Numerical Modeling and Symmetry Analysis of a Pine Wilt Disease Model Using the Mittag–Lefﬂer Kernel

: The existence of man is dependent on nature, and this existence can be disturbed by either man-made devastations or by natural disasters. As a universal phenomenon in nature, symmetry has attracted the attention of scholars. The study of symmetry provides insights into physics, chemistry, biology, and mathematics. One of the most important characteristics in the expressive assessment and development of computational design techniques is symmetry. Yet, mathematical models are an important method of studying real-world systems. The symmetry reﬂected by such a mathematical model reveals the inherent symmetry of real-world systems. This study focuses on the contagious model of pine wilt disease and symmetry, employing the q -HATM ( q -Homotopy Analysis Transform Method) to the leading fractional operator Atangana–Baleanu (AB) to arrive at better understanding. The outgrowths are exhibited in the forms of ﬁgures and tables. Finally, the paper helps to analyze the practical theory, assisting the prediction of its manner that corresponds to the guidelines when contemplating the replica.


Introduction
Forests play pivotal roles in human life. Its importance cannot be expressed by words because a forest alone can rejuvenate the Earth entirely. As such, it is important to protect trees. Trees are not only a spreading green carpet on Earth, but they provide fresh air to breathe, and give basic amenities. Unfortunately, pine wilt disease (PWD), although exquisite in its majestic appearance, causes the death of the tree within a few weeks after showing the symptoms. There are several diseases that infect the vascular system of plants that are regarded as wilt diseases. Fungi, bacteria, and nematodes can attack plants and lead to their instant elimination. Plants also can possess viruses. In woody plants, wilt diseases can be categorized into two parts: those that begin at the branches and those that begin with the roots. The ones that begin at the branches are likely to begin with the pathogens that feed on leaves or bark, and the others initiate infection by lesioning or by pathogens paving their way directly into the roots, although few spread to other plants through the root scion. PWD is a wild and perilous disease among pine wood. Pine wilt disease can damage the economy, subsequently disrupting social development. Hence, there no way to save the pine trees after they are infected; preventing infection is the only way to protect the pine forest. Pine wilt disease is caused by a pinewood nematode. It ruins the affected trees within a few weeks. This dangerous disease destroys trees in both eastern and western countries. PWD is a serious pest among pine woods. Several researchers have scrutinized the different aspects of PWD. Mamiya [1,2] discussed the history and pathology of PWD, which is caused by Bursaphelenchus xylophilus. In the year 1997, Fukuda [3] explored conceptual functions related to the progress or inhibition of PWD using pathological experiments. Proenca et al. [4] gave an explanation regarding the characteristics of the bacteria that are carried by the pinewood nematode for a better understanding of PWD. Ozair et al. [5] deliberated the Bio-stimulated analytical heuristics to examine a PWD model. In the past few decades, several investigators discussed a stability analysis of PWD, as well as its causes, in their studies [6][7][8].
The terminology mathematical modelling is used when demarcating a system using mathematical notions and language and producing a hypothetical mathematic depiction based on the evolving mathematical model. Mathematical modelling can be utilized in numerous fields to convert real world scenarios into just numbers and mathematical data. Mathematical modelling in epidemiology heightens our understanding of the system underlying the spread of pandemic diseases. The application of mathematical modelling knowledge can aid in understanding plant disease trends and develop, as well as assay, strategies to combat these trends, thus helping to avoid global food shortages and fulfilling the basic need for food. Mathematical modelling is a convenient theory to observe a disease and its proliferation. Such a model explores the non-faced measurement statistics that are generated for the microorganism. Numerous models have been developed to analyze the disease. Diverse techniques can be utilized to analyze the disease evolution. In this view, several researchers have discussed the epidemic model of pine wilt and other diseases caused by different viruses by using diverse numerical and analytical methods [9][10][11][12][13][14][15].
Leibnitz grasped a fraction in derivative, revealing that fractional calculus is far more applicable to modern, real-world problems than classical calculus. Fractional calculus interprets nature's authenticity in the most elegant and orderly way possible. The fractional differential equations, and its remarkable application, have undergone significant improvements in a number of fields such as liquid machinery, chemistry, hydrology, ecology, and manufacturing.
The fractional-order differential equations have found a reputation of modeling problems in the propagation of seismic waves, finances, viscoelastic materials, permeable media, and many other physical processes. The kernel Atangana-Baleanu fractional derivative (ABFD) relies on the standard operation of Mittag-Leffler without locality and singularity. The ABFD is well-suited to define the corporal and material realities of the world. Differences in kernel unity provide a better definition of memory within a structure at a different level. In addition, the Atangana-Baleanu operators satisfy all mathematical rules underneath the fractional calculus range. The numerous concepts of ABFD and fractional derivatives were discussed by several researchers [16][17][18][19][20][21][22][23][24][25][26][27][28].
Finding exact solutions to fractional differential equations appears to be far more challenging than finding accurate solutions to their integer-order counterparts. As a result, great emphasis has been placed on the development of effective analytical and numerical techniques for approximating solutions to this type of issue. The Homotopy perturbation method, Laplace decomposition technique, Homotopy analysis method, and Adomian decomposition method are some of these methods. The q-homotopy analysis transform approach is another really effective technique. When q ∈ 0, 1 n , this q-homotopy analysis transform method combines the Laplace transform method (LTM) with the Homotopy in the q-HATM solution ensures that it converges faster than the traditional HAM [29]. To discover an appropriate solution for this wild illness, we used the q-homotopy analysis transform method (q-HATM) to solve a system of non-linear equations. The Atangana-Baleanu operator was contracted with the Mittag-Liffler function in order to consider the new fractional operator called Atangana-Baleanu operator [30].

