The asymptotic behavior of an SIR epidemic model: collective Reed-Frost process

Abstract

The aim of this research is to provide a historical overview of the mathematical theory of epidemics and to study the asymptotic behavior of the final size of a collective Reed-Frost epidemic process with different types of infected people. This model was introduced by Picard and Lefevre [25] provides an extension of the model of Pettigrew and Weiss [24]. Under certain conditions, we show that when the number of the initial susceptible individuals is large and the number of the initial infected people is finite, the infection process is equivalent to a multitype Galton-Watson process. Our method is simple and based on Bernstein polynomials.