# Analysis of Pine Wilt Disease Model with Nonlinear Incidence and Horizontal Transmission

Authors: Muhammad Ozair
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The deterministic pine wilt model with vital dynamics to determine the equilibria and their stability by considering nonlinear incidence rates with horizontal transmission is analyzed. The complete global analysis for the equilibria of the model is discussed. The explicit formula for the reproductive number is obtained and it is shown that the “disease-free” equilibrium always exists and is globally asymptotically stable wheneverR0≤1. Furthermore, the disease persists at an “endemic” level when the reproductive number exceeds unity.


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## Body

## 1. Introduction

Pine wilt, a fatal disease of commonly planted pines brought on by the pinewood nematode (Bursaphelenchus xylophilus), causes changes to ecosystem and destructs the variety of ecosystem. Pine trees affected by pine wilt disease usually die within few months. Symptoms of pine wilt disease normally appear in late spring or summer. The most prominent symptom is the lack of resin exudation from barks wounds. The foliage becomes light grayish green, then becomes yellow, and finally it becomes reddish brown. The tree succumbs to the disease at this stage. The affected trees totally lack resin and their wood becomes dry.The long-horned pine sawyer beetles (Monochamus alternatus) are the main culprits for the spread of pinewood nematodes from infected pines to healthy or stressed pines. When new adult beetles emerge in spring, they locate a living host tree to feed on the bark of the young branches and transfer nematodes to the healthy trees through the feeding wounds produced by these sawyers. This transmission is referred to as primary transmission. The transmission of the nematodes during egg-laying activities in freshly cut timber or dying trees is referred to as secondary transmission. Nematodes, introduced during primary transmission, migrate to the resin canals of their hosts and kill these cells rendering them ineffective due to which a susceptible host can wilt and die within weeks of being infested upon the availability of favorable conditions to disease development. The principle of theBursaphelenchus xylophilus transmission and disease dissemination is reviewed by Evans et al. [1]. Pine wilt particularly kills Scots pine within few weeks to few months. Some other pine species as Austrian (Pinus nigra), jack (P. banksiana), mogo (P. mugo), and red (P. resinosa) pines are occasionally killed by pine wilt.Mathematical modeling became a considerably important tool in the study of epidemiology because it helps us to understand the observed epidemiological patterns and disease control and provides understanding of the underlying mechanisms which influence the spread of disease and may suggest control strategies. The incidence of a disease is defined as new cases occurring per unit time. It plays a vital role in mathematical epidemiology. The classical epidemiological models are developed by the assumption of bilinear incidence rateβSI and the standard incidence rate βSI/N, where β is the transmission probability per contact and S,I are susceptible and infected individuals, respectively. However, there are several reasons that require modification in these incidence rates. For example, the assumption of homogeneous mixing may be invalid and in this case a model having a particular form of nonlinear transmission may be incorporated for heterogeneous mixing and the necessary population structure.The saturation effects may also require a nonlinear incidence rate because if the proportion of the infected population is high enough so that exposure to the disease agent is virtually certain, then transmission rate may react more slowly than linear in order to increase the number of infected individuals. Capasso and Serio [2] who studied the cholera epidemic spread in Bari in 1973 and introduced the saturated incidence rate g(I)S in the epidemic model observed this effect. A variety of nonlinear incidence rates has been utilized in epidemic models [3–7]. An epidemic model with nonlinear incidence is proposed in [8], in which the authors described the dynamics of diseases spread by vector mosquitoes such as malaria, dengue, and yellow fever.The incidence of pine wilt disease depends on beetles’ density because pine sawyer beetles are the source of transmission of pinewood nematode. This incidence may approach its saturation level at very high beetle densities. The adult female pine sawyer attempts to avert from erstwhile oviposition scants. It approaches another tree before the saturation point of oviposition is reached. Thus the isolation of infected individuals results in the decrease in the number of contacts between the susceptible and infected individuals at high infective levels. These observations inspire to consider nonlinearities in the incidence rates.In this paper, based on the ideas posed in [9–11], a pine wilt disease model considering a direct mode of transmission as well as nonlinear incidence rate is formulated. The aim of this paper is to establish stability properties of equilibria and the threshold parameter R0 that completely determines the existence of endemic or disease-free equilibrium. If R0≤1, the disease-free equilibrium is globally asymptotically stable. If R0>1, a unique endemic equilibrium exists and is globally asymptotically stable. The rest of the paper is organized as follows. In Section 2, the description of the extended mathematical model is presented. Section 3 is devoted to the existence of equilibria. In Section 4, the Lyapunov function theory is used to show global stability of disease-free equilibrium and geometric approach is used to prove global stability of endemic equilibrium in Section 5. Discussions and simulations are done in Section 6.


