On a Vector-host Epidemic Model with Spatial Structure

TL;DR

This paper analyzes a reaction-diffusion vector-host epidemic model, establishing the threshold role of the basic reproduction number $R_0$ for disease persistence or eradication, and explores how spatial diffusion influences $R_0$.

Contribution

It introduces a spectral radius framework for $R_0$ in spatial models and examines the impact of diffusion rates on disease dynamics.

Findings

$R_0$ determines global stability of disease states.

$R_0$ equals the spectral radius of a product of operators.

Diffusion rates affect the relationship between $R_0$ and local $R(x)$.

Abstract

In this paper, we study a reaction-diffusion vector-host epidemic model. We define the basic reproduction number $R_0$ and show that $R_0$ is a threshold parameter: if $R_0\le 1$ the disease free steady state is globally stable; if $R_0>1$ the model has a unique globally stable positive steady state. We then write $R_0$ as the spectral radius of the product of one multiplicative operator $R(x)$ and two compact operators with spectral radius equalling one. Here $R(x)$ corresponds to the basic reproduction number of the model without diffusion and is thus called local basic reproduction number. We study the relationship between $R_0$ and $R(x)$ as the diffusion rates vary.