# Hybrid Deep Learning-Based Epidemic Prediction Framework of COVID-19: South Korea Case
## 3 Methodology of Hybrid Deep Learning Framework based on Meta-population
The migration matrix \(M\) is non-symmetric (\(m_{ij}(t)\
eq m_{ji}(t)\)) because the traffic between cities is different at any given time. The number of outflow migrants of city \(i\) at time \(t\) is denoted by Equation (14). \[m_{i}^{out}(t)=\sum_{i=1}^{K}m_{ij}(t) \tag{14}\]  
The number of inflow migrants of city \(i\) at time \(t\) is denoted by Equation (15). \[m_{i}^{in}(t)=\sum_{j=1}^{K}m_{ji}(t) \tag{15}\]  
In the SEIR model, \(\Delta S(t)\), \(\Delta E(t)\), \(\Delta I(t)\), and \(\Delta R(t)\) are defined as \(S(t)-S(t-1)\), \(E(t)-E(t-1)\), \(I(t)-I(t-1)\), and \(R(t)-R(t-1)\), respectively, where \(t\) is the daily count. As the incubation period for COVID-19 can reach 14 days, the number of exposed individuals (who show no symptoms but are infectious) is crucial for the spread of the disease. State \(E\), which is not available from official data, is an important state in this model. In a city \(i\), the four states are defined by \(S_{i}(t)\), \(E_{i}(t)\), \(I_{i}(t)\), and \(R_{i}(t)\) at time \(t\). The total susceptible population, \(N_{i}^{s}\), is the number of infected individuals in city \(i\). If city \(i\) has a total population of \(P_{i}\) and the percentage of infected population is \(\tau_{i}\), then \(N_{i}^{s}=\tau_{i}\cdot P_{i}\). The provided meta-population model incorporates human migration patterns in order to capture the epidemic dynamics. For city \(i\), the daily increase in the number of infected cases includes the inflow of infected individuals from other cities. In real situations, the inflow and outflow of exposed individuals to and from a city are important and must be estimated by the epidemic model. Thus, if \(m_{ij}(t)\) is the rate of people moving from city \(i\) to city \(j\) on day \(t\), then the number of infected individuals moving from city \(i\) to city \(j\) is given by Equation (16). \[\Delta I_{ij}^{in}(t)=I_{i}(t)m_{ij}(t) \tag{16}\]  
As the total number of migrant rates leaving the city \(j\) is \(\sum_{i=1}^{N}m_{ji}\), the number of infected cases that have migrated out of city \(j\) is determined from Equation (17), where \(P_{j}(t)\) is the population of city \(j\) on day \(t\). \[\Delta I_{j}^{out}(t)=I_{j}(t)\sum_{i=1}^{N}m_{ji}(t) \tag{17}\]  
Thus, the total number of infected cases on day \(t\) in city \(j\) is given by Equation (18), where \(\Delta I_{j}(t)=I_{j}(t+1)-I_{j}(t)\), and \(\lambda_{j}(t)\) is the infection rate in city \(j\) on day \(t\). \[\Delta I_{j}(t)=\lambda_{j}(t)E_{i}(t)-\gamma_{j}(t)I_{j}(t)+\sum_{i=1}^{N} \Delta I_{ij}^{in}(t)-\Delta I_{j}^{out}(t) \tag{18}\]Moreover, this paper introduces \(k_{I}\) (\(0<k_{I}<1\)) to the degree of migration unlikeness. In this manner, Equation (18) is converted into Equation (19). \[\Delta I_{j}(t)=\lambda_{j}(t)E_{i}(t)-\gamma_{j}(t)I_{j}(t)+k_{I}\left[\sum_{i= 1}^{N}\left(I_{i}(t)m_{ij}(t)\right)-\left(I_{j}(t)\sum\
olimits_{i=1}^{N}m_{ ji}(t)\right)\right] \tag{19}\]  
Incorporating the migration dynamics, the increase in exposed individuals on day \(t\) in city \(j\) is determined by Equation (20), where \(\Delta E_{j}(t)=E_{j}(t+1)-E_{j}(t)\)\(\beta_{j}\) is the infection rate of exposed individuals in city \(j\), and \(\alpha_{j}\) is the infection rate of susceptible individuals in city \(j\). \[\Delta E_{j}(t)=\frac{\beta_{j}(t)}{N_{j}^{s}(t)}I_{j}(t)S_{j}(t)+\frac{\alpha _{j}(t)}{N_{j}^{s}(t)}E_{j}(t)S_{j}(t)-\lambda_{j}(t)E_{i}(t)+\sum_{i=1}^{N} \left(\frac{E_{i}(t)m_{ij}(t)}{P_{i}(t)}\right)-\left(\frac{E_{j}(t)\sum_{i=1} ^{N}m_{ji}(t)}{P_{j}(t)}\right) \tag{20}\]  
Then, \(\Delta S_{j}(t)\) and \(\Delta R_{j}(t)\) are computed from Equations (21) and (22), respectively. In this study, the recovered individuals are assumed to stay in city \(j\), and the recovery rates in different cities are assumed to differ owing to the varied treatments and availability of medical facilities.