  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.
