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 Open Access
Qualitative analysis and sensitivity based optimal control of pine wilt disease
 Aziz Ullah Awan^{1},
 Takasar Hussain^{2},
 Kazeem Oare Okosun^{3} and
 Muhammad Ozair^{2}Email author
https://doi.org/10.1186/s1366201814861
© The Author(s) 2018
 Received: 10 October 2017
 Accepted: 12 January 2018
 Published: 19 January 2018
Abstract
We design a deterministic model of pine wilt affliction to analyze the transmission dynamics. We obtain the reproduction number in unequivocal form, and global dynamics of the ailment is totally controlled by this number. With a specific end goal to survey the adequacy of malady control measures, we give the affectability investigation of basic reproduction number \(R_{0}\) and the endemic levels of diseased classes regarding epidemiological parameters. From the aftereffects of the sensitivity analysis, we adjust the model to evaluate the effect of three control measures: exploitation of the tainted pines, preventive control to limit vector host contacts, and bug spray control to the vectors. Optimal analysis and numerical simulations of the model show that limited and appropriate utilization of control measures may extensively diminish the number of infected pines in a viable way.
Keywords
 dynamical system
 pine wilt disease
 stability analysis
 sensitivity analysis
 optimal control
1 Introduction
Vectorborne illnesses are the maladies that outcome from disease transmitted by the nibble of infected arthropod species, for example, mosquitoes, fleas, ticks, and bugs. These biological agents that transmit contagious pathogen are called vectors. Malaria is the most regular case of vectorborne diseases. Many occurrences of vectorborne ailments are known for plants, for instance, coconut palm disease in palms and pine wilt illness in pine trees [1].
Pine wilt disease is a deadly ailment since it slays the infected tree within a few months. Bursaphelenchus xylophilus is the nematode that causes this disease. Monochamus alternatus, pine sawyer beetle, serves as a vector for this parasite, and it spreads the nematode to pine trees [2]. It was first observed in 1905 in Japan. In United States, the pine wood nematode was first reported in 1934. Asian countries other than Japan began to report presence of pinewood nematode in the 1980s.
The first noticeable pine wilt disease symptom is reduction in the flow of oleoresin from bark wounds. Another indication of pine wilt disease is change of needle color from light grayish green to yellowish green, yellowish brown, and finally completely brown as tree succumbs to the disease [3].
Three transmission pathways of pine wilt disease are perceived. One occurs when adult beetles infested with nematode fly to healthy pine trees and begin maturation feeding and transmit nematode into the tree, and this transmission is pointed as a primary transmission. The secondary transmission occurs during egg lying activities of mature female on dead or dying, freshly cut pine tree. Horizontal transmission of nematode occurs during mating as mature male search for female beetle in bark wounds like oviposition wounds [4].
In this paper, we formulate a mathematical model based on ordinary differential equations. This model describes the infectious disease of pine trees through pine sawyer beetles. The motivation behind this paper is twooverlay. The first is to discuss the qualitative behavior of the proposed model. The second point is to accomplish awareness about the most attractive method for limiting the transmission of the disease using the sensitivity analysis. On the basis of sensitivity analysis, the model is modified by including three timedependent controls: erosion of infected trees, treeinjection, and atmospheric pesticide spray.
2 Model framework
We formulate a fourdimensional mathematical model composed of the susceptible host pine trees \(S_{h}\) at time t that are at risk of being infected by the nematode. These trees radiate oleoresin that performs as a natural barrier to beetle oviposition, infected host pine tree \(I_{h}\) at time t that have stopped exduating oleoresin, susceptible vector beetles \(S_{v}\) at time t that do not have pinewood nematode, and the infected vector beetles \(I_{v}\) at time t that carry pinewood nematode. The common transmission of nematodes among pine trees and bark beetles occur during maturation feeding of infected vectors. The pine sawyers have pinewood nematode when it emerges from infected pine trees. However, the beetles may likewise get tainted directly through copulating. Let \(N_{h}\) denote total population of host pine trees, and let \(N_{v}\) denote the total vector population consisting of adult beetles at any time t, respectively. Hence mathematically the populations are given by the equations \(N_{h}=S_{h}+I_{h}\) and \(N_{v}= S_{v}+I_{v}\).
Let \(\Pi_{h}\) be the constant recruitment rate of pine trees at time t, and let \(\Pi_{v}\) be the constant appearance rate of adults beetles at time t. We assume that the \(\delta_{1}\) represent the transmission rate per contact during maturation feeding and \(\beta_{1}\) accounts the average number of contacts per day with vector adult beetles during maturation feeding. The transmission rate of the nematode through infected vectors is denoted by \(\delta_{2}\), and \(\beta_{2}\) denotes the average number of contacts per day when adult beetles oviposit. The nematode carrying rate of adult beetles emerging from deceased trees is \(\beta_{3}\). The incidence terms for the host population are \(\beta_{1}\delta_{1}S_{h}I_{v}\) and \(\beta_{2}\delta_{2}\eta S_{h}I _{v}\) during maturation feeding and oviposition, respectively. The incidence terms for vector population are \(\beta_{3}I_{h}S_{v}\) and \(\beta S_{v}I_{v}\), where β is the rate at which beetles pass on nematode during mating. The susceptible pine trees are exploiting at the rate \(\mu_{h}\), the infected pine trees are isolating and felling at the rate σ, and \(\mu_{v}\) is the death rate of vector population.
3 Existence of equilibria

