Examination of Pine Wilt Epidemic Model through Efficient Algorithm

: Pine wilt is a dramatic disease that kills infected trees within a few weeks to a few months. The cause is the pathogen Pinewood Nematode. Most plant-parasitic nematodes are attached to plant roots, but pinewood nematodes are found in the tops of trees. Nematodes kill the tree by feeding the cells around the resin ducts. The modeling of a pine wilt disease is based on six compartments, including three for plants (susceptible trees, exposed trees, and infected trees) and the other for the beetles (susceptible beetles, exposed beetles, and infected beetles). The deterministic modeling, along with subpopulations, is based on Law of mass action. The stability of the model along with equilibria is studied rigorously. The authentication of analytical results is examined through well-known computer methods like Non-standard finite difference (NSFD) and the model’s feasible properties (positivity, boundedness, and dynamical consistency). In the end, comparison analysis shows the effectiveness of the NSFD algorithm.

eyesight, hair and skin regrowth, and red blood cell production. It can be used as a cough syrup to treat coughs and rib congestion and for sore throats. Mathematical modeling is a powerful tool for describing how the disease spreads. We can regulate those components that substantially impact the disease's transmission, and we can prepare several control measures to limit contamination's spread. Haq et al. [1] in 2017, presented a fractional-order wide-ranging model designed for the blowout of pine wilt disease using the Laplace Adomian decomposition method. Ozair et al. [2] in 2020, proposed a vector-swarm scientific model for the pine wilt disease taking local sensitivity analysis of parameters and numerical experiments to illustrate the theoretical bases for the anticipation and governor of the disease. Khan et al. [3] in 2017, suggested a mathematical classification of calculations for the pine wilt disease in two ways. Abodayeh et al. [4] in 2020, established a mathematical model to explore the result of an asymptomatic carrier to identify critical parameters to examine several intervention options. Lee investigated the best controller approach for anticipating pine wilt disease using analytical and numerical techniques acceptable to do; we put on two control approaches; treeinjecting of nematicides and the destruction of adult beetles through floating pesticide spraying [5]. Khan et al. [6] in 2018, presented the Caputo-Fabrizio fractional-order mathematical model of pine wilt disease to investigate elementary properties of the model also check numerical simulation by taking a particular parameter. Agarwal et al. in 2019, recommended a stochastic pine wilt disease model taking the primary reproductive number, and an adequate condition is provided to study the extinction and permanence of the disease. Underlying the stochastic differential equation system is analyzed, and proper Lyapunov functionals are formulated to show the stability analysis [7]. Awan et al. [8] in 2018, recommended qualitative research and sensitivity-based model taking reproduction numbers in the specific form to check the result of three control measures. Tamura et al. in 2019, proposed spatiotemporal analysis of pine wilt disease using the quantified number of PWNs and face levels of supposed pathogenesis-related (PR) genes in different positions of Japanese black pine seedling over time cured by Taqman quantitative real-time PCR (qPCR) assay. As a result, PWNs and PR levels increased drastically, leading to plant death [9]. Hirata et al. in 2017, explored the potential distribution of PWD under climate change scenarios. Pinus forests are at risk of serious harm due to environmental shifts and the spread of PWD [10]. Lee et al. [11] in 2013, studied a disease transmission model based on primary reproduction number Ro. If Ro is a smaller amount than one than disease-free equilibrium state obtained if Ro more than one then endemic equilibrium is globally asymptotically stable condition obtained. Shah et al. [12] in 2018, suggested swarm vector dynamics of pine wilt disease model taking worldwide asymptotic constancy investigation at changed equilibrium points and exposed that disease vanishes when beginning quantity falls less unity. Awan et al. [13] in 2016, studied the qualitative behavior of pine wilt disease by considering indirect and direct transmission using primary reproduction numbers. Gao et al. in 2015, studied pine wilt disease attacks happening earth possessions and pine woodland groups in the three valleys region of China. This study shows that the PWD has stuck Masson pine forest soil properties and altered forest communal structure. The disease is negatively related to Masson pine and positively associated with broad-leaved trees [14]. Shi et al. in 2013, proposed a scientific model for the spread of pine wilt disease. Primary reproduction numbers determine the global dynamics [15]. Nguyan et al. in 2016, recommended a spatially explicit model of pine wilt disease taking dispersal pattern (direction and area) of Asia such as infested neighborhoods, short-and long-distance dispersal, asymptomatic carriers, and typhoon (incorporating biological and environmental events). Using receiver operating characteristics and pair-correlation functions shows that disease occurs in both local and global aspects [16]. Khan et al. [17] in 2018, studied the changing elements of pine wilt disease is stable happening local and globally. Hussain et al. [18] investigated the dynamics of pine wilt disease with the sensitivity of parameters. Raza et al. [19] studied the structurepreserving analysis of the epidemic model with necessary properties. Some more techniques related to epidemic models are presented in [20,21]. The well-known results with different techniques are studied in [22,23]. For the best presentation, more work on the epidemiology and efficiency of the techniques are studied in [24][25][26][27][28][29][30][31][32]. In this paper, we study the dynamics of pine wilt disease via algorithms. We can observe that computational methods in literature have many problems like negativity, unboundedness, and inconsistency of solutions. These issues will resolve by our proposed idea that is a non-standard finite difference method (NSFD). Also, NSFD fulfills the properties of the biological problem. The rest of the paper is styled as follows: In Section 2, the modeling of pine wilt disease is defined. In Section 3, the construction ways of the epidemic model, equilibrium points, and computational methods and their convergence. In the last section conclusion and future problems are discussed. represents the already exposed pine trees at any time, I H (t): represents the infected pine trees at any time, N V : represents the total population of vector (beetles) at any time t, S V (t): represents the susceptible beetles at any time, E V (t): represents the exposed vector beetles at any time and I V (t): means the infected vector beetles at any time. Thus, a continuous model for populations regarding pine wilt disease is described in Fig. 1. The fixed values of the model is defined as follows: Λ H : represents the rate of recruitment of susceptible pine trees, κ 1 : represents the rate of contact during maturation, ψ: represents the average number of connections with vector beetles during development, κ 1 ψS H I V : represents the incidence rate, κ 2 : represents the probability of transmission of a nematode by an infected beetle, φ: represents the average number of contacts per day when adult beetles oviposit, α: represents the susceptible pine trees without being infected by the nematode, κ 2 φα: represents the transmission through oviposition, κ 2 φαS H I V : represents the number of new infections, δ: represents progression rate from exposed pine trees to infected trees, d 1 : represents the natural death rate, Λ v : represents the vector pine beetle's emergence rate, η: represents the measured rate, ηS V I V : represents the adult beetles escaping from dead trees carry the PWN, μ: represents the transfer rate from of infectivity, d 2 : represents the demise rate and γ : represents the disease-induced death rate. The system of differential equations can be derived from the above flow chart of the population as follows: The feasible region of the system (1-6) is as follows:

