A principled closure framework for higher-order SIS epidemic models on networks
Susceptible-infected-susceptible (SIS) epidemic models on networks are governed by hierarchical moment equations where the dynamics of smaller subsystems depend on the state of larger ones. Moment closure approximations, which truncate this hierarchy by expressing higher-order state probabilities in terms of lower-order ones, are essential for obtaining tractable reduced systems. Higher-order networks, which extend the pairwise structure to include group interactions, introduce a combinatorial explosion of closure configurations, making systematic derivation harder. Consequently, existing higher-order SIS models are derived heuristically, where structural and dynamical assumptions underpinning their closures are not always apparent from the formulation alone. We develop a bottom-up derivation of higher-order SIS dynamics, building systematically from node-level equations to pairs, triplets, and three-body interactions. Central to our approach is a network-dependent closure operator that generates topologically appropriate approximations from local pairwise and triadic structure. Using this framework, we recover three existing higher-order SIS models--Burgio et al.'s maximal clique, Malizia et al.'s pair-based and inter-order models--as special cases, each arising under specific topological and dynamical assumptions. Our derivation reveals assumptions that are invisible from heuristic approaches: for instance, Malizia et al.'s inter-order overlap parameter is insufficient alone to express the model within our framework despite performing well against simulations, with the original derivation implicitly invoking additional structural assumptions. Our framework offers both a foundation for higher-order epidemic modeling and a constructive pathway for understanding the assumptions implicit in heuristically derived mean-field closures and provides a principled method of generating new models.