\(C<0\) if and only if \(R_{0}>1\).

A is always positive.

\(B>0\) for \(R_{0}<1\).
4 Stability of equilibria
4.1 Global stability of diseasefree equilibrium
Theorem 4.1
If \(R_{0}\leq 1\), then the diseasefree equilibrium \(E_{0}\) of model (3) is globally asymptotically stable in Δ.
Proof
4.2 Global stability of endemic equilibrium
When the threshold parameter \(R_{0} > 1\), the uniform persistence of (3) can be proved by applying the technique given in [5], and the global stability of unique endemic equilibrium \(E^{\ast }\) can be proved by using the technique of geometrical approach developed by Li and Muldowney [6]. The geometric approach applied to hostvector models can be studied in [7, 8].
Theorem 4.2
([6])
Suppose that \(H_{1}\), \(H_{2}\), and \(H_{3}\) hold. The unique endemic equilibrium \(E^{\ast }\) is globally stable in Δ if \(\bar{q} _{2}<0\).
Clearly, \(\Delta =\{(S_{h},I_{h},I_{v})\in R_{+}^{3}\vert 0\leq N_{h} \leq \frac{\Pi_{h}}{\mu_{h}}, 0\leq I_{v}\leq \frac{\Pi_{v}}{\mu_{v}} \}\) is a simply connected region, so \(H_{1}\) holds. The boundedness of ξ and Lemma 5.1 given in [5] imply that system (3) has a compact absorbing set \(K \subset \Delta \). Thus \(H_{2}\) holds. \(H_{3}\) holds in the view of Theorem 3.1. The appropriate vector norm \(\vert x\vert \) in \(R^{3}\) has been chosen together with the matrixvalued function \(P(x)=\operatorname{diag}(1,\frac{I_{h}}{I_{v}},\frac{I_{h}}{I_{v}})\) of order \(3\times 3\).
5 Sensitivity analysis
Parameter values used for sensitivity analysis
Parameter  Description  Numerical Value  Reference 

\(\Pi _{h}\)  The recruitment rate of the host pine population  0.009041  [9] 
\(\Pi _{v}\)  A constant emergence rate of the vector pine sawyer beetle  0.002691  [9] 
\(\mu _{v}\)  The natural death rate of vector population  0.011764  [10] 
\(\mu _{h}\)  The natural death rate of host population  0.0000301  [11] 
β  The rate at which the beetles get directly during mating  0.00305  Assumed 
\(\beta _{3}\)  The rate in which the adult beetles have pinewood nematode when it escapes from dead trees  0.00305  [12] 
\(\beta _{1}\)  The rate in which infected beetles transmit nematode by contact  0.00166  [13] 
\(\delta _{1} \)  The number of contacts during maturation feeding period  0.2  [14] 
\(\beta _{2}\)  The rate in which infected beetles transmit nematode by oviposition  0.0004  [13] 
\(\delta _{2}\)  The number of contacts during the oviposition period  0.41  [9] 
η  The probability in which the susceptible host pine is not infectious by nematode and ceases oleoresin exudation naturally  0.0000301  [9] 
σ  The felling rate of infectious pine trees  0.004  Assumed 
Definition
Sensitivity indices of \(\pmb{R_{0}}\) , \(\pmb{I_{h}^{*}}\) , and \(\pmb{I_{v}^{*}}\) , based on the parameter values given in Table 1
Parameter  Sensitivity Index  Sensitivity Index  Sensitivity Index 