Model Equilibria
The system (1-6) has two types of equilibria in the feasible region D * is as follows: , , CMC, 2022, vol.71, no.3

Reproduction Number
In this section, we find the reproduction number R o by using the next-generation method. We introduce two types of matrices: the transition matrix and the second is transmission matrix. ⎡ , , as desired. Note that R 0 is called the reproduction number of the model.
Proof : Let us consider, Therefore, model (1-6) will be converted into a new form The general Jacobian matrix is defined as By substituting the values, we obtained , 0, 0 above matrix will become: Here we let Clearly, Trace(J(K 0 )) = T 1 + T 2 + T 3 + T 4 + T 5 + T 6 < 0 and det(J(K 0 )) > 0 This implies that disease-free point K 0 is locally Asymptotically stable.

Non-Standard Finite-Difference Algorithm
The NSFD could be developed for the system (1)-(6), the Eq. (1) of the pine wilt epidemic model may be calculated as: The decomposition of proposed method is as follows: In the same way, we decompose the remaining system into proposed NSFD method, like (9), as follows: I n+1 where the discretization gap is denoted by "h".

Linearization Process of NSFD Algorithm
In this section, we shall present the theorem at the equilibrium of the model for the process of linearization of the NSFD algorithm is as follows:  The general from of Jacobian matrix, we have The given Jacobean matrix at Disease-Free Equilibrium (DFE) K 0 = ∧ H d 1 , 0, 0, ∧ V d 2 , 0, 0 is as follows: 2022, vol.71, no.3 Now, for endemic equilibrium (EE) K 1 = (S H * , E H * , I H * , S V * , E V * , I V * ). The given Jacobean matrix is The proof is straightforward. By using the Mathematica, this is a guarantee to the fact that all values of Jacobian lie in a unit circle, as desired.

Results
In this section, we used the scientific literature presented in Tab. 1 for the simulating behavior of the system (9)- (14) at both equilibria of the model as follows:  Fig. 3 shows that the two-dimensional kernel density between years and its transmission. Fig. 4 shows the distribution analysis of illness at the given data. Fig. 5 shows the splines connectedness of disease. In Figs. 6a-6b, we used the commandbuilt software ODE-45 to simulate the model's behavior at any time t. In Figs. 7a-7b, is the true sense results affected by the non-standard finite difference method at any time step size. This computer method has the advantage over the other two methods like Euler and Runge Kutta. Independent of time step size, low-cost and effective technique. In this article, we investigated the subtleties of the numerical epidemic model with numerical strategies' effective use. We divided the entire tree population into six groups: susceptible trees, exposed trees, infected trees, easy vector beetles, exposed vector beetles, and infected vector beetles. We have calculated the reproduction number for the pine wilt disease numerical epidemic model. We have also presented the local stability at the steady states of the model, that is, at pine wilt free equilibrium and at pine wilt existing compensation, by using well-known mathematics results. We have concluded that we can control the dynamic of the pine wilt by positively using different affected techniques like, vaccination is proved to be the ultimate solution to avoid the spread of this disease. Immunization is much essential and is recommended for trees that fall under this disease. Moreover, Booster doses are recommended for disease trees to make sure of vaccination. And most importantly, a sound hygiene system is much needed to adapt, which can play a vital role in reducing the spread of pine wilt disease. In the future, we could extend this type of modeling to other complex epidemiological models and their branches.