Mathematical Optimization and Miniature
The idea of generalization of fractional calculus is the chief motto of this current study. To understand the disease and its spread, we consider mathematical modeling as an excellent factor. Such kinds of models need to unify to maintain the volumes of data that is produced on host-pathogen communication. Many factors such as the conceptual study of an inhabitant's growth of infections on vulnerable plants and animals are affected by this problem. Various methods can be adopted to analyze the spread of a disease and several models have been developed on pest-tree dynamics [10,24].
In order to capture the proper solution for the disease, the q-homotopy analysis transform method is applied to the system of nonlinear equations in this study. The new fractional operator, called Atangana-Baleanu, has been considered to produce better results.
In the current study, we deliberate the epidemiology model developed by Kamal Shah and Manar A. Alqudah for observing the transmission of the virus and obtained some results for the considered epidemic model. This extensive model is classified into non-linear equations. The susceptible class of pine trees is symbolized by S(t), the exposed pinetrees are denoted as E(t), the infected class of pine trees is symbolized by I(t), and the susceptible beetle class is referred to as R(t). In addition, the infective class of beetles are symbolized by Q(t) [6,8,[12][13][14].
Definition 3. The Laplace Transform, which is analogous to the AB operator, is written as: where K(α) is a normalising function such that K(0) = K(1) = 1.

Theorem 1.
The following Lipschitz criteria are satisfied for the Riemann-Liouville and AB derivatives, respectively [16,22,23]: and ABC Theorem 2. The solution to a fractional differential equation ABC a D α t h 1 (t) = s(t) is given as [16]: where K(α) is a normalising function such that K(0) = K(1) = 1.

The Fundamental Aspect of the q-Homotopy Analysis Transform Method
Consider the nonlinear, non-homogeneous partial differential equation of a fractional order to emphasize the basic notion of the suggested technique: where D α t v(ξ, t) signifies the function ABC derivative of the function v(ξ, t), where R designates the bounded linear differential operator in ξ and t, N indicates the operator for nonlinear differential equations, and g(ξ, t) is a concept that comes from a source.
Using the Laplace transform of Equation (9), we get the following equation: In order to illustrate Equation (10), we have: The procedure for constructing a homotopy for a non-zero primary function is as follows: where q ∈ 0, 1 n and a real function of ξ, t and q is φ(ξ, t; q). For non-zero primary functions, we create a homotopy as follows: where L is the Laplace Transform symbol, and q ∈ 0, 1 n , (n ≥ 1) appears to be the constant parameter. In fact, h = 0 is an auxiliary variable, φ(ξ, t; q) is an unspecified function, and v 0 (ξ, t) is an initial estimation of v(ξ, t; q). For q = 0 and q = 1 n , respectively, the following results are acceptable: The result φ(ξ, t; q) converges from v 0 (ξ, t) to v(ξ, t) by expanding q from 0 to 1 n . Using Taylor's theorem near q to expand the component φ(ξ, t; q) in series structure: where We may have one of the outcomes for Equation (9) by using the fundamental linear operators v 0 (ξ, t), nandh. At q = 1 n , the series φ(ξ, t; q) converges: Divide by r! and compute for q = 0 by differentiating the zeroth order deformation equation r-times with respect to q, which affords: Inverting the equation Transform of Laplace: where We have: Hence, we can obtain the v r (ξ, t) iterative term by solving the above. The q-homotopy analysis transform method's series solution is symbolized by:

q-HATM Solution for the Prediction Phase
In this segment, we will discuss the model proposed by Sing et al. [30], who developed the q-HATM solution technique using q-HAM and the Laplace Transform. As a result, several researchers have improved the technique for solving nonlinear differential equations of various kinds. Particularly when compared to other modified methodologies, it has demonstrated outstanding outcomes in a variety of circumstances such as human disease, fluid mechanics, corporal models, budget growth, optics, and others [25,29,31].
To demonstrate the behavior in Equation (2), we employed a Fractional-Order system of equations: With initial conditions: We acquire Equation (25) by applying the Laplace Transform to Equation (24): Now, the non-linear operator is projected as: The rth order deformation equation is given by expressing the given scheme and for H(x, t) = 1 : We simplify the preceding equations using the Inverse Laplace Transform as follows: We have arrived the equations by solving the above: S 2 (t) = 11.9519890h(n+h) By simplifying the preceding equations, the provided values can be obtained. As specified by the q-HATM series solution, (32)

