Brownian snails with removal: epidemics in diffusing populations

TL;DR

This paper introduces two spatial stochastic SIR models with diffusion and removal, analyzing conditions for epidemic extinction and survival, highlighting the impact of individual movement and removal rate on disease spread.

Contribution

The paper develops and analyzes two novel spatial SIR models incorporating diffusion and removal, providing conditions for epidemic extinction and survival.

Findings

Existence of a critical removal rate for epidemic survival.

Conditions under which the epidemic infects finitely many individuals.

Differentiation between delayed and immediate diffusion models.

Abstract

Two stochastic models of susceptible/infected/removed (SIR) type are introduced for the spread of infection through a spatially-distributed population. Individuals are initially distributed at random in space, and they move continuously according to independent diffusion processes. The disease may pass from an infected individual to an uninfected individual when they are sufficiently close. Infected individuals are permanently removed at some given rate $\alpha$. Such processes are reminiscent of so-called frog models, but differ through the action of removal, as well as the fact that frogs jump whereas snails slither. Two models are studied here, termed the `delayed diffusion' and the `diffusion' models. In the first, individuals are stationary until they are infected, at which time they begin to move; in the second, all individuals start to move at the initial time $0$. Using a perturbative argument, conditions are established under which the disease infects a.s. only finitely many individuals. It is proved for the delayed diffusion model that there exists a critical value $\alpha_c\in(0,\infty)$ for the survival of the epidemic.