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

Abstract

<p>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 [<i>g(t), h(t)</i>] in the model (Lin and Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. <i>J. Math. Biol</i>. 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. <i>J. Math. Biol</i>. 79, 433–466, 2019) by making use of the associated semi-wave solution, namely <i>lim_t→∞ h(t)/t = lim_t→∞ [− g(t)/t] = c_ν</i> , with c_ν</i> 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. <i>J. Math. Biol</i>. 79, 433–466, 2019): we show that <i>h(t) − c_νt</i> and <i>g(t) + c_νt</i> converge to some constants as <i>t → ∞</i>, 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. </p>