Feature Augmented SuperHyperGraph Neural Networks for Hierarchical Epidemic Modeling, Environmental Perturbations, and Source Identifiability

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Infectious disease transmission involves interactions occurring at multiple organizational levels, including individuals,households, communities, and regions. Conventional graph and hypergraph neural networks represent pairwise and higher order interactions, but they do not directly provide a recursive representation of nested population structures together with environmental factors and source identification mechanisms. We introduce the Feature Augmented SuperHyperGraph Neural Network (FA SHGNN), a mathematically formulated architecture for hierarchical epidemic modeling based on iterated powerset constructions. Individuals, population groups, communities, regions, and environmental entities are organized into multiple interconnected levels, while feature vectors encode epidemiological, contact, mobility, demographic, and environmental information. A hierarchical message passing operator propagates information across individual, hyperedge, supervertex, and environmental levels, and an SEIR type neural evolution operator with constrained transition fractions preserves positivity and population conservation. We establish well definedness and Lipschitz continuity, derive stability bounds under environmental perturbations, and characterize reduction conditions that recover mean aggregation hypergraph and pairwise message passing architectures. Source identification is posed as an inverse problem, and a separation condition yields consistency of the minimum distance estimator under a Gaussian observation model. A fully specified synthetic numerical verification confirms SEIR conservation, the environmental stability bound, and the source recovery guarantee; acontrolled protocol for larger scale and real world validation is also specified. Keywords: SuperHyperGraph; hypergraph neural networks; hierarchical epidemic modeling; higher order interactions; SEIR dynamics; source identification; environmental perturbations; Lipschitz stability—————————————————————————————————————————-