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In this paper, a reaction-diffusion system is proposed to model the spatialspreading of West Nile virus in vector mosquitoes and host birds in NorthAmerica. Infection dynamics are based on a simplified model for cross infectionbetween mosquitoes and birds, and the free boundary is introduced to model andexplore the expanding front of the infective region. The spatial-temporal riskindex $R_0^F(t)$, which involves time and characters of the region, is definedfor the simplified model with the free boundary to compare with other relatedthreshold values, including the usual basic reproduction number $R_0$.Sufficient conditions for the virus to vanish or spread are given. Our resultssuggest that the virus will be in a scenario of vanishing if $R_0\leq 1$, andthe virus will spread to the whole region if $R_{0}^F(t_0)\geq 1$ for some$t_0\geq 0$, while if $R^F_0(0)<1<R_0$, the spreading or vanishing of the virusdepends on the initial numbers of infected mosquitoes and birds, the area ofthe infected region and diffusion rates. Moreover, some remarks on the basicreproduction numbers and the spreading speed are presented and compared.

Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary