Fractional Order Modeling the Gemini Virus in Capsicum annuum with Optimal Control

: In this article, a fractional model of the Capsicum annuum ( C. annuum ) affected by the yellow virus through whiteﬂies ( Bemisia tabaci ) is examined. We analyzed the model by equilibrium points, reproductive number, and local and global stability. The optimal control methods are discussed to decrease the infectious B. tabaci and C. annuum by applying the Verticillium lecanii ( V. lecanii ) with the Atangana–Baleanu derivative. Numerical results described the population of plants and comparison values of using V. lecanni . The results show that using 60% of V. lecanni will control the spread of the yellow virus in infected B. tabaci and C. annuum in 10 days, which helps farmers to afford the costs of cultivating chili plants.


Introduction
Growth of Capsicum annuum (chili plant) [1,2] is excessive in the mid-hill region of India. It contains vitamin C, provitamin A, and calcium, which are good for health. The spicy taste and high nutritional benefit the marketing of C. annuum. Capsicum annuum is used in the pharmaceutical industries to increase immunity, antiulcer, analgesic, antidiabetic, and antihemorrhoid agents. The extracts of C. annuum employ to relieve the pain of inflammation of joints, headaches, neuralgia, and burns. The framers require advantages to yield the C. annuum in large amounts. Due to natural obstacles like soil erosion, irrigation, and diseases spread, farmers encounter heavy losses while fertilizing C. annuum.
The cause of Geminivirus (yellow virus) [3][4][5] is one of the difficulties experienced by farmers in the cultivation of C. annuum. Yellow spots appear in young leaves and shoots, and the leaves turn out to be bright yellow or mixed yellow-green, which are symptoms of the yellow virus. This virus spreads by whitefly (Bemisia tabaci) from one host to another continuously.
Controlling these viruses using overlapping cropping methods is very difficult, since insecticide must be applied to mature plants. Controlling techniques vary depending on the conditions of the plants infected by virus variety, environment, and time. To reduce the populations of white-flies, systemic insecticides are applied to control the spread of the virus, as well as to cure the infected white-fly insect and plants, like rust fungi, etc., who have the host cyclodepsipeptide toxin. This toxin was produced via the mycelium of entomopathogenic fungi (Verticillium lecanii) [6][7][8]. However, the excess use of V. lecanii generates high costs. To minimize costs of controlling the B. tabaci population, an optimal control method must be found.
Optimal control is the approach of ascertaining control and circumstances path for dynamic systems to minimize an accomplishment period. The origin of the optimal control is related to the calculus of variations. In the 1940s, the formulation of dynamic programming in the optimal control was developed by Rochard Bellman. Using the analytical method, some of the optimal control problems' solutions are difficult to find. N.H. Sweilam et al. [28] discussed the optimal control method for cancer treatment using AB-derivative. R. Amelia et al. [29] showed results to help farmers afford the costs of cultivating the red chilies by optimal control. In [30] N.H. Sweilam and S.M. AL-Mekhlafi described the fractional model of multistrain TB cure with optimal control. The optimal control problems to solve numerical procedures were investigated in [31,32].
To the best of our knowledge, the study of the C. annuum of the yellow virus with optimal control by applying the AB-derivative to the model is yet to come. This article was organized as follows: the basic results and definitions of the AB-derivative are discussed in Section 2. In Section 3, the formation of the C. annuum model with AB-derivative is explained. In Section 4, the optimality conditions demonstrate. In Sections 5 and 6, numerical results with graphs for the fractional optimal control problem have presented and conclusions.
Motivated by [22,23,28,30], this document discusses the fractional model of Geminivirus in C. annuum with AB-derivative via optimal control and stability analysis. The main contributions are organized as follows: (A) The fractional model of Geminivirus in C. annuum with AB-derivative constructed. (B) We obtained some stability results of this fractional model and discussed the equilibrium points and reproductive number of the model. (C) We derived the optimal control of this fractional model and plotted the population and comparison results of each variable in the model.

