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In dimensions $d\geq 4$, by choosing a suitable scaling parameter, we showthat the rescaled spatial SIR epidemic process converges to a super-Brownianmotion 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 asymptoticsfor the critical probability $p_c$ of the range-$R$ bond percolation on$\mathbb{Z}^d$ as $R\to \infty$.

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