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

In dimensions $d\ge 4$, 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 $d\le 3$. 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 $R\to \mathrm{\infty }$.