# Hybrid Deep Learning-Based Epidemic Prediction Framework of COVID-19: South Korea Case
## 3 Methodology of Hybrid Deep Learning Framework based on Meta-population
\[\frac{dE_{1}^{i}(t)}{dt}=\alpha_{11}^{i}\cdot S_{1}^{i}(t)-\beta_{11}^{i}\cdot E_{ 1}^{i}(t)-\delta_{E}\cdot E_{1}^{i}(t)\ m_{ij}\cdot E_{1}^{i}(t)-m_{ji}\cdot E_{1 }^{j}(t) \tag{6}\]  
\[\frac{dI_{1}^{i}(t)}{dt}=\beta_{11}^{i}\cdot E_{1}^{i}(t)-\gamma_{11}^{i}\cdot I _{1}^{i}(t)-\delta_{l}\cdot I_{1}^{i}(t)-\mu\cdot I_{1}^{i}(t)+m_{ij}\cdot I_{1 }^{i}(t)-m_{ji}\cdot I_{1}^{j}(t) \tag{7}\]  
\[\frac{dR_{1}^{i}(t)}{dt}=\gamma_{11}^{i}\cdot I_{1}^{i}(t)-\theta\cdot R_{1}^ {i}(t)-\delta_{R}\cdot R_{1}^{i}(t)m_{ij}\cdot S_{1}^{i}(t)+m_{ij}\cdot R_{1}^ {i}(t)-m_{ji}\cdot R_{1}^{j}(t) \tag{8}\]  
\[\frac{dS_{1}^{j}(t)}{dt}=\xi\cdot S_{1}^{j}(t)-\alpha_{11}^{j}\cdot S_{1}^{j} (t)-\delta_{S}\cdot S_{1}^{j}(t)+\theta\cdot R_{1}^{j}(t)-\beta_{ij}\cdot I_{1 }^{j}(t)\cdot S_{1}^{i}(t)+m_{ij}\cdot S_{1}^{i}(t)-m_{ji}\cdot S_{1}^{j}(t) \tag{9}\]  
\[\frac{dE_{1}^{j}(t)}{dt}=\alpha_{11}^{j}\cdot S_{1}^{j}(t)-\beta_{11}^{j}\cdot E _{1}^{j}(t)-\delta_{E}\cdot E_{1}^{j}(t)+m_{ij}\cdot E_{1}^{i}(t)-m_{ji}\cdot E _{1}^{j}(t) \tag{10}\]  
Figure 2.0  
Figure 2.1  
The parameters and terms of the model are provided in Figure 2.2 \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline Variables & \\ \(S_{1}^{i}\) & Number of susceptible humans in sub-population \(i\) \\ \(E_{1}^{i}\) & Number of exposed humans in sub-population \(i\) \\ \(I_{1}^{i}\) & Number of infected humans in sub-population \(i\) \\ \(R_{1}^{i}\) & Number of recovered humans in sub-population \(i\) \\ \(S_{1}^{i}\) & Number of susceptible humans in sub-population \(j\) \\ \(E_{1}^{j}\) & Number of exposed humans in sub-population \(j\) \\ \(I_{1}^{j}\) & Number of infected humans in sub-population \(j\) \\ \(R_{1}^{j}\) & Number of recovered humans in sub-population \(j\) \\ \hline Parameters & \\ \(\xi\) & Influx rate of susceptible humans \\ \(\dot{\alpha}_{11}^{i}\) & Rate from “susceptible” status to “exposed” status in sub-pop \(i\) \\ \(\beta_{11}^{i}\) & Rate from “exposed” status to “infected” status in sub-pop \(i\) \\ \(\gamma_{11}^{i}\) & Rate from “infected” status to “recovered” status in sub-pop \(i\) \\ \(\alpha_{11}^{i}\) & Rate from “susceptible” status to “exposed” status in sub-pop \(j\) \\ \(\beta_{11}^{i}\) & Rate from “exposed” status to “infected” status in sub-pop \(j\) \\ \(\gamma_{11}^{j}\) & Rate from “infected” status to “recovered” status in sub-pop \(j\) \\ \(\theta\) & Rate from “recovered” status to “susceptible” status \\ \(\delta_{S}\) & Natural death rate of susceptible human \\ \(\delta_{E}\) & Natural death rate of exposed human \\ \(\delta_{I}\) & Natural death rate of infected human \\ \(\delta_{R}\) & Natural death rate of recovered human \\ \(\mu\) & Disease-induced death rate of infected human \\ \(\dot{\beta}_{12}^{i}\) & Infection rate in the same sub-pop \(i\) \\ \hline \(\beta_{ij}\) & Infection rate from sub-pop \(i\) to sub-pop \(j\) \\ \(\beta_{ji}\) & Infection rate from sub-pop \(j\) to sub-pop \(i\) \\ \(m_{ii}\) & Migration rate in the same sub-pop \\ \(m_{ij}\) & Migration rate from sub-pop \(i\) to sub-pop \(j\) \\ \(m_{ji}\) & Migration rate from sub-pop \(j\) to sub-pop \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Notations of variables and parameters. As human-to-human transmission has been confirmed in the spreading of COVID-19, gatherings of people, contacts, or travels among cities (e.g., intra-city contact and intercity travel) of infected and exposed individuals have been the main reason for the spread of the virus. Based on the collected data, this study constructs the migration matrix at time \(t\), which is given by Equation (13). \[M(t)=\left[ \begin{array}{cccc}m_{11}(t)&m_{12}(t)&\cdots&m_{1K}(t)\\ m_{21}(t)&m_{22}(t)&\ldots&m_{2K}(t)\\ \vdots&\vdots&\ddots&\vdots\\ m_{N1}(t)&m_{N2}(t)&\cdots&m_{KK}(t)\end{array} \right] \tag{13}\]  
where \(K\) is the number of cities or regions (herein, \(K\) = 17), and \(m_{ij}(t)\) is the rate of people traveling from city \(i\) to city \(j\) at time \(t\).