Modelling the effects of social distancing, antiviral therapy, and booster shots on mitigating Omicron spread

Abstract

As the COVID-19 situation changes because of emerging variants and updated vaccines, an elaborate mathematical model is essential in crafting proactive and effective control strategies. We propose a COVID-19 mathematical model considering variants, booster shots, waning, and antiviral drugs. We quantify the effects of social distancing in the Republic of Korea by estimating the reduction in transmission induced by government policies from February 26, 2021 to February 3, 2022. Simulations show that the next epidemic peak can be estimated by investigating the effects of waning immunity. This research emphasizes that booster vaccination should be administered right before the next epidemic wave, which follows the increasing waned population. Policymakers are recommended to monitor the waning population immunity using mathematical models or other predictive methods. Moreover, our simulations considering a new variant’s transmissibility, severity, and vaccine evasion suggest intervention measures that can reduce the severity of COVID-19.

Full text

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Modelling the eects of social
distancing, antiviral therapy,
and booster shots on mitigating
Omicron spread
Jongmin Lee
1, Renier Mendoza
2, Victoria May P. Mendoza
2, Jacob Lee
3, Yubin Seo
3 &
Eunok Jung
1*
As the COVID-19 situation changes because of emerging variants and updated vaccines, an elaborate
mathematical model is essential in crafting proactive and eective control strategies. We propose
a COVID-19 mathematical model considering variants, booster shots, waning, and antiviral drugs.
We quantify the eects of social distancing in the Republic of Korea by estimating the reduction in
transmission induced by government policies from February 26, 2021 to February 3, 2022. Simulations
show that the next epidemic peak can be estimated by investigating the eects of waning immunity.
This research emphasizes that booster vaccination should be administered right before the next
epidemic wave, which follows the increasing waned population. Policymakers are recommended to
monitor the waning population immunity using mathematical models or other predictive methods.
Moreover, our simulations considering a new variant’s transmissibility, severity, and vaccine evasion
suggest intervention measures that can reduce the severity of COVID-19.
Designated by the World Health Organization as a variant of concern on November 26, 2021, Omicron has
become the dominant variant of COVID-191. Studies have shown that Omicron causes less severe infections.
However, Omicron is more transmissible2 and has a greater ability to evade immunity than Delta3. Moreover,
Omicron has a higher chance of reinfection compared to previous variants4,5.
Vaccine eectiveness and immunity induced by vaccines or a prior infection vary depending on SARS-CoV-2
variants and wane over time3,5–7. Despite these, vaccination is still an important measure in protecting the popula-
tion. Booster vaccines have been rolled out in many countries with priority given to the most vulnerable groups8.
Currently available vaccines remain eective against severe disease and death, can slow down transmission, and
may minimize the emergence of new variants9. Furthermore, non-pharmaceutical interventions (NPIs), such
as wearing masks and social distancing (SD), have been crucial in interrupting and delaying the transmission
of Omicron10.
e Republic of Korea’s COVID-19 pandemic response before vaccination started had centered on SD and
Testing/Tracing/Treatment (3T strategy)11. From February 26, 2021, vaccination rolled out with priority given
to the elderly and healthcare workers, followed by the younger age groups. A four-tier SD scheme was adopted
in July 2021 that outlined the number of people in a gathering, and operational guidelines of various facilities12.
During this time, SD was maintained at Level 2 (SD2) but was raised to Level 4 (SD4) on July 12, 2021, when the
fourth wave prompted by the Delta variant began13. By October 2021, about 75% of the population had been fully
vaccinated, and a decline in cases was observed. On November 1, 2021, the government implemented an eased
SD level as part of its ‘gradual recovery to a new normal’ (GR) policy14. A sharp rise in daily cases of COVID-19
was seen towards the end of November until early December 2021 due to the eased SD and dominance of the
Delta variant, which was detected in 96% of tested samples on December 20211. Moreover, breakthrough infec-
tions (BTI) comprised 58.2% of the cases according to the data reported on January 17, 20221. To maintain a
high population-level immunity, more than 100,000 doses of booster shots have been administered per day since
November 2021. e duration of getting a booster shot from the second dose had also been shortened from six
to three months. Cases with the Omicron variant were rst reported in Korea on December 1, 2021. On January
OPEN
1Department of Mathematics, Konkuk University, Seoul 05029, South Korea. 2Institute of Mathematics, University
of the Philippines Diliman, Quezon City 1101, Philippines. 3Division of Infectious Disease, Department of
Internal Medicine, Kangnam Sacred Heart Hospital, Hallym University College of Medicine, Seoul 07441, South
Korea. *email: junge@konkuk.ac.kr
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15, 2022, the proportion of infections with Omicron was around 26.7%1. A stronger SD policy called suspended
GR (SGR) had been reinstated on December 18, 202115. From the start of the SGR policy until February 3, 2022,
the average daily conrmed cases and severe patients were 27,442 and 274, respectively.
