  We study the susceptible-infective-recovered (SIR) epidemic on a random graph
chosen uniformly subject to having given vertex degrees. In this model
infective vertices infect each of their susceptible neighbours, and recover, at
a constant rate.
  Suppose that initially there are only a few infective vertices. We prove
there is a threshold for a parameter involving the rates and vertex degrees
below which only a small number of infections occur. Above the threshold a
large outbreak occurs with probability bounded away from zero. Our main result
is that, conditional on a large outbreak, the evolutions of certain quantities
of interest, such as the fraction of infective vertices, converge to
deterministic functions of time.
  We also consider more general initial conditions for the epidemic, and derive
criteria for a simple vaccination strategy to be successful.
  In contrast to earlier results for this model, our approach only requires
basic regularity conditions and a uniformly bounded second moment of the degree
of a random vertex.
  En route, we prove analogous results for the epidemic on the configuration
model multigraph under much weaker conditions. Essentially, our main result
requires only that the initial values for our processes converge, i.e. it is
the best possible.
