An SIS diffusion process with direct and indirect spreading on a hypergraph

Conventional graphs capture pairwise interactions; by contrast, higher-order networks (hypergraphs, simplicial complexes) describe the interactions involving more parties, which have been rapidly applied to characterize a growing number of complex real-world systems. However, such dynamics evolving on higher-order networks modeled by hypergraphs require new mathematical tools to carry out rigorous analysis. In this paper, we study a Susceptible–Infected–Susceptible-type (SIS-type) diffusion process with both indirect and direct pathways on a directed hypergraph. We choose a polynomial interaction function to describe how several agents influence one another over a hyperedge. Then, we further extend the system and propose a bi-virus competing model on a directed hypergraph. For the single-virus case, we provide a comprehensive characterization of the healthy state and endemic equilibrium. For the bi-virus setting, we further give the analysis of the existence and stability of the healthy state, dominant endemic equilibria, and coexisting equilibria. All theoretical results are supported additionally by some numerical examples.