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Recently, research that focuses on the rigorous understanding of the relationbetween simulation and/or exact models on graphs and approximate counterpartshas gained lots of momentum. This includes revisiting the performance ofclassic pairwise models with closures at the level of pairs and/or triples aswell as effective-degree-type models and those based on the probabilitygenerating function formalism. In this paper, for a fully connected graph andthe simple $SIS$ (susceptible-infected-susceptible) epidemic model, a novelclosure is introduced. This is done via using the equations for the moments ofthe distribution describing the number of infecteds at all times combined withthe empirical observations that this is well described/approximated by abinomial distribution with time dependent parameters. This assumption allows usto express higher order moments in terms of lower order ones and this leads toa new closure. The significant feature of the new closure is that thedifference of the exact system, given by the Kolmogorov equations, from thesolution of the newly defined approximate system is of order $1/N^2$. This isin contrast with the $\mathcal{O}(1/N)$ difference corresponding to theapproximate system obtained via the classic triple closure.

Exact and approximate epidemic models on networks: a new, improved closure relation