# Research on Disease Spreading Behavior Oriented to Three-Layer Complex Networks
## 4 Modeling and Analysis of Three-Layer SIS Disease Spread Model
this paper lies in the construction and visualization of three-layer network models with different intermediary layers. This algorithm greatly inherits the diversity of the multi-layer networks, which provide a good model reference for exploring the more complex and highly correlated networks. Secondly, this paper deduces the spreading threshold of the three-layer network through the spreading threshold theory of two-layer network and verifies the correctness of the theoretical results. Finally, the special properties of the intermediary layer in the three-layer network are used, and corresponding immunization strategies are formulated to effectively inhibit the spreading behavior. The three-layer network structure constructed in this paper is widely used. For example, cross-media spread of information, computer network virus spread, financial behavior spread, and so on. We construct a practical three-layer network model and formulate corresponding immunization strategy at the intermediary layer. It can save more time and effectively suppress or promote the spreading behavior based on the three-layer networks. Based on Equations (1)-(3), this paper defines the number of S-state nodes in each individual layer at time t in the three-layer network. Based on that, the univariate cubic equation is constructed. The global spreading threshold of the three-layer network is obtained by solving the eigenvalues of the univariate cubic equation. Furthermore, the correctness of the theoretical results is verified by simulations. For the network construction algorithm, the time complexity is related to the number of nodes (n) and link probability. Through the running time of the program, it is obtained that the larger the n and link probability, the slower the code-running procedure. In general, the time complexity of building a three-layer network is O(n). For the immunization strategy and spreading behavior, the program runs stably with a time complexity of O(n). Therefore, the overall algorithm time complexity is O(n2). In order to explain the contents in Figure 3 in detail and understand the spreading behavior oriented to three layer complex networks, we give the pseudocode of disease spreading oriented to three-layer complex networks, which is shown in Algorithm 1 as follows. ```
0: p: the intra-layer link probability
0: a1: the adjacency matrix of the first layer network
0: a2: the adjacency matrix of the second layer network
0: a3: the adjacency matrix of the first and second layers
0: a4: the adjacency matrix of the third layer network
0: a5: the adjacency matrix of the second and third layers
```  
Output: spreading threshold and speed etc. Contents
1. Generate the adjacency matrix of the first layer network  
1: a1(i,j) = 1;  
2: Generate the adjacency matrix of the second layer network  
2: a2(i,  
\[\left\{ \begin{array}{l}-\rho^{\pi}(t)+\lambda_{a}\langle k_{a}\rangle\rho^{\pi}(t) (1-\rho^{\pi}(t))+\lambda_{\beta\alpha}\langle k_{\beta\alpha}\rangle\rho^{ \beta}(t)(1-\rho^{\pi}(t))=0\\ -\rho^{\beta}(t)+\lambda_{\beta}\langle k_{\beta}\rangle\rho^{\beta}(t)(1- \rho^{\beta}(t))+\lambda_{\alpha\beta}\langle k_{\alpha\beta}\rangle\rho^{ \alpha}(t)(1-\rho^{\beta}(t))+\lambda_{\gamma\beta}\langle k_{\gamma\beta} \rangle\rho^{\gamma}(t)(1-\rho^{\beta}(t))=0\\ -\rho^{\gamma}(t)+\lambda_{\gamma}\langle k_{\gamma}\rangle\rho^{\gamma}(t)(1 -\rho^{\gamma}(t))+\lambda_{\beta\gamma}\langle k_{\beta\gamma}\rangle\rho^{ \beta}(t)(1-\rho^{\gamma}(t))=0\\ \end{array} \right. \tag{7}\]  
Combining the above equations, it can be seen that if an infectious disease starts to spread within any one layer, it will generate the global disease-spreading activity in the three-layer complex networks, in which there is a spreading threshold dividing the entire network into a healthy state and a spreading state. When \(\lambda_{a}<\lambda^{c}\) and \(\lambda_{\beta}<\lambda^{c}\), the spreading procedure will enter a healthy balance state, and \(\rho^{\alpha}=0\), \(\rho^{\beta}=0\).