# Influence and Control of SARS-CoV-2 Transmission under Two Different Models
## 3 Results  
### Equilibrium Stability and Basic Reproduction Number  
By using the stability analysis of the equilibrium point of the differential equation model, the long-term stable state of the model can be studied. For model 1, the following equilibrium points are found: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Definitions** & **Value** & **Unit** \\ \hline \(K\) & Carrying capacity of pathogenic population & 5000 & Quantity \\ \(b_{w}\) & Incidence rate of pathogenic population & 0.01 & Day\({}^{-1}\) \\ \(d_{w}\) & Removal rate of pathogenic population & 0.01 & Day\({}^{-1}\) \\ \(A\) & Recruitment rate of human & 1000 & Quantity \\ \(d_{h}\) & Removal rate of humans & \(3.6\times 10^{-8}\) & Day\({}^{-1}\) \\ \(\beta_{h}\) & Transmission rate among human & \(3\times 10^{-8}\) & Day\({}^{-1}\) \\ \(\beta_{w}\) & Transmission rate from pathogenic population to human & \(1\times 10^{-8}\) & Day\({}^{-1}\) \\ \(\alpha\) & Average latency period & \(\frac{1}{\Omega}\) & Dimensionless \\ \(d\) & Removal rate of infected human & 0.09817 & Day\({}^{-1}\) \\ \(N\) & Carrying capacity of inanimate-mediated & 5000 & Quantity \\ \(\beta_{v}\) & Transmission rate among inanimate-mediated & \(1\times 10^{-6}\) & Day\({}^{-1}\) \\ \(b_{v}\) & The growth rate of inanimate-mediated & 0.001 & Day\({}^{-1}\) \\ \(d_{v}\) & Removal rate of inanimate-mediated & 0.0001 & Day\({}^{-1}\) \\ \(d_{1}\) & Disinfection rate & 0-1 & Day\({}^{-1}\) \\ \(\beta\) & Transmission rate from inanimate-mediated to human & 0.01 & Day\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters in models. Figure 2: Transforming relationships with indirect transmission. \[E_{1}(0,\frac{A}{d_{h}},0,0)\]  
\[E_{2}(0,\frac{(\alpha+d_{h})(d_{h}+d)}{\beta_{h}(\alpha+d_{h}+d)},\frac{A\beta_{ h}(\alpha+d_{h}+d)-d_{h}(\alpha+d_{h})(d_{h}+d)}{\beta_{h}(\alpha+d_{h}+d)( \alpha+d_{h})},\frac{\alpha(A\beta_{h}(\alpha+d_{h}+d)-d_{h}(\alpha+d_{h})(d_ {h}+d))}{\beta_{h}(\alpha+d_{h}+d)(\alpha+d_{h})(d_{h}+d)})\]  
\[E_{3}(W^{*},S_{h}^{*},E_{h}^{*},I_{h}^{*})\]  
\[\text{where }W^{*}=\frac{K(b_{w}-d_{w})}{b_{w}},S_{h}^{*}=S_{h^{*}}^{*},E_{h}^{*}= \frac{A-S_{h}^{*}d_{h}}{\alpha+d_{h}},I_{h}^{*}=\frac{\alpha(b_{w}(A-S_{h}^{* }d_{h})-S_{h}^{*}\beta_{w}K(b_{w}-d_{w}))}{S_{h}^{*}\beta_{h}b_{w}(\alpha+d+d_ {h})}.\]  
\(S_{h}^{*}\) is the positive real root of equation:  
\[ \begin{array}{l}\beta_{h}b_{w}d_{h}(\alpha+d+d_{h})S_{h}^{*2}\\ +(-\beta_{w}K(b_{w}-d_{w})(d+d_{h})(\alpha+d_{h})-b_{w}d_{h}(\alpha+d_{h})(d+d _{h})-A\beta_{h}b_{w}(\alpha+d+d_{h}))S_{h}^{*}\\ +Ab_{w}(d+d_{h})(\alpha+d_{h})=0\end{array} \]  
To prove the stability of the disease-free equilibrium, the Jacobian matrix can be calculated:  
\[J= \begin{bmatrix}b_{w}(1-\frac{W}{K})-\frac{b_{w}W}{K}-d_{w}&0&0&0\\ -\beta_{w}S_{h}&-\beta_{w}W-\beta_{h}(E_{h}+I_{h})-d_{h}&-\beta_{h}S_{h}&- \beta_{h}S_{h}\\ \beta_{w}S_{h}&\beta_{h}(E_{h}+I_{h})+\beta_{w}W&\beta_{h}S_{h}-\alpha-d-d_{h}& \beta_{h}S_{h}\\ 0&0&\alpha&-d-d_{h}\end{bmatrix} \]  
For the disease-free equilibrium \(E_{1}(0,\frac{A}{d_{h}},0,0)\), eigenvalues of the Jacobian determinant are \(b_{w}-d_{w}\), \(-d_{h}\), \(-\alpha-d-d_{h}\) and \(\frac{A\beta_{h}-d_{h}-d_{h}^{2}}{d_{h}}\). For the eigenvalue \(b_{w}-d_{w}\), when \(b_{w}-d_{w}<0\), the disease-free equilibrium is asymptotically stable. In other words, when the incidence rate of the pathogenic population is lower than the removal rate, the disease-free equilibrium is stable. This situation requires control of the population of the pathogenic population. In other cases, if no action is taken, the pathogenic population will eventually carry the disease. Due to the number of human interventions, we also focus on disease development at the beginning of model development (60 days). We are mainly concerned with the state of transmission in its early stages. Using the stability theory of differential equations, the equilibrium points of model 2 are discussed.