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

TL;DR

This paper derives an analytical epidemic threshold for pairwise models on clustered networks by exploiting fast variables and perturbation theory, validated through numerical comparisons.

Contribution

It introduces a novel analytical method to determine the epidemic threshold in pairwise models for clustered networks, a problem previously lacking such results.

Findings

Excellent agreement between analytical threshold and numerical solutions.

The form of R0 depends on the closure choice, affecting model accuracy.

Method potentially applicable to other systems with fast variables.

Abstract

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