Implementation of Adomian Decomposition Method for Maize Streak Virus Disease Model to Reduce the Contamination Rate in Maize Plant

In this paper, the Maize Streak Virus disease model which involves the Maize and the Homopteran population is considered. This system of the differential equation is analytically solved by using Adomian Decomposition Method. The analytical results are compared with numerical simulation by assuming certain values for the parameter. Further, the demolishing and contamination rate of contagious and receptive homopteran on receptive and contagious Maize plant ) ( 2 , 1 i i parameters are analyzed to reduce the contamination rate. In addition, the death rates of contagious maize, receptive homopteran and contagious homopteran ) , , ( 3 2 1 d d d are also discussed to remove the contagious population and make them free from Maize Streak Virus.


Introduction
Maize is the major source of food, fuel, feed and fibers. Maize is a common food in many places around the world, specifically in sub-Saharan Africa. Provitamin A-biofortified maize crop is used to reduce Vitamin A deficiency in humans. Processed maize flour is used to prepare porridge. Maize (Zea mays L.) plant grows in the latitude 58N to 40S which is around agroecological areas of the continent Africa. The maize plant is a tall grass that consists of a strong central stalk surrounded by few branches. Maize  They are brown spots, common rust, downy mildew, head smut, stalk rot, banded leaf, sheath blight, ear rot, northern leaf blight, southern leaf blight and gray leaf spot. Gray leaf spot is caused by Cercospora zeae maydis. Northern and Southern leaf blight is due to Helminthosporium turcicum and Helminthosporium maydis. Banded leaf and sheath blight are because of Rhizoctonia solani. Common rust and the brown spot are caused by Puccinia sorghi and Physoderma maydis. Head smut and ear rot are due to Sphacelotheca reiliana and Aspergillus sp. Downey mildew is because of Sclerospora sorghi (Subedi, 2015;Ward et al., 1999). This paper deals with Maize Streak virus disease. The severe damage is brought on by Maize Streak Virus-A, one of the 11 different variants which are responsible for the serious damage caused to the maize plant. A family of plant viruses namely Geminiviridae is responsible for this major damage in maize plants. These viruses have one strand, circular DNA, and their heads are made up of coat proteins organized in a quasi-icosahedral pattern. Maize streak virus belongs to Geminiviridae family and the genus Mastrevirus. This disease was at first described as Mealie Blight, Mealie Yellows or Striped leaf disease in the year 1901 by Claude Fuller. The symptoms include minute circular spots in the leaf which gradually develops into yellow stripes in leaf veins and blade (Shepherd et al., 2010) ( These are the description of the maize family and the cause of the maize streak virus disease. In section 2, we describe the mathematical model of Maize Streak virus disease as proposed by Haileyesus Tessema Alemeh et.al (Alemneh et al., 2019). In section 3, we will deal with the basic concept and analytical solution of the model using Adomian Decomposition method. In Section 4, results and discussions of the various model parameters are compared with analytical and numerical simulations of the maize streak virus diseases.   The list of parameters is described in the following

Basic Concept of Adomian Decomposition Method
The above differential equation is divided into linear and non-linear terms in the Adomian Decomposition Method. The linear operator that represents the linear term of the equation is inverted and then applied to the given equation. The non-linear term is decomposed into a series which is termed as Adomian polynomials. The result creates a series whose terms are defined by the Adomian polynomials' recurrence relationship. Many mathematical models have been solved using Adomian decomposition method in various fields of science (Adomian, 1990(Adomian, , 1994 A non-linear differential equation of the form, is considered. Here F represents the non-linear differential operator and f y , are functions of t . In operator form (3.1) becomes, f Ry Ny Ly = + + (3.2) where N L, and R represents the linear, non-linear and remaining linear operator of . F When we apply (3.2) by the inverse operator The rest of the terms are determined by the recurring relationship. The non-linear term is broken down into a set of polynomials called Adomian polynomials, which are shown as: where n A is given by, The recurring relationship is given by, Thus Adomian Decomposition Method produces a series that is absolutely and uniformly convergent.

Computation using Adomian Decomposition Method
Using the Adomian Decomposition Method, the system of nonlinear differential equations defined in equation (2.1) 14) The non-linear terms are represented by Adomian polynomials.
The Adomian polynomials are derived as follows: Expanding the series we get, By assigning certain numerical values to the parameters,

RESULTS AND DISCUSSIONS
The analytical solution for the maize streak virus disease model from equation (3.23) to (3.26) is compared with numerical simulation to produce effective results. Here the initial conditions of the model are .
We analyze each parameter involved by assuming certain values of the parameter and keeping other variables fixed. The graphs which are obtained through Matlab software give us a clear understanding of the slope in the system of differential equations.        (Figure 2(a)), demolishing and contamination rate of Contagious Homopteran on Receptive maize plant 1 i ( Figure  2(b)). The analytical solution is depicted by the star (***) and the solid line (___) indicates numerical simulation accordingly.            (Figure 3(b)).

ACKNOWLEDGEMENT
The corresponding author Dr.Malinidevi Ramanathan and the first author Ms.Vaishnavi Kalirajan declare that their research has not received any funding or grant from any organizations.