On January 14, 2022, the COVID-19 antiviral drug Paxlovid was rolled out in Korea, the rst Asian country
to do so aer authorizing its emergency use in December 202116. Clinical trials show that Paxlovid has 89%
eectiveness against severity17,18. From February 3, 2022, cases soared to more than 22,000 per day, which cor-
responded to approximately 10% of the testing capacity. is prompted Korea to allow Rapid Antigen Testing
done by a doctor as an ocial conrmation19. On March 16 and March 30, 2022, Korea recorded the highest
number of daily conrmed cases and severe patients with 621,317 and 1315, respectively. From February 3 to
November 30, 2022, booster vaccines have been administered to about 14 million8.
Mathematical models have been extensively used to understand the dynamics of COVID-19 in various
countries20. ey have also been utilized in proposing strategies to ease the eects of the pandemic21. Further-
more, vaccination roll-out strategies were designed using mathematical models22. Various models considering
the Delta variant and Omicron variant have been presented23. Several papers suggested COVID-19 mathematical
modeling related to machine learning and stochastic method24–28.
In this work, we propose a deterministic mathematical model that considers several relevant factors, including
variants, vaccination, waning of natural immunity and vaccine eectiveness, booster shots, and antiviral therapy.
We will estimate the parameters of the model using the available data. e objective of this study is to explore
eective and timely vaccination strategies that can be applied to future waves of COVID-19, and taking into
account the possibility of new variants. Our research ndings enable policymakers to monitor the population’s
immunity status and identify the best time to administer vaccines.
In the next section, we present the main results of the study, including the estimated parameters, calculations
for waning and breakthrough infections, prevalence ratios of non-vaccinated and vaccinated groups, simulations
on timing of booster vaccines, and projections in the event of a new COVID-19 variant. e discussion section
highlights how the time-dependent parameter
µ
describes past situations and how it can be utilized to plan future
strategies under dierent scenarios. is section also addresses the limitations of the study and oers avenues for
future research. e conclusion section summarizes how the results of the study can be utilized to strategically
plan PIs and NPIs to manage the COVID-19 pandemic. e Methods section provides details about the data col-
lection process, mathematical model formulation, parameter estimation techniques, and experimental scenarios.
Results
Fitting results. e estimation period is covered by four SD policies: SD2, SD4, GR, and SGR. e GR
policy is an eased SD, aimed at gradual returning to normal life. We estimated
µ(t)
, which quanties the NPIs
according to the SD policies of the government. Panels (a) and (b) of Fig.1 show the best t of the model to the
daily and cumulative conrmed cases data (red circles), respectively. e vertical lines mark the changes in SD
policy. e model captured the trends in the number of cases throughout the estimation period. It showed a rise
in the daily cases during the GR phase, which peaked on December 18, 2021, and a decline shortly aer SGR was
implemented. Following the model’s trend, the cases were observed to increase sharply around February 2022.
e reproductive number
R(t)
(red curve) and the estimated
µ(t)
values (black curve) are shown in Panel
(c). e symbols and dashed curves depict the data and model t on the proportion of infections with the Delta
and Omicron variants, respectively. During SD2,
R(t)
ranged between 0.9 and 1.2 except during the two weeks
(including Chuseok, the second largest holiday in Korea) before the implementation of SD4, where
R(t)
jumped
to 1.4. On the same two-week interval, the lowest value of
µ(t)=0.58
was estimated, and the proportion of
cases with the Delta variant (green curve) increased steeply. From July 12 to October 31, 2021, Delta was the
dominant variant of SARS-CoV-2 in Korea. e
R(t)
values ranged from 0.9 to 1.1. In the last 2 weeks before
the policy was changed, the estimated
µ(t)
dropped to 0.64 and consequently,
R(t)
increased to 1.1. During the
eased GR phase,
R(t)
remained greater than 1 and the mean value of
µ(t)
was 0.54. When an enhanced policy
(SGR) was reinstated on December 18, 2021, the proportion of cases with the Omicron variant was only around
4%. In this period, the estimated
µ(t)
increased to 0.64, while
R(t)
ranged from 0.8 to 1.7. By February 3, 2022,
the proportion of Omicron infections reached 98%. Supplementary A presents the mean value of the estimates
µ(t)
obtained on each SD phase.
