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 R0<1 or the initial infected radius h0 is sufficiently small even that R0>1. Moreover, we prove that the disease will spread to the whole area if R0>1 and the initial infected radius h0 is suitably large.