Mean Field Analysis of Hypergraph Contagion Models

.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 modeled 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 [G. F. de Arruda, G. Petri, and Y. Moreno, Phys. Rev. Res., 2 (2020), 023032]. 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 [D. J. Higham and H.-L. de Kergorlay, Proc. A, 477 (2021), 20210232], and (c) numerical simulations of the microscale model.Keywordscompartmentalcollective contagionepidemiologyspectral analysissusceptible-infected-susceptibleMSC codes92D3060J27