  Epidemic models currently play a central role in our attempts to understand
and control infectious diseases. Here, we derive a model for the diffusion
limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a
heterogeneous network. Using this, we consider analytically the early
asymptotic exponential growth phase of such epidemics, showing how the higher
order moments of the network degree distribution enter into the stochastic
behaviour of the epidemic. We find that the first three moments of the network
degree distribution are needed to specify the variance in disease prevalence
fully, meaning that the skewness of the degree distribution affects the
variance of the prevalence of infection. We compare these asymptotic results to
simulation and find a close agreement for city-sized populations.
