Susceptible-infected-recovered-susceptible processes competing on simplicial complexes

Contagions do not exist in isolation but in the presence of other propagating quantities, with which they may compete. Furthermore, higher-order interactions (interactions involving more than two nodes) are widely represented in real-world systems. We propose a stochastic model for susceptible-infected-recovered-susceptible (SIRS) processes competing in a simplicial complex that accommodates higher-order interactions. We also propose deterministic microscopic Markov chain (MMC) and mean-field (MF) forms of this model, which we analyze to reveal conditions for the persistence of each infection. We verify our analysis via numerical simulations, which also unveil critical mass effects, discontinuous phase transitions, and bistability. The stochastic model exhibits eight distinct classes of dependence of final state on initial conditions. All eight of these classes can be approximately reproduced with MMC, but only seven are seen under MF. The class observed in MMC and approximated in the stochastic model, but missing from MF, involves three distinct steady states for a single set of parameter values. However, in the absence of higher-order interactions MF exhibits an additional class, not observed in MMC or the stochastic model, corresponding to an infinite spectrum of steady states, in which the ratio of infection levels remains at its initial level. Both MMC and MF models can exhibit sustained oscillations in which one contagion disappears while the other tends to a limit cycle.