This talk deals with mathematical models describing the diffusive dynamic of West Nile virus (WNV). For the spatially-independent WNv model, the usual basic reproduction number $R_0$ is given and for the diffusive WNv model in a bounded domain, the basic reproduction numbers $R_0^N, R_0^D$ are defined. To model and explore the expanding front of the infective region, a reaction-diffusion problem with free boundaries is proposed. The spatial-temporal risk index $R_0^F(t)$, which involves regional characteristic and time, is defined. Sufficient conditions for the virus to vanish or spread are given. Our results suggest that the spreading or vanishing of the virus depends on the initial number of infected individuals, the area of the infected region, the diffusion rate, and other factors. Moreover, we establish new WNv models to describe the impart of climate warming and seasonal succession.