# SIS Epidemic Propagation on Scale-Free Hypernetwork
## 1 Introduction  
Infectious diseases have been a great threat to human society since ancient times. In history, major events of infectious diseases occurred continuously, which caused great disasters to human beings [1]. With the acceleration of globalization and the development of science and technology, population flow is becoming increasingly rapid and frequent. The world is interdependent and interrelated, which provides the possibility for the rapid spread of infectious diseases. Therefore, it is of great theoretical and practical significance to explore and study the pathological principles and propagation laws of various infectious diseases. Particularly in the context of the current COVID-19 pandemic, it is increasingly important to study the pathogenesis, spread laws, evolutionary trends, and prevention and control strategies of infectious diseases. Moreover, it has become the current research field of complex systems and epidemic dynamics [2, 3]. There has been a long history of research on infectious disease modeling, propagation law, and corresponding immunization strategies. Traditional epidemiological studies have focused on mathematical models of the spread of infectious diseases using differential equations. This approach has made some success in explaining some of the spread of infectious diseases, but it focuses on uniformly mixed populations and ignores differences in infectivity among individuals. It is impossible to describe the spread process of infectious diseases in real large-scale social networks with clear heterogeneity. Whether it is a disease, a computer virus or a rumor, their spread will inevitably be affected by the networktopology structure. At the end of the 20 th century, with the emergence of small world networks and scale-free networks [4; 5], complex network theory developed rapidly. With the upsurge of complex system and network science research, people began to study the dynamic characteristics of network propagation of infectious diseases, and the method of combining complex network theory with epidemiology has become an important trend in the dynamic modeling of infectious diseases [6]. From the perspective of complex network, scholars regard individuals as nodes and the contact relationship between individuals as connecting edges to construct network structure, which can depict the spread process of disease more truly. The development and derivation of many network models provide a new direction for the study of infectious disease dynamics. Among them, there are three most classic ones: The first is the mean field model proposed by [PERSON] and [PERSON] [7; 8; 9] in 2001; the second is the seepage model proposed by [PERSON] [10] in 2002, and the related branching process method and probability generating function method; and the third is the discrete probability model proposed by [PERSON] et al. in 2003 [11]. In addition to the spread of infectious diseases, many information diffusion models have also been developed [12; 13]. In the traditional complex network structure, each edge can only be associated with two nodes, which can only describe the interaction of pairwise nodes. Namely, network models based on binary interactions are insufficient to provide a complete description of complex systems. To overcome the limitations of simple scale-free networks in describing large-scale complex systems, scholars try to depict real systems with hypernetworks based on hypergraphs. The hyperedge in the hypernetwork can contain any number of nodes, which can well express the higher-order interaction relationship between nodes. Therefore, as a form of higher-order networks, hypernetwork is more suitable for describing more complex social connections (micro-communities) captured by hyperlinks beyond standardly used pairwise couplings and the clustering characteristics of epidemic spread in the real world. At present, the research of hypernetwork mainly focuses on the analysis of static topology index and the construction of dynamic evolution model.