This paper analyzes a Susceptible-Infected-Susceptible (SIS) model of epidemic propagation over hypergraphs. We focus on simplicial complexes and refer to the model as to the simplicial SIS model. Classically, the multi-group SIS model has assumed pairwise interactions of contagion across groups and thus has been vastly studied in the literature. It is only recently that a renewed special attention has been drawn to the study of contagion dynamics over higher-order interactions and over more general graph structures, like simplexes. Previous work on mean-field approximation models of the simplicial SIS model has indicated that a new dynamical behavior domain, compared to the classical SIS model, appears due to the newly introduced higher order interaction terms: both a disease-free equilibrium and an endemic equilibrium co-exist and are both locally asymptotically stable. This paper formally establishes that bistability (as a new epidemiological behavior) also appears in the simplicial SIS model. We give sufficient conditions over the model's parameters for the appearance of this and the other behavioral domains present in the classical multi-group SIS model. We additionally provide an algorithm to compute the value of the endemic equilibrium and report numerical analysis of the transition from the disease-free domain to the bistable domain.

中文翻译：

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具有简单和高阶相互作用的多组 SIS 流行病

本文分析了超图上流行病传播的易感-感染-易感 (SIS) 模型。我们专注于单纯复形，并将模型称为单纯 SIS 模型。传统上，多组 SIS 模型假设跨组传染的成对相互作用，因此在文献中进行了大量研究。直到最近，对高阶相互作用和更一般的图结构（如单纯形）的传染动力学的研究才引起了新的特别关注。先前对单纯 SIS 模型的平均场近似模型的研究表明，由于新引入的高阶相互作用项，与经典 SIS 模型相比，出现了一个新的动力学行为域：无病平衡和地方病平衡共存并且都是局部渐近稳定的。本文正式确立了双稳态（作为一种新的流行病学行为）也出现在简单的 SIS 模型中。我们给出了模型参数的充分条件，以用于出现经典多组 SIS 模型中存在的此行为域和其他行为域。我们还提供了一种算法来计算地方性平衡的值，并报告从无病域到双稳态域过渡的数值分析。s 参数，用于显示经典多组 SIS 模型中存在的此域和其他行为域。我们还提供了一种算法来计算地方性平衡的值，并报告从无病域到双稳态域过渡的数值分析。s 参数，用于显示经典多组 SIS 模型中存在的此域和其他行为域。我们还提供了一种算法来计算地方性平衡的值，并报告从无病域到双稳态域过渡的数值分析。