Clustering for epidemics on networks: a geometric approach

TL;DR

This paper introduces a geometric method to identify and analyze clusters in large contact networks for SIS epidemic models, simplifying complex outbreak dynamics and providing new analytical solutions and approximations.

Contribution

It presents a geometric approach to classify networks with epidemic clustering, derives closed-form solutions for complete graphs, and offers low-complexity bounds for arbitrary networks.

Findings

Identifies networks where epidemics simplify to few clusters

Provides closed-form solutions for SIS on complete graphs

Develops approximations and bounds for general contact networks

Abstract

Infectious diseases typically spread over a contact network with millions of individuals, whose sheer size is a tremendous challenge to analysing and controlling an epidemic outbreak. For some contact networks, it is possible to group individuals into clusters. A high-level description of the epidemic between a few clusters is considerably simpler than on an individual level. However, to cluster individuals, most studies rely on equitable partitions, a rather restrictive structural property of the contact network. In this work, we focus on Susceptible-Infected-Susceptible (SIS) epidemics, and our contribution is threefold. First, we propose a geometric approach to specify all networks for which an epidemic outbreak simplifies to the interaction of only a few clusters. Second, for the complete graph and any initial viral state vectors, we derive the closed-form solution of the nonlinear differential equations of the N-Intertwined Mean-Field Approximation (NIMFA) of the SIS process. Third, by relaxing the notion of equitable partitions, we derive low-complexity approximations and bounds for epidemics on arbitrary contact networks. Our results are an important step towards understanding and controlling epidemics on large networks.