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. Transmission dynamics are based on a simplified model involving mosquitoes and birds, and the free boundary is introduced to model and explore the expanding front of the infected region. The spatial-temporal risk index $R_0^F(t)$, which involves regional characteristic and time, is defined for the simplified reaction–diffusion 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 to spread are given. Our results suggest that the virus will be in a scenario of vanishing if $R_0\le 1$, and will spread to the whole region if $R_{0}^F(t_0)\ge 1$ for some $t_0\ge 0$, while if $R^F_0(0)<1<R_0$, 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, some remarks on the basic reproduction numbers and the spreading speeds are presented and compared.