## 2. Model Description

The pine population, with total population size denoted byNh(t), is subdivided into two mutually exclusive compartments: susceptible pine trees Sh(t) and infectious pine trees Ih(t). Thus, Nh(t)=Sh(t)+Ih(t). The emission of oleoresin from susceptible host pines behaves like a physical barrier for beetle oviposition. Beetles can oviposit on the infected pine trees because these trees cease oleoresin. Since there are no cures for pine wilt once a susceptible tree becomes infested with pinewood nematodes, the recovered class Rh(t) has not been considered.The total vector population at any timet is denoted by Nv(t)=Sv(t)+Iv(t), where Sv(t) denotes the susceptible adult beetles that do not have any pinewood nematode at time t and Iv(t) denotes the infected adult beetles carrying pinewood nematode at time t when they emerge from dead pine trees. After emergence from the dead tree, beetles choose a healthy tree for sufficient feeding and transmit nematodes into the tree. These nematodes move through the feeding wounds and approach the xylem of the tree. When beetles are in oviposition they choose dying or dead tree and transmit nematode when they lay eggs in slits in bark. Nematodes enter these slits, feed on wood cells or fungi, and reproduce themselves. Before beetle’s emergence from dead tree the nematodes attach with the tracheae of its respiratory system. The following assumptions are made in formulating the mathematical model.(i)
The exploitation rate of pine trees infected withBursaphelenchus xylophilusis greater than the normal and susceptible pine trees.(ii)
The susceptible beetles receive nematodes directly from infectious ones through mating.(iii)
Adult beetles emerging from infected trees have pinewood nematode.(iv)
The infected vectors transmit the nematode during maturation feeding as well as via oviposition. Under these assumptions, the vector-host model with nonlinear incidence can be described by the following system of differential equations:
(1)dShdt=Πh-δ1ShIv1+α1Iv-δ2θShIv1+α1Iv-μhSh,dIhdt=δ1ShIv1+α1Iv+δ2θShIv1+α1Iv-ωIh,dSvdt=Πv-β1SvIh1+α2Ih-β2SvIv-μvSv,dIvdt=β1SvIh1+α2Ih+β2SvIv-μvIv,
where Πh is the constant increase rate of pines, Πv is the constant input rate of vectors, and μv is the mortality rate of vectors. The exploitation rate of susceptible pines is μh whereas the isolation and felling rate of infected pines is ω. The transmission between susceptible pines and infected vectors occurs when infected beetles lay eggs on those dead pines that die of natural causes or through the maturation feeding of infected vectors; the incidence terms for these transmissions are δ2θShIv/(1+α1Iv) and δ1ShIv/(1+α1Iv), respectively. The parameter θ is the probability by which susceptible pines die of natural causes and cease oleoresin exudation without being infected by the nematode, and δ2 indicates the rate at which infected vectors transmit the nematode via oviposition whereas δ1 denotes transmission rate per contact during maturation feeding. The transmission between susceptible vectors and infected hosts occurs when adult beetles emerge from dead pine trees. This transmission is denoted by β1SvIh/(1+α2Ih), where β1 is the rate at which adult beetles carry the pinewood nematode when they emerge from dead trees. The parameters α1 and α2 determine the level at which the infection is saturated. The beetles transmit nematodes directly through mating. The incidence term for this transmission is β2SvIv, where β2 is the transmission rate among beetles during mating. All parameters are assumed to be positive.The total dynamics of vector population satisfy the following equation:(2)dNvdt=Πv-μvNv.
This leads to Nv→Πv/μv as t→∞. Thus, the system (1) is reduced to the following system of differential equations:
(3)dShdt=Πh-δ1ShIv1+α1Iv-δ2θShIv1+α1Iv-μhSh,dIhdt=δ1ShIv1+α1Iv+δ2θShIv1+α1Iv-ωIh,dIvdt=β1(Πvμv-Iv)Ih1+α2Ih+β2(Πvμv-Iv)Iv-μvIv.
Considering ecological significance, we study system (3) in the closed set Ω={(Sh,Ih,Iv):Πh/ω≤Sh+Ih≤Πh/μh,0≤Iv≤Πv/μv}. It can be easily verified that Ω is positively invariant with respect to (3).


