  Moment-closure techniques are commonly used to generate low-dimensional
deterministic models to approximate the average dynamics of stochastic systems
on networks. The quality of such closures is usually difficult to asses and the
relationship between model assumptions and closure accuracy are often
difficult, if not impossible, to quantify. Here we carefully examine some
commonly used moment closures, in particular a new one based on the concept of
maximum entropy, for approximating the spread of epidemics on networks by
reconstructing the probability distributions over triplets based on those over
pairs. We consider various models (SI, SIR, SEIR and Reed-Frost-type) under
Markovian and non-Markovian assumption characterising the latent and infectious
periods. We initially study two special networks, namely the open triplet and
closed triangle, for which we can obtain analytical results. We then explore
numerically the exactness of moment closures for a wide range of larger motifs,
thus gaining understanding of the factors that introduce errors in the
approximations, in particular the presence of a random duration of the
infectious period and the presence of overlapping triangles in a network. We
also derive a simpler and more intuitive proof than previously available
concerning the known result that pair-based moment closure is exact for the
Markovian SIR model on tree-like networks under pure initial conditions. We
also extend such a result to all infectious models, Markovian and
non-Markovian, in which susceptibles escape infection independently from each
infected neighbour and for which infectives cannot regain susceptible status,
provided the network is tree-like and initial conditions are pure. This works
represent a valuable step in deepening understanding of the assumptions behind
moment closure approximations and for putting them on a more rigorous
mathematical footing.