Preliminaries
This section briefly discussed some preliminaries regarding fractional derivatives. There are few definitions for the fractional derivatives, including Riemann-Liouville, Caputo, and Caputo-Fabrizio [9,18]. Recently, a new fractional derivative with Mittag-Leffler Kernel was elaborated and implemented in a few real-world models [24][25][26][27]. We present the following definitions.
The Riemann-Liouville fractional integral (RL) is defined as follows [9,18] The Riemann-Liouville fractional order derivative (RL) is defined as follows [9,18] The Caputo fractional order derivative (C) is defined as follows [9,18] The Caputo-Fabrizio fractional order derivative in Caputo sense (CFC) is defined as follows [33] where φ ∈ H 1 (0, T), M(ζ) is a constant of normalization that depends on ζ, which satisfies that, M(0) = M(1) = 1. The Atangana-Baleanu fractional order derivative in the Riemann-Liouville sense (ABR) is defined as follows [17] The Atangana-Baleanu fractional order derivative in the Caputo sense (ABC) is defined as follows [17] ( ABC The Atangana-Baleanu fractional integral of order ζ of a function φ(t) is defined as [17] The Mittag-Leffler function of one and two parameters E α (z), E α,β (z) is defined as [9] E α (z) = The generalized Mittag-Leffler function is defined as where Γ(·) denotes the Gamma function, and note that

Modeling Framework of Gemini Virus
The fractional model based on the cure of yellow virus in C. annuum by V. lecanii with modified variables and parameters is presented. Here, the parameters depend on the fractional model, which is an extension of the integer model given in [29]. The mathematical model of C. annuum with AB fractional derivative is represented as follows: The total population is denoted by N p of C. annuum N p = S v + I v + S g + I g is taken to be constant. The total population of B. tabaci is denoted by N v = S BT + I BT . Here, the total population can be divided into 6 subgroups. The recruitment rate of C. annuum and B. tabaci is denoted by A and B respectively. The growth rate of C. annuum from vegetative to generative phase is denoted by α. β 1 , and β 2 denoted the infection rate of C. annuum in the vegetative and generative phase respectively. γ 1 , and γ 2 denoted the infection rate of B. tabaci in the vegetative and generative phase respectively. δ p stands for the rate of use of V. lecanii. The death rate of C. annuum is denoted by µ p . The natural death rate of B. tabaci is denoted by µ 1 , and the death rate of B. tabaci due to curative intervention is denoted by θ 1 .

Invariant Region
The fractional order C. annuum model of yellow virus (1) can be analyzed in the biological feasible region discussed as follows. The system (1) is split into two parts, namely the C. annuum population (N p ; with To establish the positive invariance of F i.e., solutions in F remain in F for all t > 0 . Adding the first four equations and the last two equations of the model (1) gives This can be used to show that the fractional order of the C. annuum and B. tabaci population in the system (1) shows that which implies that Hence, it is sufficient to consider the dynamics model of system (1) in F. The mathematical model is well-posed in the region F.
∴ Every solution of the basic model (1) with initial conditions in F remains in F for all t > 0. The result is summarized below.

Disease-Free Equilibrium Point
To evaluate the equilibrium points Let

Reproduction Number
For the basic reproduction number for the C. annuum model (1), suppose that y = (S v , S g , I BT ) and using next generation matrix approach [34], we have where Jacobian of F and V at E 0 , we have The basic reproduction number ψ 0 comes from the spectral radius ψ 0 = ρ[F V −1 ], given by
Proof. Endemic equilibrium point is obtained from system (1), and by putting right-hand side of each equation equal to zero, we have It is obvious from the values of S * v , I * v , S * g , I * g , S * BT & I * BT that there exists a unique positive endemic equilibrium point E * , if ψ(0) > 1. (1) is locally stable at E 0 for ψ 0 < 1 and unstable for ψ 0 > 1.

Proof. The Jacobian of system (1) is
which follows that all the eigenvalues are negative if ψ 0 < 1 and eigenvalues are positive for ψ 0 > 1. Hence, we conclude that the system (1) is locally stable under the condition ψ 0 < 1 and unstable for ψ 0 > 1. (1) is globally stable, if ψ 0 > 1 at E 0 .