## 3. Existence of Equilibria

The dynamics of the disease are described by the threshold quantityR0 which is called the reproduction number defined as “the average number of secondary infections produced by an infected individual in a completely susceptible population.” It is one of the most useful threshold parameters that characterizes mathematical problems related to infectious diseases. This metric helps to determine whether or not an infectious disease will spread through a population. The basic reproduction number of model (3) is given by
(4)R0=β2Πvμv2+β1Πvμv2Πhμhω(δ1+θδ2).
Direct calculation shows that for R0≤1, there is only disease-free equilibrium E0(Πh/μh,0,0) and for R0>1, there is an additional equilibrium E*(Sh*,Ih*,Iv*) which is called endemic equilibrium, with
(5)Sh*=Πh-ωIh*μh,Ih*=(ΠhIv*[δ1+δ2θ+(δ1α1+δ2θα1)Iv*])×([(α1μh+δ1+δ2θ)α1ωIv*2+(δ1+δ2θ+2α1μh)ωIv*+ωμh(α1μh+δ1+δ2θ)α1ωIv*2])-1,
and Iv* is the root of the following equation:
(6)AIv*3+BIv*2+CIv*+D=0,
where,
(7)A=Πhμvα2β2(α1δ1+θα1δ2),B=θωα1β2δ2μv+θα1β1δ2Πhμv+θα2β2δ2Πhμv+ωα1α1β2μhμv,C=ωβ2μhμv+α1β1δ1ΠhΠv+θα1β1δ2ΠhΠv,D=ωμhμv2(1-R0).
From (7), we see that R0>1 if and only if D<0. Since A, B, and C are always positive, there will be zero or unique positive endemic equilibrium accordingly as R0≤1 or R0>1. Thus we have the following theorem.Theorem 1.
System (3) always has the infection-free equilibrium E0. If R0>1, system (3) has a unique endemic equilibrium E*(Sh*,Ih*,Iv*) defined by (5) and (6).


## 4. Stability of Disease-Free Equilibrium

Here, we analyze stability of disease-free equilibriumE0(Πh/μh,0,0) for system (3). The linearization of the system (3) at E0 results in the following characteristic equation:
(8)(-μh-λ)[λ2+λ(ω+μv-β2Πvμv)+ωμv(1-R0)]=0.
The characteristic equation (8) has one eigenvalue -μh. The other eigenvalues can be found by the equation
(9)λ2+aλ+b=0,
where a=ω+μv-(β2Πv/μv) and b=ωμv(1-R0).We observe that the roots of the quadratic equation (9) have negative real parts if R0<1. If R0=1, one root of (9) is 0. This fact does not guarantee that all eigenvalues have negative real parts. It will only be possible in case of real roots. If R0>1, one of the roots of (9) has positive real part. The above discussion leads to the following theorem.Theorem 2.
The disease-free equilibrium of system (3) is locally asymptotically stable in Ω if R0<1 and it is unstable if R0>1.Now, we analyze the global behavior of the disease-free equilibriumE0. The following theorem provides the global property of the system.Theorem 3.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable in the interior of Ω.Proof.
The following Lyapunov function is proposed to establish the global stability of disease-free equilibrium:(10)L=β1ΠvωIh+μvIv.
Taking the time derivative of L along the solutions of (3), we have
(11)L′=β1ΠvωIh′+μvIv′=β1Πvω(δ1ShIv1+α1Iv+δ2θShIv1+α1Iv-ωIh)+μv[β1(Πvμv-Iv)Ih1+α2Ih+β2(Πvμv-Iv)Iv-μvIv]≤β1Πvω(δ1ShIv+δ2θShIv-ωIh)+μv[(β1ΠvμvIh1+α2Ih-β1IvIh1+α2Ih)+(β2ΠvμvIv-β2IvIv)]-μv2Iv<(δ1+δ2θ)β1ΠvωΠhμhIv+μvβ2ΠvμvIv-μv2Iv-β1ΠvIh+β1ΠvIh-μvβ1IvIh1+α2Ih-μvβ2Iv2=Iv[μv2(R0-1)-μvβ1Ih1+α2Ih-μvβ2Iv]≤0.
Thus L′(t) is negative if R0≤1. When R0<1, the derivative L′=0 if and only if Iv=0, while in the case R0=1, the derivative L′=0 if and only if Ih=0 or Iv=0. Consequently, the largest compact invariant set in {(Sh,Ih,Iv∈Ω),L′=0}, when R0≤1, is the singleton E0. Hence, by LaSalle’s invariance principle [12], E0 is globally asymptotically stable in Ω. This completes the proof.


