Epidemic Outbreaks in Networks with Equitable or Almost-Equitable Partitions

TL;DR

This paper models epidemic spread in community-structured networks using quotient graphs and mean-field approximation, showing that the spectral radius of the quotient graph determines epidemic thresholds and stability of the healthy state.

Contribution

It introduces a novel approach using equitable and almost equitable partitions to analyze epidemic thresholds via quotient graph spectral properties.

Findings

Spectral radius of quotient graph determines epidemic threshold.

Existence of a second steady-state above the threshold.

Model simplifies epidemic analysis in community networks.

Abstract

We study the diffusion of epidemics on networks that are partitioned into local communities. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. In community models the nodal infection probability is thus expected to depend mainly on the interaction of a few, large interconnected clusters. In this work, we describe the epidemic process as a continuous-time individual-based susceptible-infected-susceptible (SIS) model using a first-order mean-field approximation. A key feature of our model is that the spectral radius of this smaller quotient graph (which only captures the macroscopic structure of the community network) is all we need to know in order to decide whether the overall healthy-state defines a globally asymptotically stable or an unstable equilibrium. Indeed, the spectral radius is related to the epidemic threshold of the system. Moreover we prove that, above the threshold, another steady-state exists that can be computed using a lower-dimensional dynamical system associated with the evolution of the process on the quotient graph. Our investigations are based on the graph-theoretical notion of equitable partition and of its recent and rather flexible generalization, that of almost equitable partition.