics of vector population is given bydNv/dt=bv-μ2Nv. 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 limt→∞Nh(t)=ah/μ1 and limt→∞Nv(t)=bv/μ2. This implies that in our model, we assume without loss of generality that Nh(t)=ah/μ1, Nv(t)=bv/μ2 for all t≥0 provided that Sh0+Eh0+Ih0=ah/μ1, Sv0+Iv0=bv/μ2.Theorem 1.
Let(Sh,Eh,Ih,Sv,Iv) be the solution of the system (1) with initial conditions Sh(0)=Sh0, Eh(0)=Eh0, Ih(0)=Ih0, Sv(0)=Sv0, and Iv(0)=Iv0 and the compact set
(2)Ω={bvμ2(Sh,Eh,Ih,Sv,Iv)∈R+5∣0≤Sh+Eh+Ih≤ahμ1,0≤Sv+Iv≤bvμ2}.
Then, Ω is positively invariant and attracting under the flow described by (1).Proof.
Consider the following Lyapunov function:(3)V(t)=(V1(t),V2(t))=(Sh+Eh+Ih,Sv+Iv).
Its time derivative is
(4)dVdt=(dV1dt,dV2dt)=(S˙h+E˙h+I˙h,S˙v+I˙v)=(ah-μ1V1,bv-μ2V2).
With this in mind, we can get that
(5)dV1dt=ah-μ1V1≤0,forV1≥ahμ1,dV2dt=bv-μ2V2≤0,forV2≥bvμ2.
Then, it follows from (5) that dV/dt≤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
(6)0≤(V1,V2)≤(ahμ1+V1(0)e-μ1t,bvμ2+V2(0)e-μ2t),
where V1(0) and V2(0) are in the initial conditions of V1(t) and V2(t), respectively.
Thus, ast→∞, 0≤(V1,V2)≤(ah/μ1,bv/μ2) and one can conclude that Ω is an attractive set.The values ofSh and Sv can be determined correspondingly by Sv=(bv/μ2)-Iv, Sh=(ah/μ1)-Eh-Ih by the results of theorem [18]. Also, we can reduce system (1) to a 3-dimensional system by eliminating Sh and Sv, respectively, in the feasible region Ω(7)dEhdt=αϕIv1+mIv(ahμ1-Eh-Ih)-(β+μ1)Eh,dIhdt=βEh-μ1Ih,dIvdt=γIh1+nIh(bvμ2-Iv)-μ2Iv.Therefore, from now on, we will investigate the following3-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
(8)Γ={(Eh,Ih,Iv)∈R+3∣0≤Eh+Ih≤ahμ1,0≤Iv≤bvμ2,Eh≥0,Ih≥0,Iv≥0},
where R+3 denotes the nonnegative cone of R3 including its lower-dimensional faces.With respect to system (7), we have the following result.Theorem 2.
Let(Eh,Ih,Iv) be the solution of the system (7) with initial conditions Eh(0)=Eh0, Ih(0)=Ih0, Iv(0)=Iv0, and the closed set Γ. Then, Γ 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 E0=(0,0,0). The dynamics of the disease are described by the quantity R0=ahbvϕαβγ/μ12μ22(β+μ1). R0 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.
IfR0<1, the disease-free equilibrium E0 of the model (7) is locally asymptotically stable, and is unstable if R0>1.Proof.
We linearize the system (7) around the disease-free equilibrium E0. The matrix of the linearization at E0 is given by
(9)J(E0)=(-(β+μ1)0ahαϕμ1β-μ100bvβμ2-μ2).
The characteristic equation of this matrix is given by det(λI-J(E0))=0, where I is the 3×3 unit matrix. Expanding the determinant into a characteristic equation, we obtain the following equation, which is equivalent to
(10)λ3+a1λ2+a2λ+a3=0,
where
(11)a1=2μ1+μ2+β>0,a2=(μ1+μ2)(β+μ1)+μ1μ2>0,a3=μ1μ2(β+μ1)(1-R0)>0.
These three eigenvalues have negative real part if they satisfy the Routh-Hurwitz Criteria [20], such that ai>0 for i=1,2,3, with a1>0, a3>0, and a1a2>a3. If R0<1, then
(12)a1a2-a3=(2μ1+μ2+β)[(μ1+μ2)(β+μ1)+μ1μ2]-μ1μ2(β+μ1)(1-R0),=(2μ1+μ2+β)(μ1+μ2)(β+μ1)+μ1μ2(2μ1+μ2+β)-μ1μ2(β+μ1)(1-R0)=(2μ1+μ2+β)(μ1+μ2)(β+μ1)+μ1μ2(2μ1+μ2+β)-μ1μ2(β+μ1)+μ1μ2(β+μ1)R0=(2μ1+μ2+β)(μ1+μ2)(β+μ1)+μ1μ2(μ1+μ2)+μ1μ2(β+μ1)R0>0.