## 5. Stability of Endemic Equilibrium

In this section, we will discuss global stability of endemic equilibriumE* in the feasible region Ω. This is done through the geometrical approach applied by Li and Muldowney [13]. We summarize this approach below.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
(12)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)
Equation (12) has unique equilibrium x¯ in D.Let P: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). Let μ be the Lozinskiĭ measure with respect to the |·|. Define a quantity q¯2 as
(13)q¯2=limsup⁡t→∞sup⁡x0∈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 (12). The following result has been established by Li and Muldowney [13].Theorem 4.
Suppose thatH1, H2, and H3 hold; the unique endemic equilibrium E* is globally stable in Ω if q¯2<0.ObviouslyΩ is simply connected and E* is a unique endemic equilibrium for R0>1 in Ω. To apply the result of the above theorem for global stability of endemic equilibrium E*, we first prove the uniform persistence of (3) when the threshold parameter R0>1, by applying the acyclicity theorem (see [14]).Definition 5 (see [15]).
The system (3) is uniformly persistent; that is, there exists c>0 (independent of initial conditions), such that lim⁡inf⁡t→∞Sh≥c, lim⁡inf⁡t→∞Ih≥c, and lim⁡inf⁡t→∞Iv≥c.LetX be a locally compact metric space with metric d and let Ω be a closed nonempty subset of X with boundary Ω and interior Ω∘. Clearly, Ω∘ is a closed subset of Ω. Let Φt be a dynamical system defined on Ω. A set B in X is said to be invariant if Φ(B,t)=B. Define M∂:={x∈Ω:Φ(t,x)∈Ω, for all t≥0}.Lemma 6 (see [14]).
Assume that(a)
Φt has a global attractor;(b)
there existsM={M1,…,Mk} of pairwise disjoint, compact, and isolated invariant set on ∂Ω such that
(1)
⋃x∈M∂‍ω(x)⊆⋃j=1k‍Mj;(2)
no subsets ofM form a cycle on ∂Ω;(3)
eachMj is also isolated in Ω;(4)
Ws(Mj)∩Ω∘=ϕ for each 1≤j≤k, where Ws(Mj) is stable manifold of Mj. Then Φt is uniformly persistent with respect to Ω.Proof.
We haveΩ={(Sh,Ih,Iv):Πh/μh≤Sh+Ih≤Πh/ω,0≤Iv≤Πv/μv}, Ω∘={(Sh,Ih,Iv):Sh,Ih,Iv>0}, ∂Ω=Ω/Ω∘. Obviously, M∂=∂Ω. Since Ω is bounded and positively invariant there exists a compact set M in which all solutions of system (3) initiated in Ω ultimately enter and remain forever. On Sh-axis we have Sh′=Πh-μhSh which means Sh→Πh/μh as t→∞. Thus E0 is the only omega limit point on ∂Ω; that is, ω(x)=E0 for all x∈M∂. Furthermore M=E0 is a covering of Ω=⋃x∈M∂‍ω(x) because all solutions initiated on the Sh-axis converge to E0. Also E0 is isolated and acyclic. This verifies that hypotheses (1) and (3) hold. When R0>1, the “disease-free” equilibrium (DFE) E0 is unstable from theorem (3) and also Ws(M)=∂Ω. Hypotheses (4) and (5) hold. There always admits a global attractor due to ultimate boundedness of solutions.The boundedness ofΩ and the above lemma imply that (3) has a compact absorbing set K⊂Ω [15]. Now we will prove that the quantity q¯2<0. We choose a suitable vector norm |·| in R3 and a 3×3 matrix valued function
(14)P(x)=[1000IhIv000IhIv].
Obviously P is C1 and nonsingular in the interior of Ω. Linearizing system (3) about an endemic equilibrium E* gives the following Jacobian matrix: (15)J=[-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv0-Shδ1(1+α1Iv)2-Shθδ2(1+α1Iv)2Ivδ11+α1Iv+Ivθδ21+α1Iv-ωShδ1(1+α1Iv)2+Shθδ2(1+α1Iv)20β1μv(1+α2Ih)2(Πv-Ivμv)β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih].The second additive compound matrix ofJ(E*) is given by(16)J[2]=[b11Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2β1μv(1+α2Ih)2(Πv-Ivμv)b2200Ivδ11+α1Iv+Ivθδ21+α1Ivb33],where
(17)b11=-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv-ω,b22=-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv+β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih,b33=-ω+β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih.
The matrix B=PfP-1+PJ[2]P-1 can be written in block form as B=(B11B12B21B22), with
(18)B11=-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv-ω,B12=((Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2)IvIh(Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2),IvIh(Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2)),B21=(IhIvβ1μv(1+α2Ih)2(Πv-Ivμv)0),B22=(M22+Ih′Ih-Iv′Iv0Ivδ11+α1Iv+Ivθδ21+α1IvM33+Ih′Ih-Iv′Iv),
where
(19)M22=-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv+β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih,M33=-ω+β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih.
Consider the norm in R3 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
(20)g1=μ1(B11)+|B12|,g2=μ1(B22)+|B21|.
From system (3) we can write
(21)Ih′Ih=IvIh(Shδ11+α1Iv+Shθδ21+α1Iv)-ω,Iv′Iv=IhIvβ1μv(1+α2Ih)(Πv-μvIv)+β2μv(Πv-μvIv)-μv.SinceB11 is a scalar, its Lozinskiĭ measure with respect to any vector norm in R1 will be equal to B11. Thus
(22)B11=-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv-ω,|B12|=IvIh(Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2),
and g1 will become
(23)g1=-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv-ω+IvIh(Shδ1(1+α1Iv)2+Shθδ2(1+α1Iv)2)g1=Ih′Ih-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv.
Also |B21|=(Ih/Iv)(β1/μv(1+α2Ih)2)(Πv-Ivμv), |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
(24)μ1(B22)=sup⁡{M22+Ih′Ih-Iv′Iv+Ivδ11+α1Iv+Ivθδ21+α1Iv,M33+Ih′Ih-Iv′Iv}=sup⁡{-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv+β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih+Ih′Ih-IhIvβ1μv(1+α2Ih)(Πv-μvIv)-β2μv(Πv-μvIv)+μv+Ivδ11+α1Iv+Ivθδ21+α1Iv-ω+β2μv(Πv-2Ivμv)-μv-Ihβ11+α2Ih+Ih′Ih-IhIvβ1μv(1+α2Ih)(Πv-μvIv)-β2μv(Πv-μvIv)+μv},μ1(B22)=Ih′Ih-β2μv(Ivμv)-IhIvβ1μv(1+α2Ih)(Πv-μvIv)-β~1,
where
(25)β~1=min⁡{Ihβ11+α2Ih+μh,ω+Ihβ11+α2Ih}.
Hence g2≤(Ih′/Ih)-(β2/μv)(Ivμv)-β~1.Thus(26)μ(B)=sup⁡{g1,g2}≤sup⁡{Ih′Ih-μh-Ivδ11+α1Iv-Ivθδ21+α1Iv,Ih′Ih-β2μv(Ivμv)-β~1},μ(B)≤Ih′Ih-β~2,
where
(27)β~2=min⁡{μh+Ivδ11+α1Iv+Ivθδ21+α1Iv,β2μv(Ivμv)+β~1}.Since (3) is uniformly persistent when R0>1, so for T>0 such that t>T implies Ih(t)≥c, Iv(t)≥c and (1/t)log⁡Ih(t)<β~2/2 for all (Sh(0),Ih(0),Iv(0))∈K.Thus(28)1t∫0t‍μ(B)dt<log⁡Ih(t)t-β~2<-β~22,
for all (Sh(0),Ih(0),Iv(0))∈K, which further implies that q¯2<0. Therefore all the conditions of Theorem 4 are satisfied. Hence unique endemic equilibrium E* is globally stable in Ω.


