SOLUTION OF TYPHOID FEVER MODEL BY ADOMIAN DECOMPOSITION METHOD

SOLUTION OF TYPHOID FEVER MODEL BY ADOMIAN DECOMPOSITION METHOD AJIMOT FOLASADE ADEBISI1,*, OHIGWEREN AIRENONI UWAHEREN2, OLUSOLA EZEKIEL ABOLARIN3, MUSILIU TAYO RAJI4, JOSEPH A. ADEDEJI1, OLUMUYIWA JAMES PETER2 1Department of Mathematics, Osun State University, Oshogbo, Osun State, Nigeria 2Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria 3Department of Mathematics, Federal University Oye Ekiti, Ekiti State, Nigeria 4Department of Mathematics, Federal University of Agriculture Abeokuta, Ogun State Copyright © 2021 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we presents a deterministic mathematical model on the dynamics of typhoid fever disease. The Adomian Decomposition Method (ADM) is used to solve the model equations. In solving the model, the validity of the ADM is established by the classical fourth-order Runge-Kutta method implemented in Maple 18. In other to confirm the accuracy of the method, a comparison was carried out between the ADM solution and Runge-Kutta(RK4). The findings obtained confirm the precision and accuracy of the ADM to cope with the study of morden epidemics.


INTRODUCTION
Typhoid fever is a systemic infection, triggered by the ingestion of infected food or water, caused by the bacterium Salmonella Typhi. Prolonged fever, fatigue, nausea, loss of appetite, and diarrhea or even diarrhoea describe acute illness. Symptoms are often non-specific and other febrile SOLUTION OF TYPHOID FEVER MODEL BY ADOMIAN DECOMPOSITION METHOD disorders are clinically non-distinguishable [1]. There are also non-specific and clinically nondistinguishable signs of typhoid fever from other febrile diseases. Medical severity, however, varies and serious cases can lead to significant complications or even death. Recent findings indicate that an estimated 11-20 million people get sick from typhoid fever and between 128 000 and 161 000 people die from it every year. The disease is endemic in South America [2], Indian subcontinent [3], Southeast Asia [4] and mostly in Africa [5]. During year 2000, it was estimated that the disease caused illness is 21.6 million and 216,500 deaths worldwide. For many researchers, modeling the transmission dynamics of typhoid fever is an essential and important study. The study of infectious diseases in the past has concentrated primarily on their effects on the human population. Although, infectious diseases are present to some degree in human societies at all times, the results of epidemics are the most evident and spectacular [6].
George Adomian [7], an American mathematician, was the first to establish Method of Adomian decomposition. It is a form of semi-analysis that can be used in the solution of partial and ordinary differential equations in both nonlinear and linear order. It can also be used in solving higher order nonlinear differential equations. Also, it can be used in solving nonlinear differential equations of higher order. [8][9] considered Adomian Decomposition approach to solve deterministic models but not on the typhoid fever model. In fluid mechanics, [10][11][12] and in numerical analysis, [13,14].
The aim of this paper is to present the application of Adomian Decomposition Method to the proposed model and to verify the validity of the Adomian Decomposition Method in solving the model using Maple 18's classical fourth-order Runge-Kutta method as a basis for comparisone.  Susceptible population are incresesed by immigration or birth at the rate  . We assume that proportion  of susceptible class progress to carrier infected class, while the compliment   1 migrate to infected class. We assumed that the rate of transmission  for carriers is higher than the rate of transmission  of symptomatically infected individuals due to the fact that they are more likely to be unaware of their condition, and therefore continue with their regular activities.

MODEL FORMULATION
Carriers may become symptomatic at a rate  . Infectious individuals can receive treatment and recover at the rate  . Susceptible individuals receive vaccination to protect themself against infection at the rate  . 1 - is an educational parameter that caters for limiting both carriers and symptomatic individuals from spreading typhoid. This parameter lies in the interval 0 <  < 1 It means that there are no education programs in place so that vulnerable people are unaware of typhoid fever.


, then it means that all susceptible individuals are fully aware of typhoid fever, that is to say they know what causes the diseases, how it is spread and how to avoid contracting the disease. Table 1 gives a detailed summary of the parameters, while Figure 1 shows the model's compartmental flow diagram. The above description can be represented by a system of differential equations given as where   L y is the differential operator,   R y is the remainder of the differential operator,   N y is the nonlinear terms,   f t is an inhomogeneous term and L can be defined as the The composition solution series can be written as From the model (2), since this model is the system of first order differential equations then we define differential operation ( L ) and its inverse operator respectively with initial value problem. Applying (4) into (2), we have Simplifying the left hand side of (8) with the interval from 0 to t according to 1 L  definition. We

NUMERICAL SIMULATION AND GRAPHICAL ILLUSTRATION OF MODEL
In this section,we presents the numerical simulation which demonstrate the analytical results for the model. This is accomplished by using the set of parameter values based on the literature and assumptions given in table 1, as well as assumptions. By substituting the following initial conditions for the different compartments.

DISCUSSION OF RESULTS FOR ADOMIAN DECOMPOSITION METHOD
The solutions obtained by using Adomian Decomposition Method with given initial conditions compared favourably with the solution obtained by using classical fouth-other Runge-Kuta method. The solutions of the two methods follows the same pattern and behaviour. This shows that Adomian Decomposition Method is suitable and efficient to conduct the analysis of typhoid models.

CONCLUSION
We presents a deterministic mathematical model on typhoid fever transmission, Adomian Decomposition Method is used to attempt the series solution of the model. Numerical simulations were carried out to compare the results obtained with the result of classical fourth-order Runge-Kutta method. The results of the simulations were displayed graphically.The results obtained from ADM when compared with RK4 confirm the accuracy of ADM in solving the typhoid fever model.