According to the Routh-Hurwitz Criteria, the disease-free equilibrium E0 of the model (7) is locally asymptotically stable.Now, we study the global behavior of the disease-free equilibrium for system (7).Theorem 4.
IfR0≤1, the disease-free equilibrium E0 of the model (7) is globally asymptotically stable in Γ.Proof.
We construct the following Lyapunov function:(13)V(t)=a1Eh+a2Ih+a3Iv,
where
(14)a1=bvβγμ1μ22(β+μ1),a2=bvγμ1μ22,a3=1μ2.
Its derivative along the solutions to the system (7) is
(15)V′(t)=a1Eh′+a2Ih′+a3Iv′=a1[αϕIv1+mIv(ahμ1-Eh-Ih)-(β+μ1)Eh]+a2[βEh-μ1Ih]+a3[γIh1+nIh(bvμ2-Iv)-μ2Iv]≤a1[αϕIv(ahμ1-Eh-Ih)-(β+μ1)Eh]+a2[βEh-μ1Ih]+a3[γIh(bvμ2-Iv)-μ2Iv]≤bvβγμ1μ22(β+μ1)×{αϕIv(ahμ1-Eh-Ih)-(β+μ1)Eh}+bvγμ1μ22(βEh-μ1Ih)+1u2(bvγμ2Ih-μ2Iv)=Iv{(R0-1)-bvαϕβγμ1μ22(β+μ1)(Eh+Ih)}≤0.
Thus, V′(t) is negative if R0≤1. Furthermore, V>0 along the solution of the system and is zero if and only if Eh, Iv, and Iv are zero. Also, V′≤0.   If R0≤1, then V′=0 if and only if Iv=0, and in the case R0=1, V′=0 if and only if Iv=0 or Eh=Ih=0. Hence, the largest compact invariant set in {(Eh,Ih,Iv)∈Γ∣V′(t)=0} when R0≤1, is the singleton {E0}. By Lasalle’s Invariance Principle [21], then it implies that E0 is globally asymptotically stable in Γ.


## 4. The Endemic Equilibrium and Its Stability

Here, we study the existence and stability of the endemic equilibrium points. By straightforward computation, ifR0>1, then the host-vector model system (7) has a unique endemic equilibrium given by E*=(Eh*,Ih*,Iv*) in Γ, with
(16)Eh*=μ1βIh*,Ih*=μ1μ22(R0-1)μ1(μ2w+mγbv)+αϕγbv,Iv*=γbvIh*μ2(wIh*+μ2),
where
(17)w=γ+nμ2.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 E* and get the following Jacobian matrix(18)J(E*)=(-αϕIv*1+mIv*-(β+μ1)-αϕIv*1+mIv*αϕ((ah/μ1)-Eh*-Ih*)(1+mIv*)2β-μ100γ((b/μ2)-Iv*)(1+nIh*)2-γIh*1+nIh*-μ2).From the Jacobian matrixJ(E*), the second additive compound matrix is given by(19)J[2](E*)=(-αϕIv*1+mIv*-(β+2μ1)0-αϕ((ah/μ1)-Eh*-Ih*)(1+mIv*)2γ((b/μ2)-Iv*)(1+nIh*)2-αϕIv*1+mIv*-γIh*1+nIh*-β-μ1-μ2-αϕIv*1+mIv*0β-γIh*1+nIh*-μ1-μ2).The following lemma stated and proved in McCluskey and van den Driessche [24] is used to demonstrate the local stability of endemic equilibrium point E*.Lemma 5.
Let M be a3×3 real matrix. If tr(M), det(M), and det(M[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.
IfR0>1, the endemic equilibrium E* of the model (7) is locally asymptotically stable in Γ.Proof.
From the Jacobian matrixJ(E*), we have
(20)tr(M)=-(αϕIv*1+mIv*+(β+μ1)+μ1+γIh*1+nIh*+μ2)<0.
Because(21)ahμ1-Eh*-Ih*=(β+μ1)(1+mIv*)Eh*αϕIv*,Eh*=μ1βIh*,bvμ2-Iv*=(1+nIh*)μ2Iv*γIh*.
From (21), it is easy to see that
(22)αϕ((ah/μ1)-Eh*-Ih*)(1+mIv*)2×γ((b/μ2)-Iv*)(1+nIh*)2=μ1μ2(β+μ1)(1+mIv)(1+nIh)β.
