  This paper presents a novel extension of the edge-based compartmental model
for epidemics with arbitrary distributions of transmission and recovery times.
Using the message passing approach we also derive a new pairwise-like model for
epidemics with Markovian transmission and an arbitrary recovery period. The new
pairwise-like model allows one to formally prove that the message passing and
edge-based compartmental models are equivalent in the case of Markovian
transmission and arbitrary recovery processes. The edge-based and message
passing models are conjectured to also be equivalent for arbitrary transmission
processes; we show the first step of a full proof of this. The new
pairwise-like model encompasses many existing well-known models that can be
obtained by appropriate reductions. It is also amenable to a relatively
straightforward numerical implementation. We test the theoretical results by
comparing the numerical solutions of the various pairwise-like models to
results based on explicit stochastic network simulations.
