Finite or Infinite Spreading Speed of an Epidemic Model with Free Boundary and Double Nonlocal Effects

Abstract

<p>We determine the spreading speed of an epidemic model with nonlocal diffusion and free boundary. The model is evolved from a degenerate reaction-diffusion model of Capasso and Maddalena (J Math Biol 13:173–184, 1981), and was studied in Zhao et al. (Commun Pure Appl Anal 19:4599–4620, 2020) recently, where it was shown that as time goes to infinity, the population of the infective agents either vanishes or spreads successfully. In this paper, we show that when spreading is successful, the asymptotic spreading speed is finite or infinite depending on whether a threshold condition is satisfied by the kernel function governing the spatial dispersal of the agents. The proof relies on a rather complete understanding of the associated semi-wave problem and traveling wave problem. For free boundary models, the case of infinite spreading speed, also known as accelerated spreading, is only recently shown to happen in Du et al. (J Math Pure Appl 154:30–66, 2021) for a single species Fisher-KPP model" this paper is the first to show that it happens to a very different two species model with free boundary. This suggests that accelerated spreading is a rather common phenomenon for free boundary problems with nonlocal diffusion. In contrast, for the corresponding models with local diffusion, the spreading can only proceed with finite speed.</p>