Long-time dynamics of a nonlocal epidemic model with free boundaries: spreading-vanishing dichotomy

TL;DR

This paper analyzes a nonlocal epidemic model with free boundaries, establishing conditions for disease spreading or vanishing, and introduces new methods to handle the mathematical complexities arising from nonlocal effects.

Contribution

It provides a rigorous analysis of the long-term behavior of a nonlocal epidemic model with free boundaries, including sharp criteria for spreading versus vanishing.

Findings

The model exhibits a spreading-vanishing dichotomy.

New approaches are developed to handle nonlocal effects.

The model can capture accelerated disease spreading.

Abstract

In this paper, we examine the long-time dynamics of an epidemic model whose diffusion and reaction terms involve nonlocal effects described by suitable convolution operators.The spreading front of the disease is represented by the free boundaries in the model. We show that the model is well-posed, its long-time dynamical behaviour is characterised by a spreading-vanishing dichotomy, and we also obtain sharp criteria to determine the dichotomy. Some of the nonlocal effects in the model pose extra difficulties in the mathematical treatment, which are dealt with by introducing new approaches. The model can capture accelerated spreading, and its spreading rate will be discussed in a subsequent work.