European Journal of Applied Mathematics ( IF 1.9 ) Pub Date : 2023-10-13 , DOI: 10.1017/s0956792523000281 Zhiguo Wang , Hua Nie , Yihong Du
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 $[g(t), h(t)]$ 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 $\lim _{t\to \infty } h(t)/t=\lim _{t\to \infty }[\!-g(t)/t]=c_\nu$, with $c_\nu$ 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 $h(t)-c_\nu t$ and $g(t)+c_\nu t$ converge to some constants as $t\to \infty$, 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.
中文翻译:
具有自由边界的西尼罗河病毒模型解的尖锐渐近轮廓
我们考虑由具有自由边界的反应扩散系统组成的西尼罗河病毒(WNv)模型的长期行为。这样的模型描述了 WNv 的传播,自由边界代表感染区域的扩展前沿,这是一个与时间相关的区间