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In a previous paper Sharkey et al. [13] proved the exactness of closures atthe level of triples for Markovian SIR (susceptible-infected-removed) dynamicson tree-like networks. This resulted in a deterministic representation of theepidemic dynamics on the network that can be numerically evaluated. In thispaper, we extend this modelling framework to certain classes of networksexhibiting loops. We show that closures where the loops are kept intact areexact, and lead to a simplified and numerically solvable system of ODEs(ordinary-differential-equations). The findings of the paper lead us to ageneralisation of closures that are based on partitioning the network aroundnodes that are cut-vertices (i.e. the removal of such a node leads to thenetwork breaking down into at least two disjointed components or subnetworks).Exploiting this structural property of the network yields some naturalclosures, where the evolution of a particular state can typically be exactlygiven in terms of the corresponding or projected sates on the subnetworks andthe cut-vertex. A byproduct of this analysis is an alternative probabilisticproof of the exactness of the closures for tree-like networks presented inSharkey et al. [13]. In this paper we also elaborate on how the main result canbe applied to more realistic networks, for which we write down the ODEsexplicitly and compare output from these to results from simulation.Furthermore, we give a general, recipe-like method of how to apply thereduction by closures technique for arbitrary networks, and give an upper boundon the maximum number of equations needed for an exact representation.

Exact deterministic representation of Markovian SIR epidemics on networks with and without loops