# SIS Epidemic Propagation on Scale-Free Hypernetwork
## 3 Epidemic Models in Hypernetworks
### Theoretical Analysis  
In a network structure with uneven degree distribution, the hyperdegree of each node is different, and the propagation ability of different nodes is also different. Scale-free hypernetwork is a typical hypernetwork with uneven distribution of hyperdegree. Under this network structure, the dynamic process of disease propagation has its unique characteristics and laws. In the global propagation mode based on the reaction process strategy, since the infected node can propagate to any neighbor node in any hyperedge where it is located at any time, the mean field theory can be used to analyze the above global propagation model. By extending the propagation dynamics equation of BA scale-free networks in simple scale-free networks described in [7], the dynamics mean field reaction rate equation of the SIS model under the hypernetwork structure described in Section 3.1 can be expressed as:  
\[\partial_{t}\rho_{k}(t)=-\gamma\rho_{k}(t)+\beta k(m_{1}+m_{2}-1)[1-\rho_{k}( t)]\Theta(\rho(t)) \tag{2}\]  
where \(\rho_{k}(t)\) is the relative density of infected nodes with given hyperdegree \(k\). The first term in the right-hand side considers the probability that infected nodes with hyperdegree \(k\) recover to S-state. The second term in the right-hand side considers the probability that nodes with hyperdegree \(k\) are in S-state \([1-\rho_{k}(t)]\) and get infected through connected infected nodes. The probability of this term is proportional to the spreading rate \(\beta\), hyperdegree \(k\), the number of neighbors in one hyperedge \((m_{1}+m_{2}-1)\), and the probability \(\Theta(\rho(t))\) that any given hyperedge links to an infected node. Here, we do not consider the connectivity correlations. Therefore, the probability that a hyperedge encircles an infected node is only a function of the total density of infected nodes. The total density of infected nodes can be obtained by \(\rho(t)=\sum\limits_{k}P(k)\rho_{k}(t)\). In the steady state, \(\rho\) is a function of spreading rate \(\beta\) and recovering rate \(\gamma\). As a result, the probability \(\Theta\) becomes an implicit function of \(\beta\) and \(\gamma\). The system reaches its stationarity which can be described as \(\partial_{t}\rho_{k}(t)=0\). Therefore, we obtain  
\[\rho_{k}=\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{ 1}+m_{2}-1)\Theta(\beta,\gamma)} \tag{3}\]  
Figure 3: Schematic illustration of the spreading process on hypernetwork. (**a**) When \(t=0\), node 4 was set as the initial spreading seed (I-state), and the other nodes were all in S-state; (**b**) when \(t=1\), neighbor nodes 2, 3, and 10 of node 4 were infected and turned into I-state; (**c**) when \(t=2\), neighbor node 7 of node 2, neighbor node 12 of node 3, neighbor node 8 of node 4, and neighbor node 9 of node 10 were infected, at the same time, nodes 2, 4, and 10 recovered (returning to S-state). Equation (3) shows that nodes with more neighbors will have a higher probability to be infected. When defining and calculating \(\Theta(\rho(t))\), the heterogeneity of the network must be taken into account, namely, the hyperdegree of each node is different, and the probability of choosing a hyperedge to contain (related to) that node will also be different.