An SIR epidemic model with free boundary

Abstract: An SIR epidemic model with free boundary is investigated. This model describes the transmission of diseases. The behavior of positive solutions to a reaction-diffusion system in a radially symmetric domain is investigated. The existence and uniqueness of the global solution are given by the contraction mapping theorem. Sufficient conditions for the disease vanishing or spreading are given. Our result shows that the disease will not spread to the whole area if the basic reproduction number $R_{0}<1$ or the initial infected radius $h_0$ is sufficiently small even that $R_{0}>1$. Moreover, we prove that the disease will spread to the whole area if $R_{0}>1$ and the initial infected radius $h_0$ is suitably large.