Thus,(23)det(J(E*))=|-αϕIv*1+mIv*-(β+μ1)-αϕIv*1+mIv*-αϕ((ah/μ1)-Eh*-Ih*)(1+mIv*)2β-μ100γ((b/μ2)-Iv*)(1+nIh*)2-γIh*1+nIh*-μ2|=-(αϕIv*1+mIv*+β+μ1)(μ1μ2+μ1γIh*1+nIh*)-β[αϕIv*1+mIv*(γIh*1+nIh*+μ2)-μ1μ2(β+μ1)(1+mIv*)(1+nIh)β]=-(αϕIv*1+mIv*+β+μ1)(μ1μ2+μ1γIh*1+nIh*)-αβϕIv*1+mIv*(γIh*1+nIh*+μ2)+μ1μ2(β+μ1)(1+mIv*)(1+nIh*)=-μ1γIh*1+nIh*(αϕIv*1+mIv*+β+μ1)-μ1μ2αϕIv*1+mIv*-μ1μ2(β+μ1)-αβϕIv*1+mIv*(γIh*1+nIh*+μ2)+μ1μ2(β+μ1)(1+mIv*)(1+nIh*)=-μ1γIh*1+nIh*(αϕIv*1+mIv*+β+μ1)-μ1μ2αϕIv*1+mIv*-αϕIv*1+mIv*(γIh*1+nIh*)+μ1μ2(β+μ1)(1(1+mIv*)(1+nIh*)-1)<0.
Computing directly the determinant ofJ[2](E*), we can get(24)det(J[2](E*))=|-αϕIv*1+mIv*-(β+2μ1)0-αϕ((ah/μ1)-Eh*-Ih*)(1+mIv*)2γ((bv/μ2)-Iv*)(1+nIh*)2-αϕIv*1+mIv*-γIh*1+nIh*-β-μ1-μ2-αϕIv*1+mIv*0β-γIh*1+nIh*-μ1-μ2|=-(αϕIv*1+mIv*+β+2μ1)[(αϕIv*1+mIv*+γIh*1+nIh*+β+μ1+μ2)(γIh*1+nIh*+μ1+μ2)+αβϕIv*1+mIv*]-γ((bv/μ2)-Iv*)(1+nIh*)2(αβϕ((ah/μ1)-Eh*-Ih*)(1+mIv*)2)<0.
Hence, by lemma, the endemic equilibriumE* of the model (7) is locally asymptotically stable in Γ.


## 5. Global Stability of the Endemic Equilibrium

We now prove the global stability of the endemic equilibriumE*, when the reproduction number R0 is greater than the unity. For this, first we will prove the following result.Theorem 7.
IfR0>1, then system (7) is uniformly persistent; that is, there exists c>0 (independent of initial conditions), such that liminft→∞Eh(t)≥c, liminft→∞Ih(t)≥c, liminft→∞Iv(t)≥c.Proof.
Letπ be a semidynamical system (7) in (R0+)3, Let χ be a locally compact metric space, and Γ0={(Eh,Ih,Iv)∈Γ∣Iv=0}. The set Γ0 is a compact subset of Γ and Γ/Γ0 is positively invariant set of system (7). Let P:χ→R0+ be defined by P(Eh,Ih,Iv)=Iv and set S={(Eh,Ih,Iv)∈Γ∣P(Eh,Ih,Iv)<ρ}, where ρ is sufficiently small so that R0(1-(μ2/bv)ρ)/(1+nρ)>1. Assume that there is a solution x∈S such that for each t>0, we have P(π(x,t))<P(x)<ρ. Let us consider the following:
(25)L(t)=ahαβϕ(1-δ*)μ1μ2(β+μ1)Iv+Ih,
where δ*>0 is a sufficiently small constant so that R0(1-(μ2/bv)ρ)(1-δ*)/(1+nρ)>1. By a direct calculation, we have
(26)L′(t)≥μ1[ahbvαβγϕ(1-δ*)(1-(μ2/bv)ρ)μ12μ22(β+μ1)(1+nρ)-1]Ih+ahαβϕδ*μ1Iv.
Let
(27)δ=min{[ahbvαβγϕ(1-δ*)(1-μ2bvρ)μ12μ22(β+μ1)(1+nρ)-1]μ1[ahbvαβγϕ(1-δ*)(1-(μ2/bv)ρ)μ12μ22(β+μ1)(1+nρ)-1],μ2(β+μ1)δ*1-δ*[ahbvαβγϕ(1-δ*)(1-μ2bvρ)μ12μ22(β+μ1)(1+nρ)-1]}>0.
