Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 219173, 11 pages http://dx.doi.org/10.1155/2014/219173

Research Article

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

Kwang Sung Lee1 and Abid Ali Lashari2

1 Department of Mathematics, Pusan National University, 30 Geumjeong-Gu, Busan 609-735, Republic of Korea

2 School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan

Correspondence should be addressed to Kwang Sung Lee; ksl@pusan.ac.kr

Received 29 June 2013; Revised 6 November 2013; Accepted 20 November 2013; Published 6 January 2014 Academic Editor: Elena Braverman

Copyright © 2014 K. S. Lee and A. A. 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 R0. 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.

1. 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 tobe bilinear given by pSI, where p 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) = + al), p > 0, a > 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 number R0.IfR0 < 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 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 [16]. 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 t, 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 ^ 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 pi1 is the natural death rate of pine tree host and ^2 is the natural death rate of beetles as vectors. The parameter a is denoted by the probability that infectious beetles transmit nematode by means of contact and y is the probability of having pinewood nematode when the beetle emerges out in the Iv(t). And the parameter <p is the average number of contact per day of the vectors adult beetles during maturation feeding period. The parameter ¡3 denotes the transfer rates between the exposed and the infectious.

In this model, the nonlinear incidence term a<pShIvl( \ + mlv) 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 yIhSvl(l+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:

dSh_ a$ShIv

= Uh - i . ...T - Plbh>

dt h 1 + ml, dEh wpShl,

1 + ml,

f =ßEh

dSv -U _ y}hsv _ ,, c dt ~bv 1 + nlh v

dk YIhSv

dt 1 + nl

- Vik-

Considering ecological signification, we restrict our attention to the dynamics of the model in Q = {(Sh, Eh, Ih, Sv, Iv) e R+ | Sh > 0, Eh > 0,Ih > 0, Sv > 0,IV > 0}. We make some reasonable technical assumptions on the parameters of the model, namely, a > 0, $ > 0, p > 0, y > 0, ah > 0, bv > 0, > 0, ^2 > 0, in Q. The above systems for the host population and the vector are also equipped with initial conditions as follows: Sh(0) = S0h, Eh(0) = E0h, Ih(0) = I°h, Sv(0) = S°v,andIv(0) = fv.

The total host population dynamics are given by dNh/dt = ah - p1Nh.

The given initial conditions make sure that N(0) > 0.

The total dynamics of vector population is given by dNv/dt = bv - p2Nv. 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^mNh(t) = ajpi and limt^mNv(f) = bj^. This implies that in our model, we assume without loss of generality that Nh(t) = ah/^1,Nv(t) = bvl^2 for all t > 0 provided that S0h + E0h + I0h = aj^, S0V + 10 = bj^.

Theorem 1. Let (Sh, Eh, Ih, SV, IV) be the solution ofthe system (1) with initial conditions Sh(0) = S°h, Eh(0) = E°h, Ih(0) = I0, SV(0) = SV, and IV(0) = I and the compact set

Cl={(Sh,Eh,Ih,Sv,Iv)

eR5+ \0<Sh + Eh + Ih ,0<SV + IV

Then, Q is positively invariant and attracting under the flow described by (1).

Proof. Consider the following Lyapunov function:

V (t) = (V1 (t), V2 (t)) = (Sh + Eh + Ih, Sv + Iv). (3) Its time derivative is

c£/ _ (dV^ dVVi dt V dt dt

_ (Sh + Eh + h> sv + K) _ (ah - PiVlA - P2V2) ■

With this in mind, we can get that

^ _flh-PiVi < 0, for Vi , dt fa

f -,2V2 <°, forV2

Then, it follows from (5) that dV/dt < 0 which implies that Q is a positively invariant set. On the other hand, a standard comparison theorem [17] can be used to show that

0 < (V,, V2)<( ah + V (0) e-lilt, b + V2 (0) e-liA , (6)

where V1(0) and V2(0) are in the initial conditions of V1(t) and V2(t), respectively.

