Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy

TL;DR

The paper studies a nonlocal Fisher-KPP epidemic model on a dynamically expanding interval with a new free boundary condition linking boundary motion to the weighted internal population and outward flux. It proves global existence and uniqueness, develops comparison principles, and analyzes a principal eigenvalue framework to characterize the spreading-vanishing dichotomy. The work derives sharp criteria for spreading vs. vanishing based on the basic reproduction number $\\mathcal{R}_0 = \\frac{e G'(0)}{ab}$, the initial habitat size, diffusion rates, and kernel properties, including a diffusion-rate threshold $d^*$ and an initial-size threshold $L^*$. This Part 1 lays the groundwork for Part 2, which will determine the spreading speed and rates for various kernel classes. $

Abstract

This paper investigates the long-time dynamics of a nonlocal epidemic model with free boundaries, where a pathogen with density $u(t,x)$ and the infected humans with density $v(t,x)$ evolve according to a reaction-diffusion system with nonlocal diffusion over a one dimensional interval $[g(t), h(t)]$, which represents the epidemic region expanding through its boundaries $x=g(t)$ and $x=h(t)$, known as free boundaries. Such a model with free boundary conditions based on those of Cao et al. \cite{fb27} was considered by several works. Inspired by recent works of Feng et al. \cite{fb20} and Long et al. \cite{fb5}, we propose a new free boundary condition, where the expansion rate of the epidemic region, determined by $h'(t)$ and $g'(t)$, is proportional to a linear combination of the outward flux of the pathogen \(u\) through the range boundary (as in \cite{fb27}) and the weighted total population of infected individuals \(v\) within the region (as in \cite{fb5}). We prove that the system under this new free boundary condition is well-posed, and its long-time dynamical behavior is characterized by a spreading-vanishing dichotomy. Moreover, we obtain sharp criteria for this dichotomy, including a sharp threshold in terms of the initial data $(u_0,v_0)$; and by studying a related eigenvalue problem, we also find a sharp threshold in terms of the diffusion rate, which complements related results in Nguyen and Vo \cite{fb7}. This is Part $1$ of a two part series. In Part $2$, we will determine the spreading speed of the model when spreading occurs, and for some typical classes of kernel functions, we will obtain the precise rates of accelerated spreading.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4