  A stochastic SIR (susceptible $\to$ infective $\to$ recovered) epidemic model
defined on a social network is analysed. The underlying social network is
described by an Erd\H{o}s-R\'{e}nyi random graph but, during the course of the
epidemic, susceptible individuals connected to infectious neighbours may drop
or rewire such connections. Large population limits of the model are derived
giving both convergence results for the early branching process-like behaviour,
and, assuming a major outbreak, the main phase of the epidemic process which
converges to a deterministic model that is equivalent to a certain pair
approximation model. Law of large numbers results are also obtained for the
final size (i.e. total number of individuals infected) of a major outbreak. Two
results stand out (valid for a range of parameter set-ups): (i) the limiting
final fraction infected may be discontinuous in the infection rate $\lambda$ at
its threshold $\lambda_c$ (thus making a discrete jump from $0$ to a strictly
positive number) and (ii) for the situation when rewiring is necessarily to
uninfected individuals, if it is discontinuous, the limiting final fraction
infected jumps from $0$ to $1$ as $\lambda$ passes through $\lambda_c$.
