Modeling and analysis for the transmission dynamics of cotton leaf curl virus using fractional order derivatives

In this study, we examine the Atangana Baleanu Caputo fractional order for the transmission dynamics of the Cotton Leaf Curl Virus disease. The model took into account both cotton plants and vector populations. The existence and uniqueness, positivity and boundedness of the solution to the model, as well as other fundamental concepts, were examined. Additionally, the Ulam-Hyres condition stability of the suggested model was demonstrated using functional techniques. Using the Adams-Bashforth method, the numerical solution for our suggested model was computed. The numerical result shows that the disease spreads more slowly as the fractional order decreases from 1.00 to 0.72.

For India's cotton fabric region, manufacturing, and output data from 1980 to 2013, Sundar Rajan and Palanivel [12] offered six non-linear growth models: monomolecular, logistics, Gompertz, Richards, quadratic, and reciprocal growth. To forecast the size of the leaves of a cotton crop over consumption in farming systems, a multiple regression model was developed in [13]. To increase the use of water and identify the ideal soil conditioner yield levels. [14] looked into the leaf area index (LAI) algorithms and the connections among LAI, moisture content, and yield over cotton produced in Korla, Xinjiang, China, using three different nutrients for the soil. [15] explored the connection between silverleaf whitefly populations, environmental variables, and CLCuV incidence in Pakistan's farming mechanism. The relationship between the Pakistani cotton leaf curl virus outbreak and weather variables was simply modeled in [16]. In [17], the author created a model for CLCuV by dividing the total population into populations of cotton plants and vectors. Both populations include subgroups that are infected and susceptible.
Fractional calculus is crucial to create more accurate results and to better understand how the memory effect influences epidemiological models. Simply said, it is more adaptive than classical calculus because of inherited traits and memory descriptions [18]. Basically, fractional derivatives are distinguished from one another by the several kernels that can be chosen to suit the requirements of diverse applications. The main distinction between the Caputo fractional derivative [19], Caputo-Fabrizio derivative [20], and Atangana-Baleanu fractional derivative [21] is that while the Caputo derivative is characterized by a power law, the Caputo-Fabrizio derivative is characterized by a decaying law, and the Atangana-Baleanu derivative is characterized by a Mittag-Leffler law.
Many researchers have looked into the possibility that memory effect can be incorporated into fractional calculus [22][23][24]. This memory uses information from the past and present to make predictions. Because of this, it differs from integer derivatives. Numerous practical applications of fractional calculus have led to the development of both analytical and numerical approaches to solving problems [25][26][27][28][29][30][31][32] and [33]. In fractional derivatives, several studies have recently been developed. One of the best operators is the Atangana Baleanu-Caputo (ABC) operator [34]. An expanded Mittag-Leffler function with a nonsingular and nonlocal kernel serves as the foundation for this operator.
However, there haven't been any studies done to use a fractional order derivative to model the dynamics of CLCuV transmission. In this study, we extend the ABC fractional order derivatives for CLCuV diseases from the model created by [17] to other diseases. The remaining sections of the essay are organized as follows: we complete a few foundational ideas for the Atangana-Baleanu fractional operator in part two. The model is developed in part three. Part four establishes the model's existence and uniqueness, as well as positivity and boundedness for the model. The model's stability analysis using Hyers-Ulam stability is covered in part five. In parts six and seven, we use the ABC for numerical scheme and MATLAB software for simulation to the CLCuV model. In part eight, conclusions are provided.

Model formulation
The CLCuV model was split up into cotton plant and vector population for this study. The total cotton plants ( ( )) as a whole is divided into susceptible and infected subcategories. Cotton that is prone to infection is designated as and susceptible cotton as . This is defined as: ( ) = + . Within the entire vector population ( ( )), there are susceptible and infected vectors subclasses. The vectors and stand for the susceptible vector and the infected vector, respectively. This is defined as: ( ) = + . The rate of hiring of vulnerable vector 2 and the transition to infected vectors ( ) with 2 rate after consuming diseased plants or cotton were both considered in the model. When infected vectors ( ) eat susceptible cotton ( ), the diseases propagate to cotton at a rate of 1 . The susceptible cotton ( ) also replanted at a rate of 1 then the cotton was infected with the diseases. Once infected, cotton never recovers and yields either nothing at all or very little. To regulate the illness, the parameter 1 is the induced death rate and The AB derivative can be derived by changing the in the system (1) to 0 , which is defined by the differential equation system given in equation (2).

