Analysis of the Mathematical Model for the Spread of Pine Wilt Disease

This paper formulates and analyzes a pine wilt disease model. Mathematical analyses of the model with regard to invariance of nonnegativity, boundedness of the solutions, existence of nonnegative equilibria, permanence, and global stability are presented. It is proved that the global dynamics are determined by the basic reproduction number R 0 and the other value R c which is larger thanR 0 . IfR 0 andR c are both less than one, the disease-free equilibrium is asymptotically stable and the pine wilt disease always dies out. If one is between the two values, though the pine wilt disease could occur, the outbreak will stop. If the basic reproduction number is greater than one, a unique endemic equilibrium exists and is globally stable in the interior of the feasible region, and the disease persists at the endemic equilibrium state if it initially exists. Numerical simulations are carried out to illustrate the theoretical results, and some disease control measures are especially presented by these theoretical results.


Introduction
Pine wilt is a dramatic disease of pine caused by the pinewood nematode (Bursaphelenchus xylophilus), which constitutes a major threat to forest ecosystems worldwide, from both the economical point of view and the environmental (landscape) perspective [1].Pine trees are affected by pine wilt disease, wilt, and usually die within a few months.Symptoms of pine wilt disease usually become evident in late spring or summer.The first observable symptom is the lack of resin exudation from barks wounds.The foliage then becomes pale green, then yellow, and finally reddish brown when the tree succumbs to the disease.The wood in affected trees is dry and totally lacks resin.
In [2], Evans et al. reviewed the principle of the Bursaphelenchus xylophilus transmission and disease dissemination.The Bursaphelenchus xylophilus is transmitted from pine tree to pine tree by a bark beetle called the pine sawyer (Monochamus alternatus) either when the sawyer beetles are fed on the bark and phloem of twigs of susceptible live trees (primary transmission) or when the female beetles lay eggs (oviposition) in freshly cut timber or dying trees (secondary transmission).Nematodes introduced during primary transmission can reproduce rapidly in the sapwood, and a susceptible host can wilt and die within weeks of being infested if conditions are favorable to disease development.In the summer, adult pine sawyers emerge from pine trees and fly to new trees.If the beetle is carrying the pine wood nematode, it spreads to the new trees (see Figure 1).
Pine wilt disease causes significant economic losses in natural coniferous forests in Eastern Asia (especially Japan, China, and South Korea) and Western Europe (especially Portugal).As such, the pine wilt disease is among the most important pests included in the quarantine lists of many countries around the world [3].In the beginning of the 20th century, pine wilt disease was first reported in Japan.Up to the present day, pine wilt disease has become the major ecological catastrophe of pine forests in Japan.For example, it has losses reaching over 2 million m 3 /year in the 1980s.Since then, it has spread to other Asian countries and regions such as China, Taiwan, and Korea, causing serious losses and economic damage.The pinewood nematode was first detected in Portugal's Setúbal region in 1999.Immediately, several governments of the European Union prompted actions to assess the extent of the nematode's presence and to contain Bursaphelenchus xylophilus and Monochamus alternatus in an area with a 30 km radius in the Setbal Peninsula, 20 km south of Lisbon [4].Despite the dedicated and concerted actions of government agencies, this disease continues to spread.In 2006, a new strategy which is the coordination of the national program for the control of the pinewood nematode for the control and ultimately the eradication of the nematode has been announced in Portugal [4].
Experience from control actions in Japan included aerial spraying of insecticides to control the insect vector (the cerambycid beetle Monochamus alternatus), injection of nematicides to the trunk of infected trees, slashing and burning of large areas out of control, beetle traps, biological control, and tree breeding programs [1].These actions allowed not only some positive results, but also unsuccessful cases due to the pinewood nematode spread and virulence.Other Asian countries also followed similar strategies, but the nematode is still spreading in many regions.
In recent years, many attempts have been made to develop realistic mathematical models for investigating the transmission dynamics of the pine wilt disease, see [5][6][7].Previous studies have worked on modeling of population dynamics of the vector beetle (Monochamus alternatus) and the pine tree to explore expansion of the disease using an integrodifference equation with a dispersal kernel that describes beetle mobility.In this paper, we revisit these previous models but retain individuality by building a differential system model to study the relation between pine tree, Bursaphelenchus xylophilus, and its insect vector Monochamus alternatus in the long run.The following assumptions are made in formulating the mathematical model.
It follows from the system (1) that () satisfies the following differential equation: This leads to () → (/) as  → ∞.Thus, the system (1) is reduced to the following three-dimensional system: The initial conditions of the system (4) are assumed as follows: This paper is organized as follows.In Section 2, we present some preliminaries such as the positivity and the boundedness of solutions.In Section 3, we firstly calculate the basic reproduction number of system (4).Then we obtain the local and global stability of the disease-free equilibrium.In Section 4, we present the local and global stability of the endemic equilibrium of the system (4).In Section 5, we conclude with some numerical simulations and discussions.

