  Emergence of new diseases and elimination of existing diseases is a key
public health issue. In mathematical models of epidemics, such phenomena
involve the process of infections and recoveries passing through a critical
threshold where the basic reproductive ratio is 1.
  In this paper, we study near-critical behaviour in the context of a
susceptible-infective-recovered (SIR) epidemic on a random (multi)graph on $n$
vertices with a given degree sequence. We concentrate on the regime just above
the threshold for the emergence of a large epidemic, where the basic
reproductive ratio is $1 + \omega (n) n^{-1/3}$, with $\omega (n)$ tending to
infinity slowly as the population size, $n$, tends to infinity. We determine
the probability that a large epidemic occurs, and the size of a large epidemic.
  Our results require basic regularity conditions on the degree sequences, and
the assumption that the third moment of the degree of a random susceptible
vertex stays uniformly bounded as $n \to \infty$. As a corollary, we determine
the probability and size of a large near-critical epidemic on a standard
binomial random graph in the `sparse' regime, where the average degree is
constant. As a further consequence of our method, we obtain an improved result
on the size of the giant component in a random graph with given degrees just
above the critical window, proving a conjecture by Janson and Luczak.
