Abstract
In this article, the numerical and approximate solutions for the nonlinear differential equation systems, represented by the epidemic SIR model, are determined. The effective iterative methods, namely the Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM), and the Banach contraction method (BCM), are used to obtain the approximate solutions. The results showed many advantages over other iterative methods, such as Adomian decomposition method (ADM) and the variation iteration method (VIM) which were applied to the non-linear terms of the Adomian polynomial and the Lagrange multiplier, respectively. Furthermore, numerical solutions were obtained by using the fourth-orde Runge-Kutta (RK4), where the maximum remaining errors showed that the methods are reliable. In addition, the fixed point theorem was used to show the convergence of the proposed methods. Our calculation was carried out with MATHEMATICA®10 to evaluate the terms of the approximate solutions.
Highlights
In the SIR epidemic model, individuals are categorized into three groups; S is the fraction of the population that is susceptible to disease; I is the fraction of the population that is infectious at any given time; and R is the fraction of the population that has recovered after infection
We note that the error value of the proposed methods is lower than those of the Adomian decomposition method (ADM) and the variation iteration method (VIM), which indicates that the proposed methods converged faster
All the proposed methods provided an approximate solution in a number of terms
Summary
In the SIR epidemic model, individuals are categorized into three groups; S is the fraction of the population that is susceptible to disease; I is the fraction of the population that is infectious at any given time; and R is the fraction of the population that has recovered (removed) after infection. Numerous methods were studied by many researchers to solve the epidemic SIR model, as reviewed before [1]. Three iterative methods will be used to solve the epidemic SIR model and obtain a new approximate solution. The first method, namely the DJM, was proposed by Daftardar-Gejji and Jafari in 2006[9,10] This method has been used to solve various linear and nonlinear differential equations [11] and the solution of nonlinear ODEs of second order in physics [12]. The third iterative method, namely the BCM, was proposed by Daftardar-Gejji and Bhalekar in 2009 [18], which provided the required solution for various types of nonlinear equations. The following diagram shows the SIR model [20]:
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