Thus, as t ^ >x>, 0 < (V1,V2) < (ah/p1,bV/p2) and one can conclude that Q is an attractive set. □

The values of Sh and SV can be determined correspondingly by Sv = (b/fa) - Iv, Sh = (a^fa) - 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 Q

dEh _ a$Iv ( a^

dIh dt

dIv YIh

f (* "Eh-'h)-^ )E„,

_ßEh -PiIh, (7)

dt l + nIh\ fa Iv) ^

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

r_\(Eh,Ih,Iv)eR55 \0<Eh + Ih <-,

0<IV < ^ ,Eh >0,Ih >0,IV >0 p2

where R+ denotes the nonnegative cone of R 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) = Eh, Ih(0) = I0, Iv(0) = Iv, and the closed set r. Then, r 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 = ahbV<papy/p2p2,(p + fa). 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. If R0 < 1, the disease-free equilibrium E0 ofthe model (7) is locally asymptoticallystable, 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

-(ß + Pi) 0

aha$ Pi

J (Eo)_( ß -Pi 0 hß

The characteristic equation of this matrix is given by det(AI -J(E0)) = 0, where I is the 3x3 unit matrix. Expanding the determinant into a characteristic equation, we obtain the following equation, which is equivalent to

A3 + a A2 + Ü2Ä + a3 _ 0,

a _ 2 fa + p2 + ß > 0,

a2 _ (fa + P2) (ß + Pl) + >0, (11)

a3 _ №2 (ß+Pl)(l-R0) >

These three eigenvalues have negative real part if they satisfy the Routh-Hurwitz Criteria [20], such that > 0 for i = 1,2,3, with a1 > 0,a3 > 0, and a1a2 > a3. If R0 < 1, then

a~i a2 a3

= (2li + l2 + ß) [(li + l2) (ß + li ) + lil2] -lil2 (ß + li)(l - R0)' = (2li + l2 + ß) (li + l2) (ß + li) + fafa (2fa + fa + ß)- fafa (ß + fa) (1 - R0)

= (2h + l2 + ß) (ll + l2) (ß + lO + fa fa (2fa +fa + ß)- fafa (ß + fa)

+ fafa (ß+fa)R0

= (2h + l2 + ß) (ll + l2) (ß + il) + fafa (fa+fa) + fafa (ß + fa)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. If R0 < 1, the disease-free equilibrium E0 of the model (7) is globally asymptotically stable in r.

Proof. We construct the following Lyapunov function:

V(t) = a1Eh + ailh + a3Iv,

1 ill! (ß + llY 1

2~ 2 fafa

Its derivative along the solutions to the system (7) is

V' (t) = a Eh + a^lh + a3l'v aWv ( ah

_1+mIv\fa

+ a2 [ßEh - liIh] + a3

-Eh -Ih)-(ß+ li) Eh

V1+nIh\l2

-Eh -Ih ]-(ß + fa)Eh \fa /

+ a2 [ßEh - liIh] + Kßy

yIh[—-Iv)- hk y2

fal\ (ß+fa)

xfa^ -Eh-Ih)-(ß + fa)Eh}

i (hy,

+ —^2 (ßEh - lIh) + — [—h - I2K

fay2 fax fa

Vii4 (ß + ii)

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) e r | 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 r. □

4. The Endemic Equilibrium and Its Stability

Here, we study the existence and stability of the endemic equilibrium points. By straightforward computation, if R0 > 1, then the host-vector model system (7) has a unique endemic equilibrium given by E* = (E*,I*,I*) in r, with

p * - yi T

E = —rI

lill (Ro - 0

fa (faw + myhv) + a$ybv

I* = yKK V l2 (wI* +l2)'

w = y + nfa.

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

1 + mIh

J (E*) =

-(ß + fa)

J (E') =

1 + mIv

Y((b/^2)-I (1+nI")2 0

-(ß + 2fa)

1 + mIh

((ah/fa) - E' - ^ (1+mI* )2

y((*>/H2)-1*

(1 + nI*)2

1 + nI,

From the Jacobian matrix J(E*), the second additive compound matrix is given by

____yjh_

1 + mIh 1 + nI,

a$((ah/Pi)-K -Ih)

(1 + mI*)

-ß-fa-fa

1 + mIh

1 + nI,

-fa-fa

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 a 3x3 real matrix. If tr(M), det(M), and det(M[2]) are all negative, then all eigenvalues ofM have negative real part.