Comparison of the model simulations with the data from February to November 2022. Fig-
ure2 shows the plots of the daily cases and severe patients using the model and vaccination data from February
3 to November 30, 2022. Here we assume decreasing
µ
values as described in the Methods section. e red dots
represent the data for the daily cases and severe patients on the same period. Simulation results show the peaks of
the daily incidence at 773,000 on March 16 and at 397,000 on August 20. e number of severe patients peaked
at 1898 on March 23 and at 686 on August 28. In the rst peak, we note the dierence between the data and the
model simulation results in the daily cases and severe patients.
Waning and breakthrough infections. Figure3 shows the changes in the proportion of the non-infected
(Panel (a)) and infected (Panel (b)) classes as more people were vaccinated and dierent SARS-CoV-2 variants
arrived in Korea. e vertical line on February 3, 2022 marks the start of the prediction period, when the test-
ing policy was changed. From July 12 to November 1, 2021, the proportion of vaccinated individuals (fully and
partially vaccinated, waned, and ineectively vaccinated) increased considerably from around 10 to 65%. is is
consistent with the data on vaccine coverage (11.8% to 75%8). In the subsequent GR phase, the waned class (light
blue) constituted more than one-h of the non-infected population. Since the administration of booster shots
started in October 2021, the proportion of fully-protected individuals in the SGR phase increased again. In the
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Figure1. Parameter-tting results for
µ(t)
from February 26, 2021 to February 3, 2022. e vertical dotted
lines depict the transition between the dierent SD policies. Panels (a,b) show the best t of the model to the
daily and cumulative conrmed cases, respectively. e red circles represent the data on daily and cumulative
cases. Panel (c) shows the estimated values of
µ(t)
(black curve), the eective reproductive number
R(t)
(red
curve), and the proportion of Delta and Omicron variants among the conrmed cases (green and blue curves,
respectively). e green and blue circles indicate the data on the proportion of Delta and Omicron infections
among tested samples, respectively.
Figure2. Comparison of the model simulations with the data from February 3 to November 30, 2022. Panel
(a,b) show the daily conrmed cases and severe patients, respectively. e data are represented by the red circles.
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prediction, simulations show that the proportion of the fully protected and recovered classes oscillates between
70 and 90%. Towards the end of November 2022, the model shows less oscillations and the proportion of the
recovered class increased to more than 80% of the non-infected classes.
Figure3 Panel (b) shows the proportion of BTI (
SU
and
SP
) and non-BTI (
SS
) by dierent SARS-CoV-2 vari-
ants. e proportion of BTI with Delta increased until the SGR phase and thereaer, the proportion of BTI with
Omicron began to rise. Assuming that the prediction scenario based on data was continued until November 30,
2022, approximately 85% of infections were BTI, and almost one-seventh of individuals infected with Omicron
were non-vaccinated.
Prevalence ratio of non-vaccinated to vaccinated groups. Figure4 panel (a) shows that during SD2
and SD4, most of the conrmed cases are from the non-vaccinated classes, while in the SGR and prediction
phases, most are from the vaccinated classes. Panels (b) and (c) show that there are more infections and severe
cases in the non-vaccinated relative to the vaccinated. Moreover, until SGR, we can see that the prevalence of
cases only reached around 350 and severe cases remained below 10. Aer February 2022, the prevalence of cases
and severe cases increased to more than 18,000 and 150, respectively, in the non-vaccinated classes. Panel (d)
shows that the prevalence of the non-vaccinated to the vaccinated among the conrmed cases during SD2 was
around 200. In the prediction phase, the prevalence was only about 0.2. Panel (e) shows that the prevalence ratio
of cases in the non-vaccinated to the vaccinated classes during SD2 is 6.4 and declined to 2.5 in the prediction
period. Panel (f) shows that the prevalence ratio of severe cases during SD2 is less than 10, while during SD4,
GR, and SGR, prevalence ratio of severe cases is around 50. In the prediction period, this reduced to 35.
