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

TL;DR

This paper introduces a new closure relation for epidemic models on fully connected networks, significantly improving the accuracy of approximate models by reducing the error order from 1/N to 1/N^2.

Contribution

A novel closure based on moment equations and binomial distribution assumptions that enhances the precision of epidemic models on networks.

Findings

New closure reduces error from order 1/N to 1/N^2

Applicable to SIS epidemic models on fully connected graphs

Improves the accuracy of approximate epidemic models

Abstract

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