Using the above Lemma, we will study the stability of the endemic equilibrium.

Theorem 6. If R0 > 1, the endemic equilibrium E* of the model (7) is locally asymptotically stable in r.

Proof. From the Jacobian matrix J(E*), we have tr (M) = -

+(ß+fa)+fa + -y^h + to)<0.

1 + mIh

1 + nI,

Because

ah t7» t* fa-Eh -Ih ^

(ß + fa)(1+mIh)Eh

£h T*

K-I» (1 + nIh)faI

v.2 v yIh

From (21), it is easy to see that

((ah/V\) - eh -Jh) x y ((b/^2) - Ih

(1+mIhY

fafa (ß + fa)

(1 + mIv)(1+nIh)ß'

(1 + nIh)2

det (J(E»))

1 + mIh

-(ß + fa)

1 + mIh

y((b/Hi)-Ih (1 + nIh)2

1 + mIh

1 + nI,

^^ + ß + fa)(fafa + 1 + mIh ri 1 + nIh

a^((üh/fa)-el -Ih)

(1 + mI 0 yIh

1 + nI,

aU +ß+fa)(fafa + ^ )-ß

1 + mIh \ 1 + nI,

1 + mIh \ 1 +nI,

+ Pl) +

fafa (ß + fa)

T+mh [T+mh, +ß+vi)-^2 T+mF-^ (ß+^-T+mih\ 1+m,

(1+mIh)(1+nIh )ß

fafa (ß + vi) (1 + mIh) (1 + nIh)

fafa (ß + fa) _ (1 + ml*) (1 + nl*) 1 + nl*\ 1 + ml,

faiY 1 * ( аФ1,

+ ß + fa

1 + ml* 1 + ml* \ 1 +nl,

аф!* аф!,

+ fafa (ß + fa)

(1+mi; )(1+nI*)

- 1) < 0.

Computing directly the determinant of J^-2 (E*), we can get

det (j[2] (E*))

аф!,

-(ß + 2fa)

1 + mlx

Y((Mfa2)-I, (1 + nK ) 0

1 + ml*

+ ß + 2 fa

аф!,

аф1,

аф ((ajfa) -E** - V*

1 + ml* 1 + nl,

- ß-fa-fa

1 + ml* 1 + nl,

+ ß + fa+fa

Y((bv/fa2)-I*) ( <*И ((ah/^) -Eh -I1

(1 + nIif ( (1 + rnI*)2

(1 + mI* аф1*

1 + ml*

1 + nl,

■fa-fa

1 + nl,

+ fa + fa I +

aßфI^

1 + ml,*

Hence, by lemma, the endemic equilibrium E* of the model (7) is locally asymptotically stable in r. □

5. Global Stability of the Endemic Equilibrium

We now prove the global stability of the endemic equilibrium E*, when the reproduction number R0 is greater than the unity. For this, first we will prove the following result.

Theorem 7. IfR0 > 1, thensystem (7) is uniformly persistent; that is, there exists с > 0 (independent of initial conditions), such that liminft^mEh(t) > с, liminft^mIh(t) > с, liminft^mIv(t) > с.

Proof. Let n be a semidynamical system (7) in (R+)3, Let % be a locally compact metric space, and Г0 = {(Eh, Ih, Iv) 6 Г | Iv = 0}. The set Г0 is a compact subset of Г and Г/Г0 is positively invariant set of system (7). Let P : % ^ R+ be defined by P(Eh, Ih, Iv) = Iv and set S = {(Eh,Ih,Iv) 6 Г | P(Eh, Ih, Iv) < p}, where p is sufficiently small so that R0(1 -(fa/bv)p)/(\ + np) > 1. Assume that there is a solution x 6 S such that for each t > 0,we have P(n(x, t)) < P(x) < p. Let us consider the following:

fafa (ß+fa)

where 8* > 0 is a sufficiently small constant so that R0(1 -(fa/bv)p)(1 - 8*)/(1 + np) > 1. By a direct calculation, we have

L' (t) > fa

аФу^Ф (1-S*)(1- (fajbv) p) falfal (ß + fai)(1 + nP)

ahaßфS* faV

аФу^Ф (1-S*)(1- (fajbv) p)

fa2fa2 (ß+fa)(1+np)

fa2 (ß+fal)S* 1-8*

Thus, we have

L (t) > 8L (t).

