Extrapolation method for Numerical solution of a model for endemic infectious diseases

In this work we consider one endemic infectious model in the form of a nonlinear Volterra integral system. This model is the SIR one. we apply the Richardson extrapolation method for numerical solution of this model, so that the nonlinear system is solvable by an iterative process with a good accuracy. The algorithm of such systems completely described. This algorithm has a kind of nested structure, which cause we use the lag data in the future times, and it is the interesting section of programing of the algorithm. This algorithm is ready for programing with every program language, which we do this process by Mathematica programing software. Convergence and accuracy of the method is illustrated by either theoretical numerical analysis and some benchmark sample problems, for this aim by using Laplase transform, we Sketch a spectrum of sample problems, which have the analytical solution.