Mean Field Analysis of Hypergraph Contagion Model

TL;DR

This paper analyzes a hypergraph-based contagion model using mean field approximation, deriving spectral conditions for stability and comparing these with other models and simulations to understand contagion dynamics in group interactions.

Contribution

It provides new spectral conditions for stability in hypergraph contagion models and compares mean field approximations with the exact process and other models.

Findings

Spectral conditions for local and global stability of the infection-free state.

Comparison of mean field predictions with the exact stochastic process.

Numerical validation of theoretical results.

Abstract

We typically interact in groups, not just in pairs. For this reason, it has recently been proposed that the spread of information, opinion or disease should be modelled over a hypergraph rather than a standard graph. The use of hyperedges naturally allows for a nonlinear rate of transmission, in terms of both the group size and the number of infected group members, as is the case, for example, when social distancing is encouraged. We consider a general class of individual-level, stochastic, susceptible-infected-susceptible models on a hypergraph, and focus on a mean field approximation proposed in [Arruda et al., Phys. Rev. Res., 2020]. We derive spectral conditions under which the mean field model predicts local or global stability of the infection-free state. We also compare these results with (a) a new condition that we derive for decay to zero in mean for the exact process, (b) conditions for a different mean field approximation in [Higham and de Kergorlay, Proc. Roy. Soc. A, 2021], and (c) numerical simulations of the microscale model.