A FRACTIONAL DIFFERENTIAL EQUATION MODEL FOR THE SPREAD OF POTATO LEAF ROLL VIRUS (PLRV) ON POTATOES

ABSTRACT


INTRODUCTION
In Indonesia, potatoes are one of the essential commodities of vegetables.Indonesia is the largest country that produces potatoes in South East Asia [1].Potatoes are susceptible to several diseases, but Potato Leaf Roll Virus (PLRV) is the dominant one worldwide [2].Potatoes infected with the PLRV virus will experience a decrease in production up to 90%.The potato plant can be infected with the PLRV in two courses: primary and secondary infections.A primary infection is caused by the virus-carrying aphids (vector) during the growing season, while a secondary infection occurs when contaminated bulbs are planted [3].
Controlling the spread of PLRV involves various measures, including using virus-free seed potatoes, implementing strict aphid control strategies, and promoting good agricultural practices.Mathematical models can be utilized to understand the dynamics of PLRV spread better and assess the effectiveness of different control measures.Early detection and prompt action are crucial in managing and minimizing the impact of the Potato Leaf Roll Virus on potato crops.
Fractional differential equations can be used to understand the dynamics of real-life situations.Using fractional differential equations can provide better and more detailed information in approaching a problem.The fractional differential equation involves the derivative of an unknown function with fractional orders.For example, Ahmad et  .This article reviewed the research by adding another new parameter: the rate of vector death because predators are viewed as an enemy to the vector.In this article, predators can help control the spread of PLRV in potatoes because predators are considered enemies of the vectors.Also, several studies have shown that leaf predators can help control vector populations and indirectly reduce the spread of diseases caused by these vectors.

RESEARCH METHODS
The research method contains sources of data and data analysis to show how to control the spread of PLRV on Potatoes.

Sources of Data
This paper uses data from published articles from Tilahun et al. [15].The data has been collected in Table 2 to perform numerical simulations to investigate the effect of some parameters on disease control and support the theoretical analysis.

Data Analysis
This paper has developed a system of fractional differential equations of the PLRV model using fractional derivative Caputo to the ordinary differential equations.The PLRV model is analyzed qualitatively to determine the conditions of the PLRV model, and numerical simulation is performed by using MATLAB and presenting the results as a graph.

PLRV Model using Fractional Differential Equation
The fractional differential equation model of PLRV spread considers the populations of potatoes and vectors, each population consisting of susceptible and infected subclasses.The fractional differential equation model of PLRV spread has four compartments:   () representing the number of susceptible potato populations at time ,   () representing the infected potato population at time ,   () representing the susceptible vector population at time , and   () representing the infected vector population at time .
The plot and parameters of the model for the spread of PLRV in potatoes can be seen in Figure 1 and Table 1 below, respectively.Natural death rate of potato From Figure 1 and Table 1, system of differential equation are generated.
Furthermore, system Equation ( 1) is converted into a fractional differential equation system for the order 0 <  < 1 and  = 1 using a fractional Caputo derivative [6].For example, covert the first equation   on system Equation (1) into a fractional derivative.
Moreover,     ,     , and     are obtained similarly, so the following fractional differential equation was collected.

Invariant Region
The following theorem is the feasible solution that satisfies all the constraints and conditions imposed by the given system.

Positive Solutions
All solutions of the fractional differential equation model of PLRV spread in potatoes are positive for future time if all the initial values are positive.Theorem 2. If   0 ≥ 0,   0 ≥ 0,   0 ≥ 0,   0 ≥ 0 then all solution sets (  (),   (),   (),   ()) of the system of the model are positive for the future time.

Equilibrium point and basic reproduction number
There are two equilibrium points, disease free equilibrium point and endemic equilibrium point associated with basic reproduction number.The disease free equilibrium is the condition with no infection (  =   = 0).Equation (3) has a disease free equilibrium point if it does [9]:     = 0,     = 0,     = 0,     .
Basic reproduction numbers are obtained by using the next generation matrix method [10].
where  and  are the results of linearization using the Jacobian at the disease free equilibrium point which is (  ,   ,   ,   ) = ( The basic reproduction number is the dominant of eigenvalues.Therefore Furthermore, the endemic equilibrium is the condition with infection (  ≠ 0,   ≠ 0). .

