The reaction-diffusion system for an SIR epidemic model with a free boundary

The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.