Biological invasions and epidemics with nonlocal diffusion along a line

TL;DR

The paper develops a rigorous framework to quantify how a line with nonlocal diffusion, implemented via a line operator $\mathcal{J}$ with kernel $K_L$, affects front propagation in the surrounding half-plane for biological invasions and SIR-type epidemics. It establishes global spreading speeds, identifies a sharp diffusion-threshold $D_*$ separating regimes of speedup, and derives asymptotics showing $c_* \sim \sqrt{D L^2 f'(0)}$ in appropriate limits, with speed enhanced even when only $D$ or $L$ is large. The analysis relies on plane-wave supersolutions, spectral properties of $\mathcal{J}$, and Liouville-type results, and extends prior local-road results to nonlocal transport. A pure-transport variant on the road is also examined, revealing directional speeds $c_*^\pm$ that grow like $\kappa_* q$ with $\kappa_*<1$ as $q\to\infty$, highlighting nuanced transport effects on epidemic spread. Overall, the work demonstrates that long-range dispersal along networks can substantially accelerate propagation and introduces a robust mathematical treatment of nonlocal road effects on invasion and epidemic dynamics.

Abstract

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term.

Figures (1)

• Figure 1: The minimal $c$ for the existence of plane wave super-solutions: (a) $c=c_{\hbox{$K$}}$, (b) $c=c_*(D,L)$.

Theorems & Definitions (18)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• proof : Proof of Theorem \ref{['cauchy']}
• Remark 3.1
• proof : Proof of Proposition \ref{['t2.2']}
• Lemma 4.1
• proof