The above inequality (28)impliesthat L(t) ^ >x> ast ^ >x>. However, L(t) is bounded on the set r. According to Theorem 1 in [25], we complete the proof of Theorem 7. □

Here, we use the geometrical approach of Li and Mul-downey to investigate the global stability of the endemic equilibrium E* in the feasible region Q. 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 C1 map f : x ^ f(x) from an open set D c Rn to Rn such that each solution x(t,x0) to the differential equation

c' = 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 set K c D;

(H3) (29) has unique equilibrium x in D.

Let P : x ^ P(x) be a nonsingular (") x (") matrix-valued function which is C1 in D and a vector norm | • | on Rn, where N = (n2).

Let ^ be the Lozinskii measure with respect to the | • |. Define a quantity q2 as

q2 = limsupsup- I p (B (x (s, x0))) ds, (30)

t^rn x0iKt Jo

where B = PfP-1 + PJ[2]P-1,

replacing each entry p of P by its derivative in the direction of f, (pij)p and J^ 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 Q if q2 < 0.

We choose a suitable vector norm matrix valued function

P(x) =

0 Eh I 0 0

in R3 and a 3x3

Obviously, P is C1 and nonsingular in the interior of Q. Linearizing system (2) about an endemic equilibrium E* gives the following Jacobian matrix:

1 + ml

-(ß + Ui) ß

1 + ml,

( ah-E (1 + mIv)2\Ui h h

(l+nIhy\h2

yIh 1+nIh

The second additive compound matrix of J(E*) is given

1 + mf y

-(ß + Ui)-Ui

(l+nIhy\h2 0

-(ß+u,)- —

1 + mIv 1 + nIh

( ah E (1 + mIv)2\Ui h h

1 + mIv y I

1 + nI

The matrix B = P^P 1 +PJ^P 1 can be written in block form

ß = ( B11 B12

B21 B22

Bn = _

1 + ml.

_(ß + fa)_fa,

(1 +m!v) Eh\ fa

Uh P T

2 tt i ~ _ bh - 1h

"V _ T ) Eh

(l+tllh)2 \ 0

El _I_l g<j>Iv y!h

Eh Iv 1 + m!„ 1 + nh

1 + ml,,

P' T' P T fa

1 + nl,

Consider the norm in R as \(u, v,w)\ = max(\u\, |v| + \w\) where (u, v, w) denotes the vector in R3. The Lozinskii measure with respect to this norm is defined as ^(B) < sup(g1, 02), where

01 = (Bu) + \B1

02 = (B22) + \B2

From system (2), we can write

Eh 1 + mlv Eh V fa T =l3T_fa1'

{fa; _Eh_lh)_(ß + fa),

1+nlhlv \fa

and g1 will become

01 =--T^--(P + P1)-P1

1 + mlv

(\+mlv)2Eh\fa h h

1 + mlv r1

lv i ah

(1+mIv)Eh\fa Eh a$Iv

(*> _Eh _Ih ]_(ß + fa) fa1

Ei, 1 + ml,

■fa.

Also,\B21\ = (yl(\+nlh)2)((bvlfa)-lv)(Ehllv). \BU\and ^ 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 ^ norm. fa(B22) is the Lozinskii measure of 2x2 matrixB22 with respect to ^ norm in R2.Consider the following:

Since £n is a scalar, its Lozinskii measure with respect to any vector norm in R1 will be equal to B11. Thus,

B„ = _

1 + ml.

