Rescaled SIR epidemic processes converge to super-Brownian motion in four or more dimensions

TL;DR

This paper demonstrates that in four or more dimensions, the rescaled spatial SIR epidemic process converges to a super-Brownian motion with drift, extending previous results for lower dimensions and linking to percolation theory.

Contribution

It establishes the convergence of rescaled SIR epidemic processes to super-Brownian motion in high dimensions, filling a gap in the understanding of epidemic scaling limits.

Findings

Convergence of rescaled SIR processes to super-Brownian motion in $d extgreater=4$.

Scaling parameters align with asymptotics for critical percolation probability.

Extension of previous low-dimensional epidemic convergence results.

Abstract

In dimensions $d\geq 4$, by choosing a suitable scaling parameter, we show that the rescaled spatial SIR epidemic process converges to a super-Brownian motion with drift, thus complementing the previous results by Lalley (Probab. Theory Related Fields,144(2009),429--469) and Lalley-Zheng (Prob. Th. Rel. Fields,148(2010),527--566) on the convergence of SIR epidemics in $d\leq 3$. The scaling parameters we choose also agree with the corresponding asymptotics for the critical probability $p_c$ of the range-$R$ bond percolation on $\mathbb{Z}^d$ as $R\to \infty$.