  We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem
for generating exact closures (here expressing joint probabilities in terms of
their constituent marginal probabilities) for susceptible-infectious-removed
(SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and
removal processes, this enables us to obtain a systematic reduction in the
number of differential equations needed for an exact `moment closure'
representation of the underlying stochastic model. We define `transmission
blocks' as a possible extension of the block concept in graph theory and show
that the order at which the exact moment closure representation is curtailed is
the size of the largest transmission block. More generally, approximate
closures of the hierarchy of moment equations for these dynamics are typically
defined for the first and second order yielding mean-field and pairwise models
respectively. It is frequently implied that, in principle, closed models can be
written down at arbitrary order if only we had the time and patience to do
this. However, for epidemic dynamics on networks, these higher-order models
have not been defined explicitly. Here we unambiguously define hierarchies of
approximate closed models that can utilise subsystem states of any order, and
show how well-known models are special cases of these hierarchies.
