The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries

TL;DR

This paper analyzes a nonlocal diffusion epidemic model with free boundaries, establishing existence, uniqueness, and long-term behavior, revealing differences from local diffusion models, including potential accelerated spreading depending on the kernel function.

Contribution

It introduces a nonlocal diffusion epidemic model with free boundaries, providing existence, uniqueness, and criteria for spreading or vanishing, highlighting differences from traditional local diffusion models.

Findings

Unique global solution exists for all time

Long-term behavior follows a spreading-vanishing dichotomy

Nonlocal diffusion may lead to accelerated spreading

Abstract

We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin \cite{ABL2016}, with its origin tracing back to Capasso et al. \cite{CP1979, CM1981}. We prove that the problem has a unique solution defined for all $t>0$, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local diffusion model in \cite{ABL2016}. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the model of \cite{ABL2016}, where the spreading has finite speed whenever spreading happens \cite{ZLN2019}.