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

Abstract: In this paper, a reaction-diffusion system is proposed to model the spatial spreading of West Nile virus in vector mosquitoes and host birds in North America. Infection dynamics are based on a simplified model for cross infection between mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infective region. The spatial-temporal risk index $R_0^F(t)$, which involves time and characters of the region, is defined for the simplified model with the free boundary to compare with other related threshold values, including the usual basic reproduction number $R_0$. Sufficient conditions for the virus to vanish or spread are given. Our results suggest that the virus will be in a scenario of vanishing if $R_0\leq 1$, and the 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 virus depends on the initial numbers of infected mosquitoes and birds, the area of the infected region and diffusion rates. Moreover, some remarks on the basic reproduction numbers and the spreading speed are presented and compared.