## 6. Discussions and Simulations

In this paper, pine wilt disease transmission model with nonlinear incidence rates and horizontal transmission is proposed and analyzed. The basic reproduction number,R0, of the model is obtained and with the help of this reproduction number the asymptotic behavior of the model is discussed. The variation of infected hosts and infected vectors is shown in Figure 1. It is not meaningful to consider the saturation level when transmission occurred during mating. Thus bilinear incidence has been considered. By simple calculation we see that
(29)∂R0∂μv=-2ωμhμv3Πv(ωβ2μh+β1δ1Πh+θβ1δ2Πh)<0,∂R0∂ω=-1ω2β1μhΠhμv2Πv(δ1+θδ2)<0,
which means that R0 is a decreasing function of μv and ω. The question arises, which parameter is more crucial in order to decrease the reproductive number? By using the definition given in [8] and parameter values Πh=100, Πv=400, β2=0.00034, μv=0.00054, β1=0.4, μh=0.000274, ω=0.00137, δ1=0.01, θ=0.00304, and δ2=0.01, we see that the sensitivity index of the reproductive number with respect to μv is -2 and with respect to ω is -1. It means that the most sensitive parameter for R0 is μv. Increasing the mortality rate ofMonochamus alternatus by 10% decreases R0 by 20%. Also, increasing the exploitation rate of infected pines by 10%  decreases  R0 by 10%. Thus control strategies, for example, setting out beetle traps, setting vertical wood traps, and using chemicals to kill sawyer beetles, by cutting down dead pine trees and disposing of them before the emergence of beetles can be useful for eradicating the disease.Figure 1
The infected population forΠh=0.22, Πv=0.32, μh=0.00002, ω=0.003, μv=0.004, β1=0.0004, β2=0.00034, δ1=0.0016, δ2=0.00016, θ=0.00301, α1=0.001, and α2=0.001.The above mentioned measures are very effective to control pine wilt disease but they have not yet been practiced to eradicate pine wilt disease ultimately because these measures require more cost and labor and even entail danger of forest fires due to which most owners of forests hesitate to use these measures.However, we can decrease the endemic level of the disease by increasing the parametersμv and ω. Figures 2, 3, 4, and 5 show different endemic levels of Iv and Ih with respect to the parameters μv and ω. We see that by increasing these parameters the infective levels of pine trees and vectors decrease.Figure 2
The effect ofμv on infected pine trees for Πh=0.22, Πv=0.32, μh=0.02, ω=0.03, μv=0.004,0.0058,0.09,β1=0.0004, β2=0.00034, δ1=0.0016, δ2=0.00016, θ=0.00301, α1=0.001, and α2=0.001.Figure 3
The effect ofω on infected pine trees for Πh=0.22, Πv=0.32, μh=0.002, ω=0.03,0.04,0.05,0.06, μv=0.004, β1=0.0004, β2=0.00034, δ1=0.0016, δ2=0.00016, θ=0.00301, α1=0.001, and α2=0.001.Figure 4
The effect ofμv on infected vectors for Πh=0.22, Πv=0.32, μh=0.02, ω=0.03, μv=0.004, 0.0058, 0.09,β1=0.0004, β2=0.00034, δ1=0.0016, δ2=0.00016, θ=0.00301, α1=0.001, and α2=0.001.Figure 5
The effect ofω on infected vectors for Πh=0.22, Πv=0.32, μh=0.002, ω=0.003,0.03,0.07,0.15, μv=0.004, β1=0.0004, β2=0.00034, δ1=0.0016, δ2=0.00016, θ=0.00301, α1=0.001, and α2=0.001.



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*Source: 204241-2014-06-09.xml*