Rescaled SIR Epidemic Processes Converge to Super-Brownian Motion in Four or More Dimensions

In dimensions d ≥ 4 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 ( Prob. Th. Rel. Fields , 144 (2009), 429–469) and Lalley-Zheng ( Prob. Th. Rel. Fields , 148 (2010), 527–566) on the convergence of SIR epidemics in d ≤ 3 d\leq 3 . The scaling parameters we choose also agree with the corresponding asymptotics for the critical probability p c p_c of the range- R R bond percolation on Z d \mathbb {Z}^d as R → ∞ R\to \infty .