\(\boldsymbol{R_{0}}\)  \(\boldsymbol{I^{*}_{h}}\)  \(\boldsymbol{I^{*}_{v}}\)  
β  0.0385  0.0947  0.127 
\(\Pi _{v}\)  1.0  2.143  2.882 
\({\mu _{v}}\)  −2.0  −3.992  −5.370 
\(\beta _{3}\)  0.961  1.755  2.361 
\(\Pi _{h}\)  0.961  2.755  2.361 
\(\beta _{1}\)  0.961  2.048  1.755 
\(\delta _{1} \)  0.961  2.048  1.755 
\(\beta _{2}\)  0.0000143  0.0000304  0.0000261 
\(\delta _{2}\)  0.0000143  0.0000304  0.0000261 
η  0.0000143  0.0000304  0.0000261 
\(\mu _{h}\)  −0.961  −2.048  −1.755 
σ  −0.961  −2.755  −2.361 
By the analysis of sensitivity indices the most sensitive parameter is \(\mu_{v}\). The reproduction number \(R_{0}\) is inversely connected to \(\mu_{v}\). Thus, it can be said that an increase (or decrease) in \(\mu_{v}\) by 10%, \(R_{0}\) decreases (or increases) by 20%. Similarly if we increase (or decrease) σ by 10%, then \(R_{0}\) will also decrease (or increase) by 10%.
The endemic level of infected pine trees is inversely related to the mortality rate of bark beetles and exploitation rate of infected pine trees. We see that \(I_{h}^{*}\) is decreased (increased) by almost four times with respect to the parameter \(\mu_{v}\), and it is decreased (increased) almost 27% by increasing (decreasing) the exploitation rate by 10%.
The endemic level of infected vectors is again inversely related to the mortality rate of bark beetles and exploitation rate of infected pine trees. We observe that \(I_{v}^{*}\) is decreased (increased) by almost five times with respect to the parameter \(\mu_{v}\), and it is decreased (increased) almost 23% by increasing (decreasing) the exploitation rate by 10%.
The sensitivity indices of \(R_{0}\), \(I_{h}^{\ast }\), and \(I_{v}^{ \ast }\) proposed that three controls, nematicide injected into the trunk of uninfected trees, cutting down infected trees burning and burying, and spray of insecticides, can be applied for vector control.
6 Optimal control analysis
Theorem 6.1
Proof
7 Numerical simulations
Now, we investigate numerical results for the efficacy of the optimal control planning for the disease spread in a community. We have chosen the set of weight factors \(A_{1}=1\), \(A_{2}=5\), \(B_{1}=3\), \(B_{2}=7\), \(B_{3}=9\) and initial state variables \(S_{h}(0)=100\), \(I_{h}(0)=50\), \(S_{v}(0)=150\), \(I_{v}(0)=20\) and \(r_{0}=0.55\), \(r_{1}=0.1\). The other parameter values are given in Table 1. We investigate numerically the effect of the following optimal control strategies.
7.1 Application of preventive measures \((u_{1}\neq 0)\) and exploiting and disposing off infected pines \((u_{2}\neq 0)\)
7.2 Application of preventive measures \((u_{1}\neq 0)\) and spray of insecticides \((u_{3}\neq 0)\)
7.3 Use of exploitation of infected pines \((u_{2}\neq 0)\) and spray of insecticides \((u_{3}\neq 0)\)
7.4 Use of preventive measures \((u_{1}\neq 0)\), exploitation of infected pines \((u_{2}\neq 0)\) and spray of insecticides \((u_{3}\neq 0)\)
7.5 Effects of weight constants
Declarations
Authors’ contributions
All authors carried out the proofs of the main results and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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