Eect of additional vaccines on the timing and peak of the Omicron wave. Figure5 panels (a)
and (b) show the model prediction results when 14 million vaccines were administered for 1 month starting on
dierent dates in 2022. e red, green, magenta, and blue curves correspond to the scenario when the additional
booster shots were administered on February 3, April 1, June 1, and August 1, 2022, respectively. In the simula-
tions, we assume the value of
µ
, as discussed in the “Materials and methods” section, based on the vaccination
and severe patients data during this period. In all scenarios, the rst Omicron peak was in mid-March. e
maximum number of conrmed cases and severe patients were around 900,000 and 2000, respectively. However,
the second Omicron peaks occurred at dierent times between August and September 2022 since the booster
vaccination period varied. In panels (c) and (d), the maximum values of the peak in the second wave are much
smaller than the rst wave. Panels (e) and (f) show that proper vaccination timing can reduce the cumulative
daily conrmed and severe patients up to four million and two thousand when the booster timing is on June
2022. e peaks of the second Omicron waves, as shown in panels (g) and (h), occur around the same day (day
Figure3. e proportion of (a) non-infected and (b) infected classes from February 26, 2021 to November
30, 2022. e black dotted lines depict the transition between the dierent SD policies. e solid black line on
February 3, 2022 indicates the end of the estimation period and the start of the forecast period.
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153–160), except when the booster shots were administered for a month starting on June 1. In this case, the peak
was delayed to day 185.
Impact of a new variant to the daily incidence and severe patients. In Fig.6, we investigate the
eect of an emerging variant to the number of cases and severe patients assuming dierent characteristics such
as vaccine evasion, severity, and transmissibility. e simulation period covers the end of the prediction phase
until June 2023. Panels (a) to (c) show the mean daily conrmed cases and panels (d) to (f) show the mean num-
ber of severe patients from December 2022 to June 2023. If
µ=0
, the mean daily conrmed cases is between
110,000 and 130,000. e maximum is observed in panel (c) when
R0
is doubled. With the same properties as
Omicron but with
µ=0.3
, the mean daily conrmed cases is less than 100,000. If there are no changes in the
severity of the new variant, then the mean number of severe patients is expected to be less than 500. e mean
number of severe patients becomes over a thousand if the severity of the new variant is the same as that of Delta
even with
µ=0.3
.
Discussion
We divided the simulation period into three: estimation period (February 26, 2021–February 3, 2022), prediction
period (February 3, 2022–November 30, 2022), and projection period (November 30, 2022–June 30, 2023). In the
estimation period, we obtained the values of the SD-related parameter
µ
by tting the model to the cumulative
conrmed cases data. In the prediction period, we assumed the
µ
value based on the recovery policy and severe
cases since the conrmed cases are over the PCR testing capacity. We investigated the waning immunity of the
population, prevalence ratio, and did scenario-based simulations. In the projection period, we assumed there is
a new variant, possibly with dierent vaccine evasion, severity, and transmissibility properties compared to the
existing variants, and varied the
µ
from 0 to 0.3.
Figure4. Prevalence of COVID-19 in non-vaccinated and vaccinated classes during the dierent SD phases
and the prediction period. e red, blue, and gray bars show the prevalence of the vaccinated, non-vaccinated,
and the prevalence ratio of the non-vaccinated to the vaccinated classes, respectively. e estimation period is
from February 26, 2021 to February 2, 2022, and the prediction period is from February 3, 2022 to November
30, 2022. Panel (a) shows the mean daily conrmed cases, (b,c) prevalence of a case and severe case per one
million, and (d–f) prevalence ratio calculated as (prevalence of non-vaccinated)/(prevalence of vaccinated). e
insets show a magnication of the less visible bar graphs.
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Figure1 captured the main events of the COVID-19 epidemic in Korea. e rise in the daily conrmed
cases observed in July 2021 was inuenced by the spread of the Delta variant, which has about twice the basic
reproductive number compared to the pre-Delta variants. In response to the surge in cases, the Korean gov-
ernment implemented SD4, before the proportion of infections with the Delta variant reached 40%. Since the
dierence in
(1−µ)
was reduced from 0.37 (SD2) to 0.27 (SD4), the transmissibility of the disease during SD4
was reduced by about 27% compared to during SD2. Despite SD4, the reproductive number during this period
remained around 1 since the Delta variant is more transmissible than the pre-Delta variants, and the primary
vaccines may have already waned. Meanwhile, the reproductive number increased to greater than one during the
Korean anksgiving Day celebration and Halloween. On November 1, 2021, as the vaccination with primary
doses reached over 74% of the population and booster vaccination began, Korea announced the most eased SD,
the GR policy. e reproductive number during this phase remained more than one and the daily conrmed
cases reached over 10,000. en the Omicron variant emerged and SGR policy was implemented. Here, the
µ
Figure5. Simulation results when the additional 14 million vaccines were given at dierent times in 2022. e
red, green, magenta, or blue curves and bars correspond to the results when the booster shots were given for a
month starting on February 3, April 1, June 1, or August 1, 2022, respectively. Panels (a,b) show the daily cases
and severe patients for each scenario. e black bars represent the simulation results using the vaccination data
in Korea. Panels (c–h) illustrate the peak, cumulative, and timing of the second Omicron wave, respectively.