Results and Discussion
Generally, present world issues are not determinable, but rather incorporate stochastic impact, which provides a supplementary sensible method of demonstrating the viral dynamics. In this study, we used the q-homotopy analysis transform technique (q-HATM) to solve a system of non-linear equations and find an appropriate solution for an epidemic pine wilt disease model. The initial conditions for the provided model in this work are A series solution has been assessed to understand the behavior of the model. We prove and find a solution method for S(t); E(t); I(t); R(t); and Q(t) utilising different fractional orders (α) with respect to time (t) with the help of tables. From the diagram, it can be seen that the estimated model relates to the order and provides higher comfortability. As a result, using the fractional operator to anticipate the future model yields a satisfactory outcome. We can see from the diagram that the estimated model is strongly influenced by the order and provides more flexibility. In this section, we evaluated the tabular values and figures obtained by q-HATM. Tables 1-5 show the solution for S(t); E(t); I(t); R(t); and Q(t), respectively, for various fractional orders (α) with respect to time (t) obtained by q-HAT. We designate a solution for S(t); E(t); I(t); R(t); and Q(t) in Figures 1-5 correspondingly.

Conclusions
The productive model of pine wilt disease inside the complicated structure of fractional calculus is analyzed using the q-Homotopy Analysis Transform Method. The results of the model analysis revealed that the approach and fractional operator are particularly useful in analyzing real-world scenarios. Furthermore, the current method saves time and does not require any deviation while solving nonlinear models. It is concluded that the considered nonlinear model can be used to derive results for existent, non-exist- Tables 1-5 illustrate the answer to S(t); E(t); I(t); R(t); and Q(t), respectively, acquired by q-HATM. We have chosen inclined values for α to discuss the spread of disease and growth in diseased beetles. One can observe from Table 1 that enhancing values of α with respect to t increases the numerical value of the susceptible class of pine trees. Table 2 depicts the difference in exposed pine trees for varied values of α. The exposed pine tree population gradually declines with rising values of α with respect to time. The variation in the infected class of pine trees for diverse values of fractional order changes according to its time, as depicted in Table 3. One can detect from Table 3 that I(t) gradually or slightly increases with a fixed value of α with respect to a rise in t. However, a rise in values of α for a fixed time t declines I(t). Table 4 is tabulated to discuss the variation in the numerical value of the population of the susceptible class of beetles for distinct values of α. It is detected from the table that R(t) gradually increases for inclined values of α with respect to fixed. The variation in the population of the infective class of beetles for rising values of α with respect to time is tabulated in Table 5. One can observe from the table that an increase in α with respect to a fixed time reduces the numerical value of the population of the infected class of beetles. From the table, we can conclude that the fractional derivative method applied to hypothetical models allows for a noteworthy augmentation in the fitting of genuine information when compared to old-style models.
We interpret the solution by q-HATM for S(t), E(t), I(t), R(t), and Q(t) in Figures 1-5, correspondingly. Figure 1 displays the pictural illustration of the exact simulation for susceptible pine trees S(t) in a time t for diverse values of α at h = −1 and n = 1. One can observe from the plotted figure that the susceptible population of pine wilt trees increases for a rise in values of α with respect to time t. Here, S(t) acts as an increasing function of t. The visual appearance of the analytical scheme for the exposed class of pine tree in a time t for changed α at h = −1 and n = 1 is displayed in Figure 2. It is evident from the plotted graph that an increase in α with respect to time decreases the exposed class of pine trees. Further, E(t) acts as increasing function of t for a fixed α. Figure 3 shows a visual depiction of the numerical solution for the infective class of pine tree I(t) in time t for different values of α at h = −1 and n = 1. One can detect from the plotted graph that a rise in values of α increases the infective class of pine trees. Here, I(t) acts as increasing function of t for a fixed α. To verify the characteristic feature of susceptible pine trees at various fractional orders at h = −1 and n = 1, Figure 4 is portrayed. For an inclined fractional order, it is observed that the population of susceptible pine trees grows. Furthermore, R(t) acts as a deceasing function of t for a fixed α. One can observe that the fractional differential co-efficient order at h = −1 and n = 1 enhances the rate of growth in the infective class of beetles, as revealed in Figure 5. Here, Q(t) acts as an increasing function of t for a fixed α.

Conclusions
The productive model of pine wilt disease inside the complicated structure of fractional calculus is analyzed using the q-Homotopy Analysis Transform Method. The results of the model analysis revealed that the approach and fractional operator are particularly useful in analyzing real-world scenarios. Furthermore, the current method saves time and does not require any deviation while solving nonlinear models. It is concluded that the considered nonlinear model can be used to derive results for existent, non-existent, and other phenomenal factors.