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

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 $R\to \infty$.