# Research on Disease Spreading Behavior Oriented to Three-Layer Complex Networks
## 4 Modeling and Analysis of Three-Layer SIS Disease Spread Model
Conversely, when both \(\lambda_{a}\) and \(\lambda_{\beta}\) exceed the spreading threshold \(\lambda^{c}\), the network will globally spread the disease through the links in the intermediary layer, and the density of the infected nodes in the network will increase, even if it increase up to 1. By ignoring the \((1-\rho^{\alpha}(t))\), \((1-\rho^{\beta}(t))\), \((1-\rho^{\gamma}(t))\) in Equation (7), Equation (8) can be obtained as follows. \[ \begin{bmatrix}\lambda_{a}\langle k_{a}\rangle-1&\lambda_{\beta\alpha}\langle k _{\beta\alpha}\rangle&0\\ \lambda_{a\beta}\langle k_{\alpha\beta}\rangle&\lambda_{\beta}\langle k_{ \beta}\rangle-1&\lambda_{\gamma\beta}\langle k_{\gamma\beta}\rangle\\ 0&\lambda_{\beta\gamma}\langle k_{\beta\gamma}\rangle&\lambda_{\gamma}\langle k _{\gamma}\rangle-1\\ \end{bmatrix}\begin{bmatrix}\rho^{\alpha}(t)\\ \rho^{\beta}(t)\\ \rho^{\gamma}(t)\\ \end{bmatrix} =0 \tag{8}\]  
By rectifying the above equations, we can get Equation (9). \[\left( \begin{bmatrix}\lambda_{a}\langle k_{a}\rangle&\lambda_{\beta\alpha} \langle k_{\beta\alpha}\rangle&0\\ \lambda_{a\beta}\langle k_{a\beta}\rangle&\lambda_{\beta}\langle k_{\beta} \rangle&\lambda_{\gamma\beta}\langle k_{\gamma\beta}\rangle\\ 0&\lambda_{\beta\gamma}\langle k_{\beta\gamma}\rangle&\lambda_{\gamma}\langle k _{\gamma}\rangle\\ \end{bmatrix} - \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix} \right) \begin{bmatrix}\rho^{\alpha}(t)\\ \rho^{\beta}(t)\\ \rho^{\gamma}(t)\\ \end{bmatrix} =0 \tag{9}\]  
\[\text{And }R=\left[ \begin{matrix}\lambda_{\alpha}\langle k_{a}\rangle&\lambda_{ \beta\alpha}\langle k_{\beta\alpha}\rangle&0\\ \lambda_{a\beta}\langle k_{a\beta}\rangle&\lambda_{\beta}\langle k_{\beta} \rangle&\lambda_{\gamma\beta}\langle k_{\gamma\beta}\rangle\\ 0&\lambda_{\beta\gamma}\langle k_{\beta\gamma}\rangle&\lambda_{\gamma}\langle k _{\gamma}\rangle\\ \end{matrix} \right],\rho=\left(\rho^{\alpha}(t),\rho^{\beta}(t),\rho^{\gamma}(t )\right)^{T}\!\!The global spreading threshold of the three-layer complex networks is calculated as follows. \[\lambda^{c}{}_{\text{min}}=\frac{-b+\sqrt{A}(cos\frac{\theta}{3}-\sqrt{3}sin \frac{\theta}{3})}{3a} \tag{11}\]  
and \(\theta=\arccos T,T=\frac{2b^{3}-9 abc+27a^{2}d}{2\sqrt{\left(b^{2}-3 ac\right)^{3}}}\), \(A=b^{2}-3 ac\). Without considering inter-layer spread, we only analyze the disease-spreading thresholds in intra-layer. For the three single layers \(\alpha,\ \beta,\ \gamma,\) the disease spreading form is described as follows. \[\left\{ \begin{array}{l}\frac{d\rho^{\alpha}(t)}{dt}=-\rho^{\alpha}(t)+ \lambda_{\alpha}\langle k_{\alpha}\rangle\rho^{\alpha}(t)(1-\rho^{\alpha}(t)) \\ \frac{d\rho^{\beta}(t)}{dt}=-\rho^{\beta}(t)+\lambda_{\beta}\langle k_{\beta} \rangle\rho^{\beta}(t)(1-\rho^{\beta}(t))\\ \frac{d\rho^{\gamma}(t)}{dt}=-\rho^{\gamma}(t)+\lambda_{\gamma}\langle k_{ \gamma}\rangle\rho^{\gamma}(t)(1-\rho^{\gamma}(t))\end{array} \right. \tag{12}\]  
Similarly, if we ignore the \(1-\rho^{\alpha}(t)\), \(1-\rho^{\beta}(t)\), and \(1-\rho^{\gamma}(t)\) terms in Equation (12), we get  
\[\left\{ \begin{array}{l}-\rho^{\alpha}(t)+\lambda_{\alpha}\langle k_{\alpha} \rangle\rho^{\alpha}(t)=0\\ -\rho^{\beta}(t)+\lambda_{\beta}\langle k_{\beta}\rangle\rho^{\beta}(t)=0\\ -\rho^{\gamma}(t)+\lambda_{\gamma}\langle k_{\gamma}\rangle\rho^{\gamma}(t)=0 \end{array} \right. \tag{13}\]  
The final spreading threshold for the single layer network is obtained as \(\lambda_{\alpha}^{c}=\frac{1}{\langle k_{\alpha}\rangle}\), \(\lambda_{\beta}^{c}=\frac{1}{\langle k_{\beta}\rangle}\), \(\lambda_{\gamma}^{c}=\frac{1}{\langle k_{\gamma}\rangle}\). When \(\lambda^{c}<\lambda_{\alpha}^{c}<\lambda_{\beta}^{c}<\lambda_{\gamma}^{c}\), we get \(\langle k_{\alpha}\rangle>\langle k_{\beta}\rangle>\langle k_{\gamma}\rangle\).