Thus, we have
(28)L′(t)≥δL(t).
The above inequality (28) implies that L(t)→∞ as t→∞. However, L(t) is bounded on the set Γ. According to Theorem 1 in [25], 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 equilibriumE* 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 aC1 map f:x↦f(x) from an open set D⊂Rn to Rn such that each solution x(t,x0) to the differential equation
(29)x′=f(x)
is uniquely determined by the initial value x(0,x0). We have the following assumptions:(H1)
D is simply connected;(H2)
there exists a compact absorbing setK⊂D;(H3)
(29) has unique equilibrium x¯ in D.LetP:x↦P(x) be a nonsingular (n2)×(n2) matrix-valued function which is C1 in D and a vector norm |·| on RN, where N=(n2).Letμ be the Lozinskiĭ measure with respect to the |·|. Define a quantity q¯2 as
(30)q¯2=limsupt→∞supx0∈K1t∫0t‍μ(B(x(s,x0)))ds,
where B=PfP-1+PJ[2]P-1, the matrix Pf is obtained by replacing each entry p of P by its derivative in the direction of f, (pij)f, and J[2] is the second additive compound matrix of the Jacobian matrix J of (19). The following result has been established in Li and Muldowney [16].Theorem 8.
Suppose that(H1), (H2), and (H3) hold; then the unique endemic equilibrium E* is globally stable in Ω if  q¯2<0.We choose a suitable vector norm|·| in R3 and a 3×3 matrix valued function
(31)P(x)=(1000EhIv000EhIv).Obviously,P is C1 and nonsingular in the interior of Ω. Linearizing system (2) about an endemic equilibrium E* gives the following Jacobian matrix:(32)J=(-αϕIv1+mIv-(β+μ1)-αϕIv1+mIvαϕ(1+mIv)2(ahμ1-Eh-Ih)β-μ100γ(1+nIh)2(bvμ2-Iv)-γIh1+nIh-μ2).The second additive compound matrix ofJ(E*) is given by(33)J[2]=(-αϕIv1+mIv-(β+μ1)-μ10-αϕ(1+mIv)2(ahμ1-Eh-Ih)γ(1+nIh)2(bvμ2-Iv)-αϕIv1+mIv-(β+μ1)-γIh1+nIh-μ2-αϕIv1+mIv0β-μ1-γIh1+nIh-μ2).The matrixB=PfP-1+PJ[2]P-1 can be written in block form as
(34)B=(B11B12B21B22),
where(35)B11=-αϕIv1+mIv-(β+μ1)-μ1,B12=(0-αϕ(1+mIv)2IvEh(ahμ1-Eh-Ih)),B21=(γ(1+nIh)2(bvμ2-Iv)EhIv0),B22=(Eh′Eh-Iv′Iv-αϕIv1+mIv-(β+μ1)-γIh1+nIh-μ2-αϕIv1+mIvβEh′Eh-Iv′Iv-μ1-γIh1+nIh-μ2).Consider the norm inR3 as |(u,v,w)|=max(|u|,|v|+|w|) where (u,v,w) denotes the vector in R3. The Lozinskiĭ measure with respect to this norm is defined as μ(B)≤sup(g1,g2), where
(36)g1=μ1(B11)+|B12|,g2=μ1(B22)+|B21|.From system (2), we can write
(37)Eh′Eh=αϕ1+mIvIvEh(ahμ1-Eh-Ih)-(β+μ1),Ih′Ih=βEhIh-μ1,Iv′Iv=γ1+nIhIhIv(bvμ2-Iv)-μ2.
Since B11 is a scalar, its Lozinskiĭ measure with respect to any vector norm in R1 will be equal to B11. Thus,
(38)B11=-αϕIv1+mIv-(β+μ1)-μ1,|B12|=αϕ(1+mIv)2IvEh(ahμ1-Eh-Ih),
and g1 will become
(39)g1=-αϕIv1+mIv-(β+μ1)-μ1+αϕ(1+mIv)2IvEh(ahμ1-Eh-Ih)≤-αϕIv1+mIv-μ1+αϕ(1+mIv)IvEh(ahμ1-Eh-Ih)-(β+μ1)≤Eh′Eh-αϕIv1+mIv-μ1.
