Spatial Epidemics and Local Times for Critical Branching Random Walks in Dimensions 2 and 3

TL;DR

This paper studies the critical behavior of spatial epidemic models in 2D and 3D, showing their scaled limits converge to super-Brownian motion or variants, and proves a conjecture about local time processes.

Contribution

It establishes the convergence of epidemic-related measure-valued processes to super-Brownian motion and proves Adler's conjecture on local times for branching random walks.

Findings

Convergence of scaled epidemic processes to super-Brownian motion.

Differentiation between models with and without location-dependent killing.

Proof of Adler's conjecture on local time process convergence.

Abstract

The behavior at criticality of spatial SIR (susceptible/infected/recovered) epidemic models in dimensions two and three is investigated. In these models, finite populations of size N are situated at the vertices of the integer lattice, and infectious contacts are limited to individuals at the same or at neighboring sites. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. It is shown that the measure-valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit, either to a standard Dawson-Watanabe process (super-Brownian motion) or to a Dawson-Watanabe process with location-dependent killing, depending on the size of the the initially infected set. A key element of the argument is a proof of Adler's 1993 conjecture that the local time processes associated with branching random walks converge to the local time density process associated with the limiting super-Brownian motion.