NUMERICAL SOLUTION FOR MATHEMATICAL MODEL OF EBOLA VIRUS

* R. Hussain, A.Ali, A. Chaudary , F. Jarral, T. Yasmeen, and F. Chaudary. Mirpur University of Science and Technology (MUST), Mirpur-10250(AJK ), Pakistan. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History


Materials and Method:-
The motivation of this study was to solve the well-knownMathematical model called Susceptible-Infected-Recovery (SIR) model. The real physical problems in the world usually exhibited nonlinear mathematical models which includes biological issues. It was extremely challenging to obtainthe exact solutions for such problems that actually represented such phenomena. It was a big task for scientific community to search for appropriate methods such as numerical or perturbation method to solve nonlinear problems [4], but the numerical methods were considered to best for such problems. Therefore numerical techniques Euler, Runge-Kutta 2 and Runge-Kutta 4 were applied to solve the proposed Susceptible-Infected-Recovery(SIR) model in this study.

Description of Model:-
The total population in a specific place was divided into three groups, the susceptible group S(t), the infected group I(t) and the recovered group R(t), "t" was anytime interval ( [2], [3], [14]). The total population was represented by "N" and was taken to be constant for a short time interval and was given by N=S (t)+I (t)+R (t).
Population of the susceptible group reduced as the infected peoples come into contact with them by the rate of infection β. Therefor the change in population of susceptible group was equal to the negative product of -β with S (t) and I (t): Population of the infected group was changed according to two different ways: (a) Susceptible group who joined the infected group by adding the total population of infected group with the term βS (t) I (t) (b) Infected group who joined the recovered group in any time interval, in such way that the total population of infected group was decreased by a term µI (t). Therefore the differential equation of infected group was written as: Finally, the differential equation of recovered group that based on those peoples who recovered from the Ebola virus by a rate µ. The successive iterations of Euler method were given in Table-1.

Runge-Kutta-2 Method
Solution of the SIR model by RK-2 Heun's method is presented as.  The successive iterations of RK-2 Heun's method are given in Table-2. Similarly, the successive iterations of the RK-4 method were presented in Table-3.

Results and Discussion:-
Results obtained in case of all three numerical techniques were found to be exactly the same as were found by homptopyperturbation method in [4]. Solution of the model is presented graphically in Fig-1. Moreover we solve the model analytically by linearization as follows: The system of equations 1-3 was nonlinear which was linearized as: Let  1538 Different numerical methods like Euler, Runge-Kutta 2, RK-2 and RK-4 were used to solve the model numerically, the results were presented in Tables 1, 2 and 3 were exactly the same as in Equations 4, 5 and 6 obtained by analytical method .

Conclusionand Future Work:-
Solution of model by three numerical methods Euler, Runge-Kutta-2 (Heuns) and Runge-Kutta-4 were presented in Tables 1, 2 and 3. Conclusion was drawn that most optimistic estimates for each group of individuals were obtained like the susceptible group did not change and remained 460, but the infected group gradually decreased its values, due to the high mortality rate of infected group, and the recovered group slightly increased its values. This study can be extended further for parameter estimation and sensitivity analysis of the model.