Longtime behaviors of an epidemic model with nonlocal diffusions and a free boundary: spreading-vanishing dichotomy

TL;DR

This paper analyzes a nonlocal epidemic model with a moving boundary, establishing conditions for the epidemic to spread or vanish, and studying the long-term behavior and eigenvalues of the associated system.

Contribution

It introduces a novel nonlocal epidemic model with a free boundary and provides criteria for spreading versus vanishing, including eigenvalue asymptotics without assuming self-adjointness.

Findings

Established well-posedness of the model

Proved a spreading-vanishing dichotomy for long-term behavior

Derived criteria for epidemic spreading and vanishing

Abstract

We propose a nonlocal epidemic model whose spatial domain evolves over time and is represented by $[0,h(t)]$ with $h(t)$ standing for the spreading front of epidemic. It is assumed that the agents can cross the fixed boundary $x=0$, but they will die immediately if they do it, which implies that the area $(-\infty,0)$ is a hostile environment for the agents. We first show that this model is well posed, then prove that the longtime behaviors are governed by a spreading-vanishing dichotomy and finally give some criteria determining spreading and vanishing. Particularly, we obtain the asymptotical behaviors of the principal eigenvalue of a cooperative system with nonlocal diffusions without assuming the related nonlocal operator is self-adjoint, and the steady state problem of such cooperative system on half space $[0,\infty)$ is studied in detail.