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value was similar to the value of
µ
in SD2. e Omicron epidemic trend diered from that of Delta because the
variants have dierent transmissibility and vaccine evasion properties.
From February 26, 2021 to February 3, 2022, there were three surges of COVID-19 infections: at the end of
SD2, GR, and SGR. e surge in SD2 and SGR can be explained by the emergence of a more transmissible vari-
ant and by vaccine evasion. For the surge in GR, we investigated the immunity state of the population. Figure3
shows the proportion of the population in the non-infected and infected classes. We observed that the waned
population increased to 20% during GR phase, which made the non-infected population more susceptible to the
disease. A similar phenomenon occurred between July and September 2022, which caused the second Omicron
wave. Since early 2022, the recovered population had increased to 80%. For the model, this meant that most
of the population do not infect other classes, there is a reduced population that can be infected, and thus, the
peak of the next wave is reduced. Meanwhile, breakthrough infections increased to more than 80% during the
Omicron wave, unlike in the Delta and pre-Delta periods.
In panels (a) and (d) of Fig.4, the mean daily conrmed cases among the vaccinated is about ve times more
than the non-vaccinated during the prediction phase. is gap is reversed during SD2, where prevalence was
six times greater in the non-vaccinated compared to the vaccinated. Meanwhile, the number of severe patients
is directly related to the medical capacity. In panel (f), the prevalence of severe patients in the SGR period is 50
times more in the non-vaccinated than in the vaccinated. ese results illustrate that although the number of
cases among the vaccinated is much larger than the non-vaccinated, vaccines eectively prevent infection and
severe symptoms. At the end of the SGR period, Korea approved the use of an antiviral drug for COVID-19.
Hence, in the prediction period, we observe that the prevalence ratio reduced to 30. e question we now ask is,
when is the most favorable timing to give the vaccines to the population?
Figure5 depicts the projected daily cases and severe patients under dierent timing of administration of
booster vaccines. All scenarios show a second Omicron wave between August and November 2022. e gray
bars represent the data. e main objective of nding the appropriate booster timing is to delay the peak or
reduce its size. Booster administration between April and June results to a high peak size in severe patients as
shown in panel (d). Giving booster shots aer September is inappropriate since the second omicron peak already
occurred before September.
e smallest peak size in daily conrmed cases and severe patients is obtained when the booster shots are
administered starting on August 1, 2022 (blue). However, under this scenario, the peak of the second Omicron
wave is expected to occur 153 days aer the rst Omicron peak. e peak can be delayed the most when booster
administration is started on June 1, 2022 (magenta), which is two months before the next epidemic wave is
expected to occur assuming that no booster shots were given to the population. is scenario has the lowest
cumulative conrmed cases and the lowest cumulative severe patients, but the peak size could reach 430,000.
Considering the size and date of occurrence of the second Omicron peak, the most favorable time to administer
Figure6. Simulation results characterizing new variants given varying vaccine evasiveness, disease severity,
and
R0
from December 2022 to June 2023 and
µ
∈
(0, 0.3)
. Vaccine evasion ratio of 1 means that the vaccine
eectiveness against the variant is the same as the original Omicron variant, while a ratio of 2 means that
vaccines are 27% eective against the new variant. e severity and
R0
ratios indicate the ratio of severity rates
and transmissibility of the new variant compared to the original Omicron variant. Panels (a–c) show the mean
daily incidence, and panels (d–f) show the mean severe patients. e red dotted curves indicate the contours of
the heatmap.
