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We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered)epidemic defined on a configuration model random graph, in which infectiveindividuals can infect only their neighbours in the graph during an infectiousperiod which has an arbitrary but specified distribution. Central limittheorems for the final size (number of initial susceptibles that becomeinfected) of such an epidemic as the population size $n$ tends to infinity,with explicit, easy to compute expressions for the asymptotic variance, areproved assuming that the degrees are bounded. The results are obtained for boththe Molloy-Reed random graph, in which the degrees of individuals aredeterministic, and the Newman-Strogatz-Watts random graph, in which the degreesare independent and identically distributed. The central limit theorems coverthe cases when the number of initial infectives either (a) tends to infinity or(b) is held fixed as $n \to \infty$. In (a) it is assumed that the fraction ofthe population that is initially infected converges to a limit (which may be$0$) as $n \to \infty$, while in (b) the central limit theorems are conditionalupon the occurrence of a large outbreak (more precisely one of size at least$\log n$). Central limit theorems for the size of the largest cluster in bondpercolation on Molloy-Reed and Newman-Strogatz-Watts random graphs followimmediately from our results, as do central limit theorems for the size of thegiant component of those graphs. Corresponding central limit theorems for sitepercolation on those graphs are also proved.

Central limit theorems for SIR epidemics and percolation on configuration model random graphs