Competing spreading dynamics in simplicial complex.

Interactions in biology and social systems are not restricted to pairwise but can take arbitrary sizes. Extensive studies have revealed that the arbitrary-sized interactions significantly affect the spreading dynamics on networked systems. Competing spreading dynamics, i.e., several epidemics spread simultaneously and compete with each other, have been widely observed in the real world, yet the way arbitrary-sized interactions affect competing spreading dynamics still lacks systematic study. This study presents a model of two competing simplicial susceptible-infected-susceptible epidemics on a higher-order system represented by simplicial complex and analyzes the model's critical phenomena. In the proposed model, a susceptible node can only be infected by one of the two epidemics, and the transmission of infection to neighbors can occur through pairwise (i.e., an edge) and high-order (e.g., 2-simplex) interactions simultaneously. Through a mean-field (MF) theory analysis and numerical simulations, we show that the model displays rich dynamical behavior depending on the 2-simplex infection strength. When the 2-simplex infection strength is weak, the model's phase diagram is consistent with the simple graph, consisting of three regions: the absolute dominant regions for each epidemic and the epidemic-free region. With the increase of the 2-simplex infection strength, a new phase region called the alternative dominant region emerges. In this region, the survival of one epidemic depends on the initial conditions. Our theoretical analysis can reasonably predict the time evolution and steady-state outbreak size in each region. In addition, we further explore the model's phase diagram both when the 2-simplex infection strength is symmetrical and asymmetrical. The results show that the 2-simplex infection strength has a significant impact on the system phase diagram.