### INTRODUCTION

### MATERIALS AND METHODS

### Estimation of transmission rate matrix between age groups using maximum likelihood estimation

*Y*infects an individual in age group

*X*is β

*, then the probability that individual*

_{XY}*i*in age group

*X*at time

*t*is not infected at time

*t*+1 is

*p*(

_{sur,X,i}*t*)=

*I*(

_{Y}*t*) is the number of infected people in age group Y that can transmit the disease at time

*t*, and

*N*is the total population. Conversely, the probability of being infected at time

*t*+1 is

*p*(

_{inf,X,i}*t*)=

### Mathematical modeling for the COVID-19 pandemic

*S*), Latent or Exposed (

*E*), Infectious (

*I*), Isolated (

*Q*), and Recovered (

*R*) groups. Three additional groups (

*U, V, P*) were considered to reflect vaccination: the group who is vaccinated but not effectively (Unprotected:

*U*), the group that had been effectively vaccinated but not yet immunized (Vaccinated:

*V*), and the group that was effectively vaccinated and became immunized after a certain period (Protected:

*P*). Therefore, the groups

*S, U*, and

*V*can be infected, and it was assumed that all had the same susceptibility. The mathematical model flowchart is shown in Figure 1, and the governing differential equations are as follows.

*X*denotes the age group, and β

*was the estimated transmission rate between age groups using MLE, which represented the average transmission among age groups during the third wave of the epidemic. It was assumed that the transmission rates between age groups were the same during the entire simulation period. However, epidemic trends change depending on intervention policies and transmission patterns, so an adjusting constant, β*

_{XY}_{0}(

*t*), was considered to the force of infection to account for intervention policy and transmission pattern that changed every period. Vaccination was reflected in the model using νX, and the actual amount of vaccination was reflected until the estimated simulation period of β

_{0}(

*t*) [6]. The phase dependent parameter β

_{0}(

*t*) was estimated by mini mizing the square error of the number of cumulative confirmed cases in the model (

*∫∑αI*) and the reported data. The period of estimation was from October 12, 2020, the start of social distancing phase 1 before the third wave in Korea, to May 25, 2021 [16]. The parameter,

_{X}dt*p*, represents the average vaccine effectiveness. The amount of vaccination and vaccine effectiveness of each type of vaccine administered in Korea were calculated as a weighted average and set to 0.84 [17]. The values of the model parameters are listed in Table 1, and the mortality rate by age group was calculated based on the data of individual confirmed cases who were released from isolation. The effective reproductive number R(

*t*) calculated using the next-generation method was presented in the results section [18].

### 2021 Vaccination scenarios

_{0}(

*t*) were used.

■ Scenario 1 (S1): Phase 1 is applied with the largest value of β

_{0}(*t*) during the third wave (when R(*t*)= 1.68).■ Scenario 2 (S2): Phase 1 is applied with the value of β

_{0}(*t*) during the period right before the third wave (when R(*t*)= 1.22, October 12 to November 3, 2020).

_{0}(

*t*) during the third wave both in scenarios 1 and 2 (when R(

*t*)= 0.71).

### RESULTS

### Estimation of transmission rate matrix between age groups using maximum likelihood estimation

*) is visualized in Figure 2. Figure 2A is the value of the log-scaled transmission rates, and Figure 2B is the result of calculating the risk of being infected adjusted according to the population of each age group. The entries of the matrix are listed in Supplementary Material 1. The transmission rate was estimated to be high in the diagonal components; the highest was for the group aged 18-29 years (II) at 0.79, and the second highest was for the group aged ≥ 75 years (V) at 0.78.*

_{XY}### Analysis of the effective reproductive number and the ratio of confirmed cases by age group

_{0}(

*t*), according to the government’s intervention policy. Figure 3A is the effective reproductive number, Figure 3 are the mathematical model simulation results (dark curve) and the KDCA press release data (red squares) for the number of daily and cumulative confirmed cases for all age groups, respectively. The graphs comparing the number of confirmed cases by age group and the model simulation results are shown in Figure 3D and E.

_{0}(

*t*), was estimated to be 1.09 (effective reproductive number of 1.22) between October 12 and November 3, 2020, when phase 1 of social distancing was implemented at the start time of the model, and it was estimated to be the highest at 1.50 (effective reproductive number of 1.68) between November 4 and November 24, 2020. It was estimated to be the smallest at 0.64 (effective reproductive number of 0.71) between December 23, 2020 and January 18, 2021, when the social distancing was strengthened to phase 2.5, and when the ban on private gatherings with five or more people began in the metropolitan area.

### 2021 Vaccination scenario simulation results

_{0}(

*t*) is varied according to the social distancing level. The dotted line is the simulation result of scenario 1, and the solid line is scenario 2. If there is a point when the effective reproductive number is consistently below 1, the time is marked with an asterisk. The results of vaccinating 60% and 80% of the total population and varying the vaccine effectiveness to 65%, 70%, 75%, 80%, and 90% are summarized in Supplementary Material 2.

### DISCUSSION

*t*)= 1.68), or whether it would be similar to phase 1 of distancing implemented on October 12, 2020 (S2, at the time when R(

*t*)= 1.22)). In S1, with the elderly having vaccination priority, even if 70% of the population is vaccinated during 2021, it is not until December that the effective reproductive number can start to be consistently below 1. This is not, however, conclude that priority vaccination for the elderly is not appropriate. Rather, when the elderly is vaccinated on priority, mortality number is always lower than of the scenarios when the vaccine is administered in equal proportion to the total population, excluding 588 people in the worst-case scenario (vaccination rate of 60%, scenario 1) (Table 2). Simulation results showed that if social distancing is maintained as in S2, it is possible that the effective reproductive number begins to remain consistently below 1 in August when 70% of the population is vaccinated by the end of December (Figure 5D and E solid lines).

*t*)= 1.68) can also be interpreted as one reflecting the increase in transmission caused by variants. Further studies will be conducted considering second dose vaccination, transmission rate by epidemic period, waning immunity, detailed vaccine effectiveness, and variants.