# The spatial-temporal risk index and spreading dynamics for a time-periodic diffusive WNv model

## TL;DR

This study models West Nile virus spread in a heterogeneous, time-periodic environment using a diffusive epidemic model with a free boundary, analyzing how spatial and temporal factors influence virus persistence or eradication.

## Contribution

It introduces a novel spatial-temporal risk index and provides conditions for virus spreading and vanishing in a heterogeneous, periodic environment.

## Key findings

- Defined the basic reproduction number $R_0^D$ and risk index $R_0^F(t)$ considering heterogeneity and periodicity.
- Established sufficient conditions for virus spreading and vanishing.
- Analyzed the impact of spatial heterogeneity and temporal periodicity on virus dynamics.

## Abstract

This paper is concerned with a simplified epidemic model for West Nile virus in a heterogeneous time-periodic environment. By means of the model, we will explore the impact of spatial heterogeneity of environment and temporal periodicity on the persistence and eradication of West Nile virus. The free boundary is employed to represent the moving front of the infected region. The basic reproduction number $R_0^D$ and the spatial-temporal risk index $R_0^F(t)$, which depend on spatial heterogeneity, temporal periodicity and spatial diffusion, are defined by considering the associated linearized eigenvalue problem. Sufficient conditions for the spreading and vanishing of West Nile virus are presented for the spatial dynamics of the virus.