  We study the spreading speed of a diffusive epidemic model proposed by Li et
al. \cite{LL}, where the Stefan boundary condition is imposed at the right
boundary, and the left boundary is subject to the homogeneous Dirichlet and
Neumann condition, respectively. A spreading-vanishing dichotomy and some sharp
criteria were obtained in \cite{LL}. In this paper, when spreading happens, we
not only obtain the exact spreading speed of the spreading front described by
the right boundary, but derive some sharp estimates on the asymptotical
behavior of solution component $(u,v)$. Our arguments depend crucially on some
detailed understandings for a corresponding semi-wave problem and a steady
state problem.