Stability of equilibrium point
There are two main types of stability associated with equilibrium points: local stability and global stability.Local stability and global stability of the equilibrium points is given in the following theorem.
Theorem 3. The disease free equilibrium point is locally asymptotical stable if  0 < 1 and unstable if  0 > 1.
Previously, it has shown that  1 ,  2 > 0 if  0 < 1, thus polynomial Equation ( 19) has the negative real root and satisfies |(  )| =  >  2 for any 0 <  ≤ 1.Thus, disease free equilibrium point is locally asymptotically stable if  0 < 1.   Figure 2 shows that fractional order which is  related to how fast the system is heading towards an endemic PLRV point on potatoes, that is  * = (16.257,1.804, 67.857, 41.856).Figure 2 also shows that the susceptible potato population and infected potato population is decreasing steadily.The susceptible vector population gets an extreme decrease at initial time and then increase of susceptible vector population follows.Otherwise, infected vector population gets an extreme increase at initial time and then the infected vector population getting decrease in future time.Figure 3 shows that fractional order which is  related to how fast the system is heading towards a free PLRV point on potatoes, that is  0 = (20, 0, 32.7586, 0).Fractional order  changes will affect the complexity interaction in the system.Figure 3 also shows that susceptible potato population and infected potato population is getting decreasing steadily.Susceptible vector population getting extreme decrease at an initial time and then increase of susceptible vector population followed.Otherwise, infected vector population gets an extreme increase at initial time and then in future time, infected vector population gets decrease faster than the endemic PLRV condition on potatoes.Figure 6 shows that when the elimination rate of infected potato ( 2 ) rises, the infected potato population is reduced.It means, increasing the elimination rate of infected potato can be control the spread of PLRV.
al. used fractional differential equations for the reaction-diffusion model [16], and Singh et al. applied them to obtain a hyperbolic-type solution for a particular equation [17].Fractional differential equations form a mathematical model that can describe the propagation of PLRV.Mapinda et al. have used a typical differential equations system to model the spread of Banana Xanthomas Wilt bacteria (BXW) on bananas.Studies have shown that pruning bananas infected with BXW and sterile farming tools are necessary strategies to control the distribution of BXW [4].Shah et al. applied a fractional differential equation to study the spread of pests in tea plants.The findings suggest that selecting predators as the enemy of problems has reduced the spread of pests in tea plants [5].Ali et al. described regression modeling strategies to predict PLRV disease [18].Furthermore, Bonyah formulated a potato disease model in a fractional-order derivative [19].Tilahun et al. have also researched the mathematical model of PLRV deployment using differential fractional equations and stability [15]

Figure 4 .Figure 4
Figure 4. Numerical Solution of Variation of  on PLRV modelFigure4shows that as the infection rate of potato () rises, so does the infected potato population.It means, the virus will spread faster as the infection rate of potato increases.

Figure 5 .Figure 5
Figure 5. Numerical Solution Of Variation of  on PLRV modelFigure5shows that as infection rate of the vectors () rises, so does the infected potato population.

Figure 6 .
Figure 6.Numerical Solution Of Variation of   on PLRV model

Table 1 . Parameters Description of the PLRV model
Population of susceptible potato   Natural death rate of vector   Population of susceptible vector  1 Replanting rate of potato   Population of infected potato  2 Recruit rate of vector   Population of infected vector  Infection rate of potato  1 Virus induced death rate  Infection rate of vector  2 Elimination rate of infected potato   Death of vector by predators

Theorem 5 .
[14]disease free equilibrium point is globally stable if  0 < 1. Next, calculate   (  ,   ,   ,   ) to show    ≤ 0 at disease free equilibirum point[14].Then the endemic equilibrium of the fractional order model is globally stable in the interior of Ω.