  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.