Positivity and Boundedness of Solutions
It is important to prove that the solutions of the system (4) are positive and bounded with the positive initial conditions (5) for the biology meaning. Let Proposition 1.All solutions (, , ) of the system (4) are nonnegative.Moreover,  is a global attractor in R 3 + and positively invariant for the system (4).
Proof.The first statement is trivial.It easily follows from the argument for reduction in the last equation that 0 ≤  ≤ /.It follows from the first two equations of the system (4) that for all  ≥ 0, then the second statement follows immediately.

Stability of the Disease-Free Equilibrium
For all infectious diseases, the basic reproduction number, sometimes called the basic reproductive rate or the basic reproductive ratio, is one of the most useful threshold parameters which characterize mathematical problems concerning infectious diseases.This metric is useful because it helps determine whether or not an infectious disease will spread through a population.In this section, we will calculate the basic reproduction number of the system (4).Moreover, we will obtain the local and global stability of the disease-free equilibrium.

Journal of Applied Mathematics
We can get Submitting  0 into F, then and giving is the next generation matrix for the system (4).It then follows that the spectral radius of matrix FV −1 is (FV −1 ) = √/.According to Theorem 2 in [8], the basic reproduction number of model ( 4) is In the following, we will discuss the local and global stability of the disease-free equilibrium.From above and [8], we can obtain the following theorems.
Theorem 2. The disease-free equilibrium  0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.
Proof.Define a Lyapunov function  of the system (4) as follows: Its derivative along a solution of the system (4) is It is clear from ( 16) that for R  ≤ 1, / ≤ 0 as / ≤  +  ≤ /.Furthermore, if  is the set of solutions of the system where / = 0, then the Lyapunov-Lasalle Theorem [9] implies that all paths in  approach the largest positively invariant subset of the set . Here,  is the set where  = 0. On the boundary of  where  = 0, we have  = 0 and / =  − .So  → / as  → ∞.Hence, all solution paths in  approach the disease-free equilibrium  0 .
Remark 4. This above result is of outmost importance because it shows that if at any time, through appropriate interventions, we are able to lower R 0 and R  to less than 1, then the pine wilt disease will disappear.Obviously, R 0 < R  .
The condition of global stability of disease-free equilibrium is stronger than that of local stability.In fact, since R 0 < 1 < R  , it is possible to be no outbreak appearance, see Figure 2(a).R  > 1 indicates that pine wilt disease could occur, while R 0 < 1 shows that the outbreak will stop since the infected pine decreases to the disease-free equilibrium.We know that R 0 and R  depend on , the mortality of Monochamus alternatus.Thus, if we are able to increase the value of , R 0 and R  will decrease.
In the following, we will consider the locally asymptotical stability of the positive equilibrium when R 0 > 1.
Hence, thanks to the Routh-Hurwitz criterion all eigenvalues of ( * ) have negative real part, and consequently  * is locally asymptotically stable.
We have shown that R 0 > 1 implies the existence and uniqueness of the endemic equilibrium  * .In the following, we provide sufficient conditions leading to a globally asymptotically stable infected steady state when R 0 > 1.The stability analysis of  * will be here performed through the geometric approach to global stability due to Li and Muldowney [10][11][12].Firstly, we will summarize the main facts related to our research.
Hirsch [13] and Smith [15,16] proved that threedimensional competitive systems that live in convex sets have the Poincare-Bendixson property [17]; that is, any nonempty compact omega limit set that contains no equilibria must be a closed orbit.
We recall additional definitions that we will use later.We first recall the basic definitions in [18].Suppose that (24) has a periodic solution  = () with minimal period  > 0 and orbit  = {() : 0 ≤  ≤ }.This orbit is orbitally stable if and only if, for each  > 0, there exists a  1 > 0 such that any solution (), for which the distance of (0) from  is less , remains at a distance less than  from  for all  ≥ 0. It is asymptotically orbitally stable, if the distance of () from  also tends to 0 as  goes to ∞.This orbit  is asymptotically orbitally stable with asymptotic phase if it is asymptotically orbitally stable and there exists a  > 0, such that, any solution (), for which the distance of (0) from  is less than , satisfying |() − ( − )| → 0 as  → +∞ for some  which may depend on (0).Definition 6.We say that the system (24) has the property of stability of periodic orbits if and only if the orbit of any periodic solution (), if it exists, is asymptotically orbitally stable.
The following lemma is the main tool to prove the global stability of the endemic equilibrium with disease.
Lemma 7 (see [19]).Assume that  = 3 and  is convex and bounded.Suppose that (4) is competitive and persistent and has the property of stability of the periodic orbits.If  0 is the only equilibrium in int() and if it is locally asymptotically stable, then it is globally asymptotically stable in int().
In order to apply this lemma to prove the globally asymptotically stability of the endemic equilibrium, we will prove the persistence of the system (4).Theorem 8. On the boundary of , the system (4) has only one -limit point which is the equilibrium  0 .Moreover, for R 0 > 1,  0 cannot be the -limit of any orbit in int().
Proof.The vector field is transversal to the boundary of  except in the -axis, which is invariant with respect to (4).On the -axis we have which implies that  → / as  → ∞.Therefore,  0 is the only -limit point on the boundary of .
To prove the second part of the position, we consider the function the derivative of which along solutions is given by Therefore, there exists a neighborhood  of  0 such that for (, , ) ∈  ∪ int() the expression inside the brackets is positive.In this neighborhood, we have  1 > 0 unless  =  = 0.Moreover, the level sets of  1 are the planes which go away from the -axis as  increase.Since  1 increases along the orbits starting in  ∪ int(), we conclude that they go away from  0 .This proves the proposition and therefore the persistence of system (4) when R 0 > 1.
By looking at the Jacobian matrix of the system (4) and choosing the matrix  as we can see that the system (4) is competitive in , with respect to the partial order defined by the orthant  1 = {(, , ) ∈ R 3 :  ≤ 0,  ≤ 0,  ≤ 0}.Our main results will follow from this observation and the above theorems.
Theorem 9.If R 0 > 1, then the positive equilibrium  * of the system (4) is globally asymptotically stable.
Proof.Since the system (4) is competitive permanent if R 0 > 1 holds and the only equilibrium point  * of the system (4) is locally asymptotically stable.Furthermore, in accordance with Lemma 7, Theorem 9 would be established if we show that the system (4) has the property of stability of periodic orbits.In the following, we prove it.