_(ß + fa)_fa,

Iv ( ah

(1 +m!v) EhKfa

_ Eh _ Ih

fa (B22) = suP

E'h l'v

Ei !„

1 + mlv

1 + ml,

E'h Ii

I„ 1 + ml,

1 + nIh

1 + nIh

1 + nl,

Hence,

Eh K , a$h Yh

Iv 1 + mlv

1 + nh

(l+nlh)2 V fa

K _j\Eh

Eh £ a$h ,. Yh

--— + - _ fai — -

Eh Iv 1 + mlv 1 + nlh

(1+nIh)2 \ fa2

K _j\Eh

h ^ Hh ,, yh _ — + --- — fa

Eh Iv 1 + mlv 1 + nIh

"v _ t ) E'h , h

1 I + I

(1+nIhy\ fa2

1+nIh\ fa E'h a$Iv

K _I\Ih

Eh 1 + mlv

1 + nh

K _j\Eh

(1+nIhy\ fa2 >h 1+mh\fa2

K _j\h

E'h a$Iv

Eh 1 + mlv

V Eh K

1 + mh (1 + nihy h^2

_ -2 ^h ----+ -J/i

(1 + nIh)2 1 + nIhIvfa 1+nIh

Eh 1 + mf

(1 + nlh)2 h fa

y E y h K

(1 + nIh) h 1 + nIhIvfa

< Ek , ^ fa , y__

Eh 1 + mh 1 (1+nIh)2 hfa

1h) V h k,

2 ' 2 t

(1 + nIh)2 (1 + nIh)2 h fa

<$ + _fa +

Eh 1 + mlv

(1 + mh) fah fa

v c bv

(1+nIh)2 h (1+nIh)2 Iv fa

E'h a^Iv

g2 < — +

Eh 1 + mlv

_fai K ( ah

(1 + mh)2 fa\ fai

(1 + nIh)

< + ---fa

Eh 1 + mlv

< Eh. + _u Eh m fa1

Ei aSM Eh m

if fa > ah/c and M = bv/fa2. Thus,

fa(B) = sup[gi,g2] <

fa 2 '

if fa > 2a$M/m.

Since (2) is uniformly persistent when R0 > 1,sofor T> 0 such that t > T implies that Eh(t) > c, Ih(t) > c, IV(t) > c, and (1/t) log Ih(t) < faj4 for all (Sh(0),Ih(0),Iv(0)) e K. Thus,

if' fa(B)dt< log Eh(t)_fa- <_fa-, t Jq t 2 4

for all (Sh(0), Ih(0), IV(0)) e K. The condition (H3) is satisfied provided that fa1 > max[ah/c, 2a<pM/m\. Therefore, all the conditions of Theorem 7 are satisfied. Hence, unique endemic equilibrium E* is globally stable in Q.

6. Discussion

We know that the basic reproduction number of the model R0 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 a and vector infectious rate p, 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 E*, infected host I* decreases as m increases (see Figures 1 and 2), and the steady state value of the infective vector I* decreases as n increases (see Figure 3). The numerical simulations are carried out using Sh(0) = 300, Eh(0) = 30, 4(0) = 20, SV(0) = 65, IV(0) = 20, ah = 0.009041, bV = 0.002691, a = 0.00166, 0 = 0.2, ¡3 = 0.057142, y = 0.00305, fa1 = 0.0000301, fa2 = 0.011764, m1 = 0.01, m2 = 0.03, m3 = 0.07, m4 = 0.09, n = 0.01, fa = 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.

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

Time (day)

• m = 0.01 - m = 0.07

- m = 0.03 ----m = 0.09

Figure 1: Plot of the exposed host population.

Time (day)

• m = 0.01 - m = 0.07

- m = 0.03 ----m = 0.09

Figure 2: Plot of the infected host population.

to be determined by the basic reproduction number R0. More specifically, by constructing suitable Lyapunov function, we proved that if R0 < 1, then disease-free equilibrium E0 is globally asymptotically stable in r, and thus the disease always dies out. If Rq > 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.

Conflict of Interests

The authors declared that there is no conflict of interests.

Time (day)

• n = 0.01 — n = 0.03

— n = 0.02

Figure 3: Plot of the case infected vector population.

Acknowledgments

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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