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the booster shots is between June and August 2022. ese two scenarios have lower cumulative daily cases and
severe patients compared to the data (gray). is strategy supports the vaccination plan of Denmark in 202129.
e main reason for the surge in infections is the emergence of new variants with high transmissibility or
vaccine evasion. Figure6 illustrates scenarios considering a new variant that can have dierent properties, as
in the case of 1H-2023 variant of Omicron. Without any variants and NPIs, Korea suered about 110,000 cases
per day and 300 severe patients in a day when 1H-2023 emerged. Transmissibility-related parameters cannot
increase the severe cases up to 500 without NPIs. e severity-related parameter increases the number of severe
patients by more than a thousand, even with the NPIs. is means that strategies which can reduce severity, such
as the use of antiviral drugs or vaccines targeting the specic variant, are more important than maintaining NPIs.
ere are several limitations to this study. Since we did not consider any inhomogeneity like regional or
population structure, we cannot realize the dierences in contact and characteristics of each group. So our study
does not consider age-related dierences in disease severity and prioritization in vaccination. Nevertheless, this
study investigates the infection dynamics by various vaccines and variants. With the results, we emphasize the
impact of vaccination, the importance of vaccination timing, and the strategies to minimize the impact of future
variants. Unreported cases are another limitation of this research since during the rst Omicron epidemic, the
number of conrmed cases was greater compared to the daily PCR testing. For this reason, we used the severe
data from the February 3, 2022 and divided the estimation and prediction periods.
Our future studies will be about intervention strategies considering heterogeneity in the population or region.
We also plan to include unreported cases and study their eect in the epidemic situation. In formulating
µ
, we
assumed that the parameter values change every 2 weeks. One can explore using
µ
values on a weekly interval.
However, this will double the number of unknown parameters and signicantly increase the computational cost.
is may require the use of dierent optimization algorithms (like metaheuristic methods) since local search
methods (using the Matlab built-in function lsqcurvet) may not work for high-dimensional problems. We
also plan to develop a dashboard based on a mathematical model which dierentiates between those who had
COVID-19 and those who had not. In the dashboard, users can change the properties of the new variants and
observe what may happen in the future. It can be useful for citizens, governments, and policymakers.
Conclusion
In this research, we propose a mathematical model considering NPIs and PIs. Results suggest that vaccination
is benecial for citizens to protect themselves from severe symptoms of COVID-19. Moreover, vaccination
campaigns should be implemented just before the expected surge in cases driven by the waning immunity of the
population. Policymakers are advised to use mathematical modeling or other predictive methods in monitoring
the waning of immunity. To address the potential threat of new variants, reducing the severity rate of the variants
is a critical measure to prevent overwhelming the medical system.
Materials and methods
Data. From February 26, 2021 to December 31, 2022, we compiled the data on conrmed cases, severe cases,
primary vaccinations, and booster vaccinations8. In the parameter estimation period (February 26, 2021 to Feb-
ruary 3, 2022), we utilized the data on conrmed cases, primary and booster vaccinations. For the prediction
period (February 3, 2022 to December 31, 2022), we used the primary and booster vaccination data. Severe
patients data were used to compare with the simulation results of the model. To calculate the eectiveness of the
vaccines administered in Korea, we considered the proportion of the population who were administered with a
certain vaccine type30. e data on the proportion of Delta and Omicron variants were obtained from Korea Dis-
ease Control and Prevention Agency (KDCA)31,32 and compared with the model simulations as shown in Fig.1.
Mathematical model. We use a deterministic SEIQR model to describe the transmission of COVID-19
considering the variants, vaccination, waning, and booster shots. e model diagram in Fig.7 shows that once
susceptible individuals (S) are vaccinated, they are categorized into eectively (V) or ineectively (U) vaccinated
groups. e parameter
θv
denotes the number of primary doses of vaccines administered daily, according to
available data. e parameter
ev
j
represents the adjusted eectiveness of primary vaccines. Here,
j∈{1, 2, 3}
denote infection with pre-Delta, Delta, and Omicron, respectively. We assume that individuals in U cannot get
immunity against infection but have immunity against getting severe from vaccines. Aer an average of
1/ω
days
from the primary vaccination, individuals in V become protected (
Pj
) against pre-Delta, Delta, or Omicron vari-
ants. We assume that individuals protected from Omicron are also protected from the other variants, and indi-
viduals protected from Delta are also protected from the pre-Delta variants. For example, individuals in
P1
may
be infected by Delta or Omicron variants. As the immunity provided by vaccines wanes over time, the protected
individuals (
Pj
) move to the waned W at a rate
τv
j
. When individuals in W receive booster shots, they move to
the
Vb
compartment and become protected again aer an average of
1/ω
days. Similarly, according to the data,
θb
represents the number of daily administered booster shots. e parameters
eb
j
denote the adjusted eectiveness
of booster doses. Vaccine-induced immunity is assumed to wane exponentially. e calculated values of
ev
j
,
eb
j
,
τv
j
, and f are shown in Table1 and the details of the calculation are given in Supplementary C.
