  A Markovian SIR (Susceptible-Infectious-Recovered) model is considered for
the spread of an epidemic on a configuration model network, in which
susceptible individuals may take preventive measures by dropping edges to
infectious neighbours. An effective degree formulation of the model is used in
conjunction with the theory of density dependent population processes to obtain
a law of large numbers and a functional central limit theorem for the epidemic
as the population size $N \to \infty$, assuming that the degrees of individuals
are bounded. A central limit theorem is conjectured for the final size of the
epidemic. The results are obtained for both the Molloy-Reed (in which the
degrees of individuals are deterministic) and Newman-Strogatz-Watts (in which
the degrees of individuals are independent and identically distributed)
versions of the configuration model. The two versions yield the same limiting
deterministic model but the asymptotic variances in the central limit theorema
are greater in the Newman-Strogatz-Watts version. The basic reproduction number
$R_0$ and the process of susceptible individuals in the limiting deterministic
model, for the model with dropping of edges, are the same as for a
corresponding SIR model without dropping of edges but an increased recovery
rate, though, when $R_0>1$, the probability of a major outbreak is greater in
the model with dropping of edges. The results are specialised to the model
without dropping of edges to yield conjectured central limit theorems for the
final size of Markovian SIR epidemics on configuration-model networks, and for
the giant components of those networks. The theory is illustrated by numerical
studies, which demonstrate that the asymptotic approximations are good, even
for moderate $N$.
