Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this talk, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation and we formalise lumping and its direct link to graph automorphism. Lumping is a powerful technique that exploits graph symmetry and allows to keep the model exact while considerably reducing the number of equations. Finally, for pairwise model two different closures are presented, one well established and one that has been recently proposed. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models work well for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness. Keywords: network, epidemic, Markov chain, moment closure.