Individuals in the susceptible and vaccinated compartments may be exposed (
EX
j
) to any variants with forces
of infection
j(t)
. Here,
X∈{S,U,P}
where S; U; and P are infected from S, V; U;
P2
,
P1
,
Vb
, and W, respectively.
e parameter
j(t)
is dened as follows:
(1)

j(t)=(1−µ(t))R0,jαj
�
X
I
X,j
(t)
N.
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e expression
(1−µ)
in
j(t)
accounts for the reduction in transmission induced by NPIs such as social
distancing. e basic reproduction numbers of pre-Delta, Delta, and Omicron are denoted as
R0,1,R0,2
, and
R0,3
, respectively. For brevity, we omit (t) notation on each compartment and parameter. All the compartments
are time-dependent, and
θv
,
θb
,
j
,
µ
are time-dependent parameters. Exposed individuals become infectious
(
IX
,
j
) aer
1/κj
days. Once they were identied by contract tracing or symptom onset, it will take about
1/αj
days to be conrmed and distinguishedas mild quarantined (
Qm
X
,
j
) or severe quarantined (
Qs
X
,
j
) cases according
Figure7. e model owchart depicts the transmission of COVID-19 considering dierent variants,
vaccinations, waning, and booster shots. e large round box group the compartments of non-infected people
with dierent vaccine statuses. e compartments enclosed by red, green, and blue round boxes indicate
individuals who can be infected by any variant, only Delta and Omicron, and only Omicron, respectively. e
Pj
compartments represent the vaccinated individuals who are protected from the variants. e compartments
inside the gray box denote infected individuals who are in one of several stages of infection, such as latent,
infectious, isolated, recovered, or deceased.
Table 1. e symbols and values used in the mathematical model. e subscripts
j
=
1, 2, 3
refer to infection
with pre-Delta, Delta, and Omicron variants, respectively. *Assumed value.
†
Calculated value).
Symbol Description (units) pre-
δ
δ
oRef.
R
0,
j
Basic reproduction number 2.87 5.08 9.5 2,33,34
1/κj
Mean latent period (days) 4 2 2 35,36
1/α
Mean infectious period (days) 6 4 4 36,37
pj
Proportion of severe cases 2.28% 3.39%
0.68%∗
4,8,38
fMean fatality rate among severe cases
60.7%†
60.7%†
60.7%†
8
1/γ m
Mean duration of hospitalization for mild cases (days) 11.7 11.7 7 39,40
1/γ s
Mean duration of hospitalization for severe cases (days)
11∗
11∗
11∗
40
ev
j
Adjusted eectiveness of primary vaccines
91%†
90%†
63%†
3,6,30
eb
j
Adjusted eectiveness of booster vaccines –
95%†
66%†
3,6,30
es
Vaccine eectiveness against severe infections
97%
93%
93%
3,6
1/τ n
Mean waning rate of infection-induced immunity for non-vaccinated (days) 350 350 350 7
1/τ n,v
Mean waning rate of infection-induced immunity for vaccinated (days) 480 480 480 5
1/τ v
j
Mean waning rate of vaccine-induced immunity (days)
349†
349†
138†
3,6,30
1/ω
Mean duration to have immune aer vaccines (days) 14 14 14 3,6
1/edrug
Eectiveness of antiviral against severity 89% 89% 89% 41
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to severe rate (
pj
). Severity can be reduced by vaccines (
es
) and antiviral drugs (
edrug
), which are multiplied to
the severity rate. An isolated individual either recovers (
RX
,
j
) aer
1/γ m
j
or
1/γ s
days on average, or dies (
DX
,
j
)
from the disease with fatality rate f. Individuals who recover from the disease develop natural immunity against
COVID-19 until their immunity wanes. In this study, we set the natural recovery waning period for the non-
vaccinated (
1/τ n
) as 350 days and for the vaccinated (
1/τ n,v
) as 480 days5,7. e model parameters are listed in
Table1 and the model equations are in Supplementary B.
