SERIES SOLUTION OF TYPHOID FEVER MODEL USING DIFFERENTIAL TRANSFORM METHOD

1,4,5 Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria. Department of Mathematical Sciences, Adekunle Ajasin University Akungba,Ondo State. Department of Mathematics, National Open University of Nigeria Jabi, Abuja Nigeria peterjames4real@gmail.com, oluwaseun.akinduko@aaua.edu.ng, cishola@noun.edu.ng, 4 afolabiahmed79@gmail.com , aahfees11@yahoo.com *Corresponding author: peterjames4real@gmail.com +2348033560280


Introduction
Typhoid fever, a communicable disease which infects human only and occurs due to systemic infection mainly by salmonella typhi organism that causes symptoms. It is an acute generalized infectious disease of the intestinal lymphoid tissue and the gall bladder. Incubation period of typhoid fever, usually 10-14 days but it may be as short as 3 days or as long as 21 days. The disease is transmitted from person-to-person as a result of improper hygiene and unsafe food and water handling practices. Recent report, however, suggests that individuals may be indirectly infected with typhoid through contact with fecal and urine contamination in their immediate environment. (Shanahan, 1998).
The disease is endemic in many developing countries where water supply sanitation and waste treatment is inadequate. The disease remains a substantial public health problem. Globally, the disease burden was estimated to be over 16 million cases of illness each year, resulting in over 600,000 mortality rates Mushayabasa (2011).
Several mathematical models has been developed on the transmission dynamics of typhoid fever, these includes, (Adetunde, 2008;Cvjetanovic et al, 2014;Kalajdzievska, 2011;Lauria et al 2009;Moatlhodl & Gosaamang, 2017;Chamuchi et al, 2014;Joshua, 2011;Muhammad, et al 2015;Mushayabasa, 2011;Nthiiri, 2016;Virginia et al, 2014;Watson & Edmunds, 2015;Peter, et al 2017). In this paper, we extend previous efforts by introducing a model that includes educated infected and uneducated infected class. This work focused on the application of differential transform method to the proposed model and to verify the validity of the method in solving the model equations using computer inbuilt Maple 18 classical fourth-order Runge-Kutta method as a base. In recent years, the differential transform method (DTM) is mostly used for solving non-linear ordinary and partial differential equations. It is a semi-analytic technique that formalizes the Taylor series in a totally different approach. The concept of (DTM) was first introduced by Zhou, (1986) in a study to solve nonlinear problems of electrical circuits. The DTM obtains an analytical solution in form of polynomial. DTM has been successfully applied to solve many nonlinear problems arising in engineering, mathematics, physics and mechanics. Abazari et al, (2010). The main advantage of DTM is that it can be applied directly to solve linear and nonlinear Ordinary Differential Equations without requiring linearization, discretization or perturbation. (Hassan, 2008;Akinboro, et al 2014).
We employ the (DTM) to the system of differential equations which describe the proposed model and approximate the solutions in a sequence of time intervals. In other to verify the accuracy and validity of the (DTM), we compared the obtained results with fourth-order Runge-Kutta Method.

Methodology
This section describes the formulation of the model.
Individuals are recruited into the susceptible class by either immigration or birth at the rate  . Susceptible individuals acquire typhoid infection at per capita rate  . We assume that proportion  progress to educated infectious class, while the compliment   1 progress to uneducated infectious compartment class. Susceptible individuals received vaccination to protect themselves against the disease at the rate  . Since vaccine wanes with time, then after its expiry, the protected individuals return back to susceptible class at the rate  . We assume that individuals in each compartment undergo a natural death at the rate  . Let 1  , 2  , and 3  be transmission rates for uneducated infectious, educated infectious and treated individuals respectively then the susceptible population S(t), is exposed to force of infection denoted by Detailed description of parameters and variables are shown in Table 1 while the compartmental flow diagram of the model is shown in Figure 1. 3. Susceptible individual can be infected through a direct contact with educated infected or uneducated infected. 4. All parameters are non-negative. 5. No treatment failure. All treated individuals recoverd.
Where the force of infection  is given as:

Existence and Uniqueness of Solution
The implementation of any mathematical model largely based on whether the given system of equations has a solution, and if the solution is unique, we shall use the Lipchitz condition to verify the existence and uniqueness of solution for the system of the model equation 3.
Let the system of equations of the model be as follows:

R T
These partial derivative exist, continuous and are bounded, similarly for 4 A through to 6 A . Hence, by theorem 2, the model has a unique solution

Result and Analysis
The processes involved in DTM is given as follows: Given an arbitrary function of x , suppose

Solution of the Model
In this section, we apply the steps involved in differential transform method as follows: Using the operational properties (1), (2), (6) and (7) in Table 2 and applying them to the systyem of differential equations in (1) we obtain the following system of transformed equations below,

Numerical Simulation and Graphical Illustration of the Model.
We demonstrated the numerical simulation which illustrate the analytical results for the proposed model . This is achieved by using some set of parameter values given in the

Discussion of Results
The solutions obtained by using Differential Transform 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 Differential Transform Method is suitable and efficient to conduct the analysis of epidemic models.

Conclusion
In this paper, (DTM) is employed to attempt the series solution of the model. Numerical simulations were carried out to compare the results obtained by (DTM) with the result of classical fourth-order Runge-Kutta method. The results of the simulations were displayed graphically.The results shows that (DTM) is in good agreement with RK-4 and produced accurately the same behaviour thus, validating the reliability of (DTM) in finding the approximate solution of epidemic model.