  In this paper, we outline the theory of epidemic percolation networks and
their use in the analysis of stochastic SIR epidemic models on undirected
contact networks. We then show how the same theory can be used to analyze
stochastic SIR models with random and proportionate mixing. The epidemic
percolation networks for these models are purely directed because undirected
edges disappear in the limit of a large population. In a series of simulations,
we show that epidemic percolation networks accurately predict the mean outbreak
size and probability and final size of an epidemic for a variety of epidemic
models in homogeneous and heterogeneous populations. Finally, we show that
epidemic percolation networks can be used to re-derive classical results from
several different areas of infectious disease epidemiology. In an appendix, we
show that an epidemic percolation network can be defined for any
time-homogeneous stochastic SIR model in a closed population and prove that the
distribution of outbreak sizes given the infection of any given node in the SIR
model is identical to the distribution of its out-component sizes in the
corresponding probability space of epidemic percolation networks. We conclude
that the theory of percolation on semi-directed networks provides a very
general framework for the analysis of stochastic SIR models in closed
populations.
