# SIS Epidemic Propagation on Scale-Free Hypernetwork
## 3 Epidemic Models in Hypernetworks  
### Description of Model  
The dynamics of infectious disease nodes on the network can actually be seen as information dissemination on a specific network structure, and the evolution of different types of nodes caused by information over time, which depends not only on the topology of the network, but also on the way of information spread. The spread of infectious diseases at the group level can be considered as the spread of infectious viruses or bacteria on social networks, which depends on both the way of contact between people (which constitutes a social contact network) and the infectivity of the virus. Based on the evolution model in [17], we construct a social contact hypernetwork, in which nodes represent people and hyperedges represent the complex interaction relationship between these individuals. The model construction parameters are described as follows: At the initial moment, there is a hyperedge containing \(m_{0}\) nodes in the network, and \(m_{1}\) new nodes are added to the network at each time step, \(m_{2}\) old nodes in the network are selected to form a hyperedge. A total of \(m\) hyperedges are formed in a time step. After \(t\) time steps, when the total number of nodes in the network reaches the preset value \(|N|\), the cycle ends and the evolution terminates. We set \(m_{0}=m_{1}+m_{2}\), and get an \(m_{0}\)-uniform hypernetwork with a total of \(|N|=m_{0}+m_{1}t\) nodes and \(|E|=1+mt\) hyperedges finally. When \(m_{1}=m_{2}=1\) in the model, the hypernetwork unified model degenerates into the BA model (Barabasi-Albert scale-free network) of the simple scale-free network, thus the classic BA model can be regarded as a special case of this model. When the evolution is terminated, the hyperdegree distribution of the nodes in the hypernetwork is irrelevant with time \(t\), it depends on the number of nodes and hyperedges that have been added in the creation process of the network. The hyperdegree distribution of the hypernetwork generated by the above model is \(P(k)=1/m[(m_{1}/m_{2})+1](m/k)^{\frac{m_{1}}{m_{2}}+2}\)[17].  
The reactive process strategy (RP) [26; 27] shows global propagation, namely, the nodes enclosed in all hyperedges where the initial spreading node is located are the neighbor nodes of the initial spreading node, and will be affected by the initial spreading node. It is a form of higher-order interaction in hypernetworks based on hypergraph. The global propagation mechanism of infectious diseases based on RP strategy and SIS model under the hypernetwork structure can be described as follows:  
1. Construct a hypernetwork. The propagation process starts from a single infection source which is in infected state(I), and the rest of the nodes are all susceptible state(S).
2. At each time step, all nodes in infected state(I) spread the disease to all neighboring nodes that are in susceptible state(S) in their hyperedges with probability \(\beta\), and the susceptible nodes(S) turn into infected state(I), while nodes in infected state(I) themselves recover to susceptible state(S) with probability \(\gamma\).  
Figure 2: State transition diagram of the SIS propagation model, in which green and orange represent the susceptible and infected states respectively, and the arrows indicate the direction of the state transition.  
With the passage of time, the disease will continue to spread in the network, and eventually the system will reach a stable state, and the density of nodes in the states S and I will reach a relatively stable value and fluctuate slightly around this value.  
The spreading process of infectious diseases under the above hypernetwork structure is shown in Figure 3, where the numbers are used to identify nodes. Red nodes represent infected individuals and hollow nodes represent susceptible individuals.