The free boundary problem of an epidemic model with nonlocal diffusions and nonlocal reactions: spreading-vanishing dichotomy

TL;DR

This paper studies a nonlocal epidemic model with free boundaries, analyzing how nonlocal diffusion and reaction terms influence the spreading or vanishing of the disease, and establishing criteria for these outcomes.

Contribution

It introduces a novel approach to analyze principal eigenvalues in a nonlocal epidemic model with free boundaries, revealing the impact of nonlocal reactions on spreading dynamics.

Findings

Established existence and asymptotic behavior of principal eigenvalues.

Proved a spreading-vanishing dichotomy with explicit criteria.

Found that increased nonlocal reactions hinder disease spreading.

Abstract

This paper concerns the free boundary problem of an epidemic model. The spatial movements of the infectious agents and the infective humans are approximated by nonlocal diffusion operators. Especially, both the growth rate of the agents and the infective rate of humans are represented by nonlocal reaction terms. Thus our model has four integral terms which bring some diffculties for the study of the corresponding principal eigenvalue problem. Firstly, using some elementray analysis instead of Krein-Rutman theorem and the variational characteristic, we obtain the existence and asymptotic behaviors of principal eigenvalue. Then a spreading-vanishing dichotomy is proved to hold, and the criteria for spreading and vanishing are derived. Lastly, comparing our results with those in the existing works, we discuss the effect of nonlocal reaction term on spreading and vanishing, finding that the more nonlocal reaction terms a model has, the harder spreading happens.