Also, |B21|=(γ/(1+nIh)2)((bv/μ2)-Iv)(Eh/Iv). |B12| and |B21| are the operator norms of B12 and B21 which are mapping from R2 to R and from R to R2, respectively, and R2 is endowed with the l1 norm. μ1(B22) is the Lozinskiĭ measure of 2×2 matrix B22 with respect to l1 norm in R2.Consider the following:
(40)μ(B22)=sup{Eh′Eh-Iv′Iv-αϕIv1+mIv-μ1-γIh1+nIh-μ2,Eh′Eh-Iv′Iv+αϕIv1+mIv-μ1-γIh1+nIh-μ2},=Eh′Eh-Iv′Iv+αϕIv1+mIv-μ1-γIh1+nIh-μ2.Hence,(41)g2=Eh′Eh-Iv′Iv+αϕIv1+mIv-μ1-γIh1+nIh-μ2+γ(1+nIh)2(bvμ2-Iv)EhIv=Eh′Eh-Iv′Iv+αϕIv1+mIv-μ1-γIh1+nIh+γ(1+nIh)2(bvμ2-Iv)EhIv-μ2=Eh′Eh-Iv′Iv+αϕIv1+mIv-μ1-γIh1+nIh+γ(1+nIh)2(bvμ2-Iv)EhIv+Iv′Iv-γ1+nIh(bvμ2-Iv)IhIv=Eh′Eh+αϕIv1+mIv-μ1-γIh1+nIh+γ(1+nIh)2(bvμ2-Iv)EhIv-γ1+nIh(bvμ2-Iv)IhIv=Eh′Eh+αϕIv1+mIv-μ1-γIh1+nIh+γ(1+nIh)2EhIvbvμ2-γ(1+nIh)2Eh-γ1+nIhIhIvbvμ2+γ1+nIhIh=Eh′Eh+αϕIv1+mIv-μ1+γ(1+nIh)2EhIvbvμ2-γ(1+nIh)Eh-γ1+nIhIhIvbvμ2≤Eh′Eh+αϕIv1+mIv-μ1+γ(1+nIh)2EhIvbvμ2-γ(1+nIh)2Eh-γ(1+nIh)2IhIvbvμ2≤Eh′Eh+αϕIv1+mIv-μ1+γ(1+nIh)2ahμ1Ivbvμ2-γ(1+nIh)2Eh-γ(1+nIh)2cIvbvμ2.So,(42)g2≤Eh′Eh+αϕIv1+mIv-μ1+γ(1+nIh)2Ivbvμ2(ahμ1-c)-γ(1+nIh)2Eh<Eh′Eh+αϕIv1+mIv-μ1<Eh′Eh+αϕIvm-μ1<Eh′Eh+αϕMm-μ1,
if μ1>ah/c and M=bv/μ2.Thus,(43)μ(B)=sup{g1,g2}≤Eh′Eh-μ12,
if μ1>2αϕM/m.Since (2) is uniformly persistent when R0>1, so for T>0 such that t>T implies that Eh(t)≥c, Ih(t)≥c, Iv(t)≥c, and (1/t)logIh(t)<μ1/4 for all (Sh(0),Ih(0),Iv(0))∈K. Thus,
(44)1t∫0t‍μ(B)dt<logEh(t)t-μ12<-μ14,
for all (Sh(0),Ih(0),Iv(0))∈K. The condition (H3) is satisfied provided that μ1>max{ah/c,2αϕM/m}. Therefore, all the conditions of Theorem 7 are satisfied. Hence, unique endemic equilibrium E* is globally stable in Ω.


## 6. Discussion

We know that the basic reproduction number of the modelR0 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 α and vector infectious rate β, respectively. The basic reproduction number R0 does not depend on m, n definitely; numerical simulations indicate that when the disease is endemic, the steady state value of the exposed host Eh*, infected host Ih* decreases as m increases (see Figures 1 and 2), and the steady state value of the infective vector Iv* decreases as n increases (see Figure 3). The numerical simulations are carried out using Sh(0)=300, Eh(0)=30, Ih(0)=20, Sv(0)=65, Iv(0)=20, ah=0.009041, bv=0.002691, α=0.00166, ϕ=0.2, β=0.057142, γ=0.00305, μ1=0.0000301, μ2=0.011764, m1=0.01, m2=0.03, m3=0.07, m4=0.09, n=0.01, n1=0.02, and n2=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.Figure 1
Plot of the exposed host population.Figure 2
Plot of the infected host population.Figure 3
Plot of the case infected vector population.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 numberR0. More specifically, by constructing suitable Lyapunov function, we proved that if R0≤1, then disease-free equilibrium E0 is globally asymptotically stable in Γ, and thus the disease always dies out. If R0>1, the unique endemic equilibrium E* exists and is globally asymptotically stable, so that the disease persists at the endemic equilibrium if it is initially present.



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*Source: 219173-2014-01-06.xml*