  We consider a stochastic SIR (susceptible $\to$ infective $\to$ recovered)
epidemic defined on a configuration model random graph, in which infective
individuals can infect only their neighbours in the graph during an infectious
period which has an arbitrary but specified distribution. Central limit
theorems for the final size (number of initial susceptibles that become
infected) of such an epidemic as the population size $n$ tends to infinity,
with explicit, easy to compute expressions for the asymptotic variance, are
proved assuming that the degrees are bounded. The results are obtained for both
the Molloy-Reed random graph, in which the degrees of individuals are
deterministic, and the Newman-Strogatz-Watts random graph, in which the degrees
are independent and identically distributed. The central limit theorems cover
the 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 of
the 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 conditional
upon 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 bond
percolation on Molloy-Reed and Newman-Strogatz-Watts random graphs follow
immediately from our results, as do central limit theorems for the size of the
giant component of those graphs. Corresponding central limit theorems for site
percolation on those graphs are also proved.
