Long-time dynamics and threshold Phenomena for a free-boundary SIS Model with asymmetric kernels in advective periodic environments

TL;DR

This paper analyzes a complex nonlocal SIS epidemic model with asymmetric dispersal kernels, advection, and free boundaries, establishing thresholds for disease spread and long-term behavior using advanced spectral theory.

Contribution

It introduces a novel approach employing generalized principal eigenvalue theory to handle non-symmetric nonlocal operators in a free-boundary epidemic model, addressing key analytical challenges.

Findings

Identified sharp threshold for disease spreading and vanishing.

Characterized long-time dynamics of susceptible and infected populations.

Established the impact of asymmetric kernels and advection on epidemic thresholds.

Abstract

We study a nonlocal SIS epidemic model with free boundaries, advection, and spatial heterogeneity, where the dispersal kernels are not assumed to be symmetric. The model describes the evolution of susceptible and infected populations in a bounded infected habitat whose endpoints move according to nonlocal boundary fluxes. Our goal is to determine the sharp threshold between disease spreading and vanishing, and to characterize the long-time behavior of solutions. The analysis faces several essential difficulties. The linearization around the disease-free equilibrium gives rise to a genuinely coupled nonlocal system with drift, so the relevant spectral quantity cannot be reduced directly to a standard scalar eigenvalue problem. In addition, the presence of advection terms and possibly non-symmetric kernels destroys self-adjointness, so no useful variational characterization is available; in particular, classical Rayleigh quotient and minimax arguments cannot be applied. To overcome these difficulties, we employ the generalized principal eigenvalue theory for nonlocal operators developed by Coville and Hamel, together with the Harnack inequality for non-symmetric nonlocal operators established therein. This non-variational framework is particularly well suited to our setting. Combined with comparison principles, sub- and supersolution constructions, and uniform estimates on time-dependent spatial intervals, it allows us to derive the precise asymptotic behavior of the generalized principal eigenvalue with respect to the spatial domain and the diffusion rate, identify the sharp threshold and the critical habitat size, and determine the long-time dynamics of $S$ and $I$ via an $\omega$-limit set approach. To the best of our knowledge, this is the first work on a free-boundary SIS epidemic model with non-symmetric nonlocal dispersal kernels, advection, and spatial periodicity.