# A nonlocal West Nile virus model with nonlocal free boundary conditions driven by both mosquitoes and birds

## TL;DR

This paper introduces a new model for West Nile virus that considers the spread influenced by both mosquitoes and birds, with novel boundary conditions.

## Contribution

The model introduces nonlocal free boundary conditions and considers the impact of infected mosquitoes, offering new insights into virus dynamics.

## Key findings

- The model introduces new challenges in analyzing spreading-vanishing dynamics when the basic reproduction number is less than or equal to one.
- New conclusions are derived from considering the impact of infected mosquitoes on the free boundary conditions.
- Numerical schemes provide quantitative insights into how various factors affect transmission dynamics.

## Abstract

This paper presents a novel West Nile virus model that has more extensive free boundary conditions and also takes into account the impact of infected mosquitoes on the free boundary, both of which are firsts in West Nile virus modeling. Specifically, the free boundary conditions independent of the dispersal kernel functions in the equations, bring new challenges to the dynamical analysis of spreading-vanishing, especially for the case where the basic reproduction number \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal R_0\le 1$$\end{document}R0≤1, which involves new ideas and techniques for dynamics analysis. Moreover, due to the consideration of the impact of infected mosquitoes in the free boundary conditions, new conclusions have been obtained. Numerical schemes have been developed, which not only verify qualitative theoretical results, but also provide novel quantitative insights into the effects of various factors on transmission dynamics. Overall, our results not only differ significantly from the local diffusion version presented in Lin and Zhu (2017) but also extend all the conclusions from the nonlocal diffusion version in Du and Ni (2020), with some conclusions obtained under more general conditions.