Abstract
Complex contagion models that involve contagion along higher-order structures, such as simplicial complexes and hypergraphs, yield new classes of mean-field models. Interestingly, the differential equations arising from many such models often exhibit a similar form, resulting in qualitatively comparable global bifurcation patterns. Motivated by this observation, we investigate a generalised mean-field-type model that provides a unified framework for analysing a range of different models. In particular, we derive analytical conditions for the emergence of different bifurcation regimes exhibited by three models of increasing complexity-ranging from three- and four-body interactions to two connected populations which simultaneously includes both pairwise and three-body interactions. For the first two cases, we give a complete characterisation of all possible outcomes, along with the corresponding conditions on network and epidemic parameters. In the third case, we demonstrate that multistability is possible despite only three-body interactions. Our results reveal that single population models with three-body interactions can only exhibit simple transcritical transitions or bistability, whereas with four-body interactions multistability with two distinct endemic steady states is possible. Surprisingly, the two-population model exhibits multistability via symmetry breaking despite three-body interactions only. Our work sheds light on the relationship between equation structure and model behaviour and makes the first step towards elucidating mechanisms by which different system behaviours arise, and how network and dynamic properties facilitate or hinder outcomes.