  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.