Theorem 3. The system
Proof. First, we construct the Lyapunov function L(t), for the system as: Then, differentiating the Equation (5) with respect to time, we have By manipulating along the point E 0 , we get d dt (L(t)) = −(θ 1 δ p N p + µ 1 ) Again differentiating the above equation, we have By manipulating along the point E 0 , we get Therefore, if ψ 0 > 1, then d dt (L(t)) < 0, which implies that the system (1) is globally stable for ψ 0 > 1 at E 0 .

Remark 1.
In the case of ψ 0 < 1 at E * , it is an interesting problem to find an effective strategy to prevent the disease.

Optimal Control
The purpose of the dynamic red chili model is to minimize the population of plants infected during vegetative or generative period and insects infected by optimizing V. lecanni using AB-derivative [28][29][30].
The objective functions used are as follows: where u 1 is the late of giving V. lecanni and A i ≥ 0, for i = 1, 2, . . ., 4 is the cost coefficient and t f is end time in [0, T f ]. Therefore, by using u 1 V. lecanni in the Equation (1), it becomes Now, to minimize the objective functional: subject to the constraints where, ξ i = ξ(S v , I v , S g , I g , S BT , I BT , u 1 , t), i = 1, 2, . . ., 6, with initial conditions:

Numerical Results
Here, we examine the mathematical model of C. annuum with AB fractional derivative and optimal control numerically. We assume initial conditions and parameter values in Table 1 with ζ = 0.9 and N(ζ) = 1.
The optimal control is since 0 ≤ u 1 ≤ 1. Consider u 1 = 0.6 and using the parametric values in (7) then, which gives the numerical values plotted in Figures 1-6.
In Figures 1-6, we show that the use of 60% V. lecanni for 5 days changed the population of susceptible and infected plants in vegetative and generative phases, as well as the variation of population of susceptible and infected white bugs.       Table 1. Parametric representation with approximation values of (1) in [29].

Variable with Values Definition
The death rate of B. tabaci due to curative intervention When u 1 = 0, i.e., without V. lecanni, the control system reduce to the model given below: In Figures 7-12, we compare the numerical values of the variables S v (t), I v (t), S g (t), I g (t), S BT (t) and I BT (t) with and without V. lecanii u 1 . The comparison of these variables of ordinary differential equations is shown in [29] of the figures are Figures 2-7. The results with V. lecanni in each stage of plants and insects are more effective and accurate in Atangana-Baleanu fractional derivative than ordinary differential equations stated in [29].       Figures 7 and 9 show that the susceptible C. annuum in vegetative and generative phases, which increased the population by 1% with V. lecanni compared to that of without V. lecanni because the infected B. tabaci cannot transmit the virus through chili plants. Figures 8 and 10 show the comparison of infected C. annuum in vegetative and generative phases, which decrease the population by 1% with V. lecanni compared to that of without V. lecanni.
In Figure 11, the comparison of susceptible B. tabaci population decreases with V. lecanni by 50% compared to that of without V. lecanni because the infected B. tabaci cannot infect the healthy one with an antidote. In Figure 11, the comparison of infected B. tabaci population decreases with V. lecanni by 4% compared to that of without V. lecanni because the infected ones were either cured or dead due to curative intervention.
In Figure 13, the measure of implementing 60% of V. lecanni per day will reduce 1% of infected C. annuum and 1% of infected B. tabaci. By continuing this process, the 60% of V. lecanni control the spread of the yellow virus within 10 days, which helps the farmers to afford the costs of cultivating the C. annuum.

Conclusions
In this study, we described the C. annuum model of the yellow virus in two discrete aspects. First, we examined the C. annuum model and applied the optimal control. Second, we analyzed the C. annuum model using the Atangana-Baleanu derivative. The threshold quantity is less than one when the presented model is locally stable. Furthermore, the model is globally stable when ψ 0 > 1. With the help of V. lecanii (an entomopathogenic fungus), u 1 (t) optimal control reduced the population of infected B. tabaci and C. annuum. The numerical results of optimal conditions of the C. annuum model with AB-derivative are described detailly by successive approximation method. The infected population of C. annuum increases and decreases according to V. lecanii use and vice-versa for susceptible. The results show that using 60% of V. lecanni controls the spread of the yellow virus in infected B. tabaci and C. annuum over 10 days, which helps farmers to afford the costs of cultivation.