One Iteration method in Numerical solution of Endemic Models with arbitrarily distributed periods of infection for long time

In this paper, we consider endemic models with arbitrary distributed periods of infection. Some samples of such diseases are: HIV/AIDS, rubella, influenza, and so on. Since beginning of 20th century, there are many authors that investigate about mathematical modelling, existence of solutions and stability of models. Many of these models are in the form of differential equations, integral equations, algebraic equation or hybrid of these. In this work we convert the existence model to a system of integral equations with a nonlinear algebraic equation. We solve numerically this model by an iterative process. Organization of the given algorithm is such that the problem is solved by a numerical iteration on union of short time intervals, and the process forward interval by interval. Convergence of the method is given expansively. In the numerical results section, according to the structure of the problem and by using Laplace transform, we sketch a spectrum of sample problems that have analytical solutions. Finally, we illustrate accuracy and applicability of the method by two benchmark sample problems.