Dynamics of an epidemic model with nonlocal di?usion and a free boundary

TL;DR

This paper analyzes an epidemic model with nonlocal diffusion and a free boundary, providing insights into eigenvalues, steady states, and long-term behaviors including spreading speed and conditions for epidemic spread or vanishing.

Contribution

It introduces a novel approach to analyze the eigenvalues and long-term dynamics of epidemic models with nonlocal diffusion and free boundaries.

Findings

Existence and asymptotic behavior of principal eigenvalues established.

Criteria for spreading and vanishing of the epidemic identified.

Spreading speed is finite under certain kernel function conditions.

Abstract

An epidemic model, where the dispersal is approximated by nonlocal diffusion operator and spatial domain has one ?xed boundary and one free boundary, is considered in this paper. Firstly, using some elementary analysis instead of variational characterization, we show the existence and asymptotic behaviors of the principal eigenvalue of a cooperative system which can be used to characterize more epidemic models, not just ours. Then we study the existence, uniqueness and stability of a related steady state problem. Finally, we obtain a rather complete understanding for long time behaviors, spreading-vanishing dichotomy, criteria for spreading and vanishing, and spreading speed. Particularly, we prove that the asymptotic spreading speed of solution component (u; v) is equal to the spreading speed of free boundary which is ?nite if and only if a threshold condition holds for kernel functions.