Asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment

TL;DR

This paper investigates how the basic reproduction number in a spatially heterogeneous epidemic model behaves as diffusion rates vary, revealing its limits as diffusion approaches zero or infinity.

Contribution

It establishes the asymptotic limits of the basic reproduction number for general reaction-diffusion epidemic models in heterogeneous environments.

Findings

As diffusion rates approach zero, the reproduction number converges to the maximum local value.

As diffusion rates approach infinity, the reproduction number converges to the spectral radius of the averaged next generation matrix.

The results apply broadly to various spatially heterogeneous epidemic models.

Abstract

The effect of diffusion rates on the basic reproduction number of a general compartmental reaction-diffusion epidemic model in a heterogeneous environment is considered. It is shown when the diffusion rates tend to zero, the limit of the basic reproduction number is the maximum value of the local reproduction number on the spatial domain. On the other hand when the diffusion rates tend to infinity, the basic reproduction number tends to the spectral radius of the "average"~next generation matrix. These asymptotic limits of basic reproduction number hold for a class of general spatially heterogeneous compartmental epidemic models, and they are applied to a wide variety of examples.