A Hermite Polynomial Approach for Solving the SIR Model of Epidemics

Aydin Secer,Mustafa Bayram,Neslihan Ozdemir

doi:10.3390/math6120305

Aydin Secer, Mustafa Bayram + Show 1 more

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https://doi.org/10.3390/math6120305

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Abstract

In this paper, the problem of the spread of a non-fatal disease in a population is solved by using the Hermite collocation method. Mathematical modeling of the problem corresponds to a three-dimensional system of nonlinear ODEs. The presented scheme reduces the problem to a nonlinear algebraic equation system by expanding the approximate solutions by using Hermite polynomials with unknown coefficients. These coefficients of the Hermite polynomials are computed by using the matrix operations of derivatives together with the collocation method. Maple software is used to carry out the computations. In addition, comparison of our method with the Homotopy perturbation method (HPM) and Laplece-Adomian decomposition method (LADM) proves accuracy of solution.

Highlights

Systems of ordinary differential equations are useful in representing some real life problems in terms of the mathematical expressions, which abound in the fields of biological, physical, engineering, financial or sociological fields
We know that exact solutions of most realistic systems of ordinary differential equations cannot be found, so we need numerical and approximate methods for finding approximate solutions.There are a lot of methods that have been studied by many researchers to solve the systems of ordinary differential equations
Some of these methods are the multi-step method proposed by Hojjati et al [1], the collocation method presented by Mastorakis [2], the Adomian decomposition method improves [3], the exponential Galerkin method introduced by Yüzbaşı and Karaçayır [4], the exponential collocation method proposed by Yüzbaşı [5], the Galerkin finite element method given by Al-Omari et al [6]

Summary

Introduction

Systems of ordinary differential equations are useful in representing some real life problems in terms of the mathematical expressions, which abound in the fields of biological, physical, engineering, financial or sociological fields. Finding exact solutions of SIR models is important because biologists could use it to design and run experiments to observe the spread of infectious diseases by introducing natural initial conditions. Through these experiments, as well as through mathematical modelling, one can learn the ways on how to control the spread of epidemics. To obtain approximate solutions of Equation (1), some authors have studied this model using different methods. Hermite collocation method (HCM) has been used to solve systems of nonlinear ordinary differential equations with special initial conditions.

Error Estimate for the Solution

Illustrative Example

Conclusions