A bottom-up derivation of higher-order SIS dynamics, building systematically from node-level equations to pairs, triplets, and three-body interactions and provides a principled method of generating new models.
Abstract
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.
We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.
Amit Tiwari, C. Hens, Prosenjit Kundu· 0 citations
This work investigates the susceptible-infected-susceptible epidemic model on clustered regular networks using the pair mean-field approach, where higher-order effects are explicitly considered in dynamic analysis and proves the effectiveness of the pair mean-field method for analyzing higher-order network dynamics.
Qing-Chu Wu, Zhao-Yan Wu· Journal of Mathematical Biol...· 0 citations
This paper studies a susceptible--infected--susceptible (SIS) epidemic model coupled with opinion dynamics over a network of communities with higher-order interactions. Unlike standard networked SIS models, which account only for pairwise transmission, the proposed model incorporates group-level infection mechanisms and feedback between epidemic prevalence and community opinions. We establish local stability and instability conditions for a particular healthy equilibrium, derive a sufficient condition for global exponential eradication of the infection state, and identify conditions under which higher-order interactions induce bistability in the reduced dynamics on a positively invariant synchronous set. These results characterize how higher-order interactions alter the dynamics of opinion-dependent epidemic systems. Numerical simulations illustrate the predicted eradication, bistable, and endemic regimes.
Saba Samadi, José I. Caiza, S. Gracy et al.· 0 citations
This paper provides the rigorous mathematical foundations and algorithmic details underlying the CPMC framework, and details a dynamic programming scheme to exactly implement complex boundary conditions - including patient-zero and macroscopic outbreak-size constraints - enabling the rejection-free generation of valid trajectories.
Thomas Barthel, Jiazheng Sun, Jhao-Hong Peng· 1 citation
Transmission through temporary groups does not necessarily preserve complete information about the group-size distribution in aggregate epidemic data. We show that, in finite-population SIS and SIR models, the group-size distribution enters the dynamics only through a finite set of moments selected by the order of the nonlinear transmission kernel. Consequently, markedly different distributions can generate identical stochastic dynamics when their relevant moments coincide. This equivalence extends to transient evolution, fluctuations, extinction-time statistics, and final outbreak sizes. In the deterministic limit, the first moment sets the invasion threshold, whereas the second controls the nature of the transition and the emergence of bistability and hysteresis. The two distributions become distinguishable only when a higher-order transmission mechanism activates their first unmatched moment; even a weak additional channel can shift the phase boundary and place the systems in different dynamical regimes. Solutions of the master equation and stochastic simulations support these analytical predictions. These results establish an intrinsic limit on epidemic inference: a single aggregate dynamical protocol can identify only an equivalence class of group-size distributions, rather than uniquely reconstructing the full distribution.
Abstract.
In the early stages of a newly emerged or reemerged disease, there is a rapid increase in new infections, which can potentially lead to a healthcare crisis due to constraints of available medical resources. It is important to note that the recovery period of these emerging diseases typically follows a Gamma distribution rather than being narrowly centered around the mean. In this study, we propose a susceptible-infectious-recovered (SIR) network model that incorporates a general recovery rate and a saturation treatment function. We establish the well-posedness and global stability of the disease-free equilibrium in the model by employing semigroup theory and the standard comparison principle, respectively. From an epidemiological perspective, when the delayed treatment effect exceeds a significant threshold, a phenomenon known as backward bifurcation emerges near the disease-free equilibrium. This assertion is supported by an updated version of the Lyapunov–Schmidt approach. Additionally, we conduct numerical simulations to investigate how network topology and non-Markovian processes affect the patterns of disease transmission.
Jun-Yuan Yang, Maia Martcheva, Jun Zhang et al.· SIAM Journal on Applied Math...· 0 citations
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