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The epidemic threshold is probably the most studied quantity in the modellingof epidemics on networks. For a large class of networks and dynamics theepidemic threshold is well studied and understood. However, it is less so forclustered networks where theoretical results are mostly limited to idealisednetworks. In this paper we focus on a class of models known as pairwise modelswhere, to our knowledge, no analytical result for the epidemic thresholdexists. We show that by exploiting the presence of fast variables and usingsome standard techniques from perturbation theory we are able to obtain theepidemic threshold analytically. We validate this new threshold by comparing itto the numerical solution of the full system. The agreement is found to beexcellent over a wide range of values of the clustering coefficient,transmission rate and average degree of the network. Interestingly, we findthat the analytical form of $R_0$ depends on the choice of closure,highlighting the importance of model choice when dealing with real-worldepidemics. Nevertheless, we ex- pect that our method will extend to othersystems in which fast variables are present.

Epidemic threshold in pairwise models for clustered networks: closures and fast correlations