In this paper, we outline the theory of percolation networks and their use in the analysis of stochastic epidemic models on undirected contact networks. We then show how the same theory can be used to analyze epidemic models with random mixing. In the percolation network for a random-mixing model, undirected edges disappear in the limit of a large population, so the percolation network is purely directed. In a series of simulations, we show that 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 conditions under which percolation network models are equivalent to branching processes and use percolation networks to re-derive several classical results from different areas of infectious disease epidemiology. In an appendix, we show how percolation networks can be defined for any time-homogeneous stochastic epidemic model. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR epidemic models in closed populations, which are an important part of theoretical infectious disease epidemiology.