Numerical Simulations and Disease Control
In this paper, we investigate the dynamical behavior of a pine wilt disease model that incorporates a standard incidence rate.Qualitative analysis of the model is presented.The model has two equilibria, the disease-free equilibrium and endemic equilibrium.The behavior of the system (4) near each equilibrium has been studied.Threshold value of the relative basic reproductive number R 0 which determines the spread of infection has been worked out.
We obtain the basic reproduction number of the system (4), R 0 = √/.Furthermore, we obtain that the diseasefree equilibrium  0 ( 0 , 0, 0) of the system (4) is globally asymptotically stable when R 0 < R  < 1 (see Figure 2(c)).And if R 0 > 1, the endemic equilibrium  * ( * ,  * ,  * ) is globally asymptotically stable (see Figure 2(b)).That is to say, if R 0 and R  are lower than the threshold 1, the system of forest insect pest appears in disease-free state, that is, Bursaphelenchus xylophilus will be eradicated ultimately, which is what we hope; if R  > 1, while R 0 < 1, though the pine wilt disease could occur, the outbreak will stop since the infected pine decreases to the disease-free equilibrium.If R 0 is larger than 1, the system of forest insect pest have insect pests equilibrium, that is Bursaphelenchus xylophilus will not disappear, and with the time to develop, susceptible pine, infected pine, Bursaphelenchus xylophilus, and its vector will be extend to a stable state, that is, pine wilt disease will in an endemic steady state.
By simple calculation, we can find R  / < 0. Hence, we can make R  < 1 by increasing .Also from Figure 2, we can find that R  = 17.3205 > 1 in Figure 2(b) and the pine wilt disease is not disappearing.As  increases from 0.005 to 0.5, we calculate the R  = 0.5477 < 1 in Figure 2(c), keeping all the other parameters the same as in Figure 2(b).Thus, the disease-free equilibrium  0 ( 0 , 0, 0) of system (4) is globally asymptotically stable, and Bursaphelenchus xylophilus will be eradicated ultimately.From the Remark 4, we know that the pine wilt disease will not break out when R 0 < 1.So we only need to lower R 0 to less than 1 to control the pine wilt disease.We can also find R 0 / < 0 and R 0 / < 0 by a simple calculation.Hence, we can make R 0 < 1 by increasing  or (and) .Since  and  are the mortality of Monochamus alternatus and the percent isolated and felled of pine which has been infected by Bursaphelenchus xylophilus, respectively, we can increase  or (and)  by various control efforts in reality.On the one hand, we can take a set of measures to increase the mortality of Monochamus alternatus, for example, setting out beetle traps, setting vertical wood traps, releasing the natural enemies of Bursaphelenchus xylophagous (such as releasing the Scleroderma guanior), using chemicals to kill sawyer beetles to reduce the number of longhomed beetle.On the other hand, we can also take a set of techniques for specialized treatment of timber to increase the mortality of infected longhorned beetle; removal

Figure 3 :
Figure 3: (a) Simulation results showing the effect of decreasing infected pine trees to parameter  for  = 0.0005, 0.001, 0.0015, 0.002, 0.0025, and other parameter values are as given in Figure 2(b).(b) Simulation results showing the effect of decreasing infected pine trees to parameter  for  = 0.008, 0.016, 0.024, 0.032, 0.04, and other parameter values are as given in Figure 2(b).

𝑔 2 (
Fortunately, this lack of differentiability can be remedied by using the right derivative of (), denoted as  + ().