Spatial Propagation in an Epidemic Model with Nonlocal Diffusion: the Influences of Initial Data and Dispersals

TL;DR

This study investigates how initial data and nonlocal dispersal strategies influence the spatial spread of an epidemic, revealing effects on spreading speeds, directionality, and stability of solutions.

Contribution

It introduces a detailed analysis of nonlocal dispersal effects on epidemic propagation, including asymmetry impacts and conditions for traveling wave solutions.

Findings

Spreading speeds depend on initial data decay rates.

Asymmetry in dispersal influences propagation direction.

Minimum spreading speed matches that of exponentially fast decaying initial data.

Abstract

This paper studies an epidemic model with nonlocal dispersals. We focus on the influences of initial data and nonlocal dispersals on its spatial propagation. Here the initial data stand for the spatial concentrations of infectious agent and infectious human population when the epidemic breaks out and the nonlocal dispersals mean their diffusion strategies. Two types of initial data decaying to zero exponentially or faster are considered. For the first type, we show that the spreading speeds are two constants whose signs change with the number of elements in some set. Moreover, we find an interesting phenomenon: the asymmetry of nonlocal dispersals can influence the propagating directions of solutions and the stability of steady states. For the second type, we show that the spreading speed is decreasing with respect to the exponentially decaying rate of initial data, and further, its minimum value coincides with the spreading speed for the first type. In addition, we give some results about the nonexistence of traveling wave solutions and the monotone property of solutions. Finally, some applications are presented to illustrate the theoretical results.