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Pairwise models are used widely to model epidemic spread on networks. Theseinclude the modelling of susceptible-infected-removed (SIR) epidemics onregular networks and extensions to SIS dynamics and contact tracing on moreexotic networks exhibiting degree heterogeneity, directed and/or weighted linksand clustering. However, extra features of the disease dynamics or of thenetwork lead to an increase in system size and analytical tractability becomesproblematic. Various `closures' can be used to keep the system tractable.Focusing on SIR epidemics on regular but clustered networks, we show that evenfor the most complex closure we can determine the epidemic threshold as anasymptotic expansion in terms of the clustering coefficient.We do this byexploiting the presence of a system of fast variables, specified by thecorrelation structure of the epidemic, whose steady state determines theepidemic threshold. While we do not find the steady state analytically, wecreate an elegant asymptotic expansion of it. We validate this new threshold bycomparing it to the numerical solution of the full system and find excellentagreement over a wide range of values of the clustering coefficient,transmission rate and average degree of the network. The technique carries overto pairwise models with other closures [1] and we note that the epidemicthreshold will be model dependent. This emphasises the importance of modelchoice when dealing with realistic outbreaks.

Fast variables determine the epidemic threshold in the pairwise model with an improved closure