Sharp asymptotic profile of the solution to a West Nile virus model with free boundary

Abstract

We consider the long-time behaviour of a West Nile virus (WNv) model consisting of a reaction-diffusion system with free boundaries. Such a model describes the spreading of WNv with the free boundary representing the expanding front of the infected region, which is a time-dependent interval in the model (Lin and Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381-1409, 2017). The asymptotic spreading speed of the front has been determined in Wang et al. (Spreading speed for a West Nile virus model with free boundary. J. Math. Biol. 79, 433-466, 2019) by making use of the associated semi-wave solution, namely limt→∞ h(t)/t = limt→∞ [-g(t)/t] = cv, with cv the speed of the semi-wave solution. In this paper, by employing new techniques, we significantly improve the estimate in Wang et al. (Spreading speed for a West Nile virus model with free boundary. J. Math. Biol. 79, 433-466, 2019): we show that h(t)-cvt and g(t) + cvt converge to some constants as t → ∞, and the solution of the model converges to the semi-wave solution. The results also apply to a wide class of analogous Ross-MacDonold epidemic models.