General SIS diffusion process with indirect spreading pathways on a hypergraph

TL;DR

This paper develops a comprehensive framework for SIS diffusion processes on directed hypergraphs, incorporating both direct and indirect pathways, and analyzes their equilibrium states and stability, extending to bi-virus competition.

Contribution

It introduces a novel SIS diffusion model on hypergraphs with both direct and indirect pathways, including a bi-virus competing system and a general interaction function, with rigorous theoretical analysis.

Findings

Characterization of healthy and endemic states for single-virus models

Analysis of stability and coexistence of multiple equilibria in bi-virus models

Numerical validation of theoretical results

Abstract

While conventional graphs only characterize pairwise interactions, higher-order networks (hypergraph, simplicial complex) capture multi-body interactions, which is a potentially more suitable modeling framework for a complex real system. However, the introduction of higher-order interactions brings new challenges for the rigorous analysis of such systems on a higher-order network. In this paper, we study a series of SIS-type diffusion processes with both indirect and direct pathways on a directed hypergraph. In a concrete case, the model we propose is based on a specific choice (polynomial) of interaction function (how several agents influence each other when they are in a hyperedge). Then, by the same choice of interaction function, we further extend the system and propose a bi-virus competing model on a directed hypergraph by coupling two single-virus models together. Finally, the most general model in this paper considers an abstract interaction function under single-virus and bi-virus settings. For the single-virus model, we provide the results regarding healthy state and endemic equilibrium. For the bi-virus setting, we further give an analysis of the existence and stability of the healthy state, dominant endemic equilibria, and coexisting equilibria. All theoretical results are finally supported by some numerical examples.