Theorem 1. The eco-epidemiologically feasible region of AB model
Using Lemma 3, we can demonstrate that the set Ω is positively invariant.
Due to (5), all of (5)'s solutions are nonnegative and remain in ℝ 4 + , making the set Ω established in (4) positively invariant for the equation system (2). Finally, given that all of the parameters are positive, we move on to the next step, the boundedness of the fractional model's solutions (2), by adding up all of the model's equations for cotton and the vector population, which results in: Using the Laplace transform to the equation (6) gives , Using the work which is done by [41] and applying the inverse transform, the solution will be provided by: where , refers to the Mittag-Leffler function. Given that the Mittag-Leffler function exhibits asymptotic behavior it is not difficult to observe that ( ) → 1 for goes to ∞. Also, applying Laplace transform to the equation (6) gives Similarly by using the same procedures to equation (7) we obtain ( ) → 2 for goes to ∞. Thus, (4) is the biologically feasible region of system of equation (2).

Existence and uniqueness of solutions
This part investigates the existence and uniqueness of the solutions to the fractional-order model (2). We apply the well-known Banach fixed point theorem to show the existence of the solution to model (2), a detailed analysis of fixed points and contractions, using [42] applying and the references within it. We now go through the steps to show that the solution exists and is distinct. Model (2) yields the following when the AB fractional integral is used.
] is the Banach space of real-valued continuous functions defined on an interval = [0, ] with the corresponding norm defined by ‖ , , ,
In (6), we have the norm on both sides of each equation Moreover, the first equality in (11) can be simplified as follows:

Theorem 3. The ABC fractional model given in (2) has a solution if we determine Π 0 such that it satisfies the inequality.
( Proof. From (12) and (13) we get , , ] .
From (14) we get

Hyers-Ulam (HU) stability
This part evaluates the stability of the fractional model. Different stability types have been used to analyze the epidemic model. Because HU-type stability has the advantage of providing an approximative solution when a problem is complex, many researchers have recently used it for epidemic models. [43] and [44] for more information about the HU stability. In this analysis, we use HU-type stability for CLCuV. For quick proof, we first demonstrate that model (2) is HU stable.
Proof. Since we are aware that ( ) ∈ holds for (16), let > 0. Additionally, using Theorem 4, we assume that ( ) is the only solution to the suggested model (2). Thus, Consequently, we calculate using Lemma 4 and the properties of the triangular inequality to (21).

Numerical simulation
In this part, the effects of different fractional order values on the model are explored. To this end, we present various results of the model using a numerical method created by Toufic and Antagana, as shown in equations (30)- (33). Following initial conditions are applied during simulations and analysis: (0) = 700, (0) = 150, (0) = 100, (0) = 180, and the parameter values are given in Table 1. Fig. 2a shows that increasing the fractional order ( ), from 0.72 to 1.00 leads to decreasing number susceptible cotton plants ( ). Fig. 2b shows that increasing the fractional order ( ), from 0.72 to 1.00 leads to increasing number of infected plants ( ). Fig. 3a shows that increasing the fractional order ( ), from 0.72 to 1.00 leads to decreasing number susceptible vector population ( ). Fig. 3b shows that increasing the fractional order ( ), from 0.72 to 1.00 leads to increasing number of infected vector population ( ).
In other words, based on Figs. 2 -3, we can conclude that decreasing , significantly reduces the number of and cases. Besides that, as shown in the figures, the curves for each compartment , and compress as decrease from 1.00 to 0.72. We can conclude that as gets closer to zero from the right, diseases in , and decreases. This actually means that the memory impact of the dynamic system is larger when the derivative order is decreased from 1.00 to 0.72, and the infection in each compartment progressively increases for a long time. According to the data, crop infection growth is greatly decreased when the derivative's fractional order is decreased, and no change in rate of change is observed when the