Parameter estimation. e time-dependent parameter
µ(t)
is tted by minimizing the square error of the
cumulative number of cases using the model
3

j=
1
αj(IS,j+IU,j+IP,j
)
and the data from February 26, 2021 to
February 3, 20228. We assume that the value of
µ(t)
changes every two weeks. e least-squares formulation is
solved using the MATLAB built-in function lsqcurvet. e initial values of individuals exposed to Delta
and Omicron variants when the rst cases of each variant are conrmed are set to 1. We set the date of rst con-
rmed case of the Delta and Omicron variants to April 27, 2021 and November 24, 2021, respectively42–44.
e eective reproductive number
R(t)
is calculated using next-generation method45,46. We obtain
Bootstrapping. Because the estimation process has some uncertainty caused by oscillatory data or the esti-
mation method, we performed an uncertainty analysis by resampling the cumulative conrmed data using a
Poisson distribution for each data point. is process, called bootstrapping, is a method used for uncertainty
analysis to obtain statistical information about the estimated parameters such as standard deviation or con-
dence interval47. e algorithm starts with the estimated model results from the data. We then apply perturba-
tions to the model simulation results obtained from data tting to obtain a certain number of simulated datasets.
In this study, we re-estimated the parameters from 1000 dierent datasets that we generated using the Poisson
error of each simulation point. As the model was tted to the cumulative conrmed data, we generated cumula-
tive conrmed datasets from the model simulation. Supplementary A shows the distribution of the re-estimated
parameters. e mean value, standard deviation, and 95% condence interval of each
µ
value are also given.
Most of the condence intervals are less than 1.0e−3.
Modelling scenarios. We divided the simulation period into estimation, prediction, and projection peri-
ods. e modeling starts on February 26, 2021, the date when vaccination started. e estimation period includes
the data-tting process to quantify the social distancing level
µ(t)
and proportion of the variants. From February
3, 2022, we set the
µ
value based on the gradual easing of policies and used the vaccination data until November
30, 2022. e last estimated
µ
value was kept until March, then the
µ
value is set to 0.5 from April to July 2022,
when the SD levels were lied. e value of
µ
was decreased every month from August 2022 when outdoor mask
writing was lied. Figure2 shows the model simulations under these assumptions on
µ
.
Next, we investigate the proportion of the population of non-infected and infected people, and the prevalence
of the disease in the non-vaccinated and vaccinated groups. Based on the model, the non-infected population can
be divided into ve groups: fully protected against all the variants (
P3
), partially protected against the variants
(
P1,P2
), vaccine waned population (
W,Vb
), ineectively vaccinated (U), and unprotected (S,V). e infected
groups are divided according to the variant and whether it is a BTI (subscript U, P) or non-BTI (subscript S).
We look at the dynamics of the proportions of the non-infected and infected groups as more individuals got
vaccinated with primary and booster doses, and as the variants emerged. Results shown in Fig.5 uses the same
assumption on SD as in the previous scenario, but a dierent assumption about vaccination. e simulation with
dierent vaccination timing was performed to investigate the importance of timely vaccination.
e projection period is from November 30, 2022 to June 30, 2023. Here, we examine the impact of vaccine
evasion, disease severity, and Omicron’s transmissibility relative to Delta. We are interested in the mean number
of daily cases and severe patients under dierent assumptions on the value of
µ
. We vary the values for vaccine
evasion from
eb
3
to
eb
3
−
(eb
2
−
eb
3)
. e range of severity is from
p3
to
p2
. We assume that the reproductive number
of the new variant could be twice that of the Omicron variant.
Data availability
e study uses the soware MATLAB 2022b. All the data used in this study is available in the references8,30–32.
(2)
R
(t)=R0,1(1−µ(t))
S+V+U+W+V
b
N
I
S,1
+I
U,1
+I
P,1
3

j=1
(IS,j+IU,j+IP,j)
+R0,2(1−µ(t)) S+V+U+W+Vb+P
N
IS,2 +IU,2 +IP,2
3

j=1
(IS,j+IU,j+IP,j)
+R0,3(1−µ(t)) S+V+U+W+Vb+P+Pδ
N
IS,3 +IU,3 +IP,3
3

j
=1
(IS,j+IU,j+IP,j)
.
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Received: 3 February 2022; Accepted: 25 April 2023
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Acknowledgements
is paper is supported by the Korea National Research Foundation (NRF) Grant funded by the Korean govern-
ment (MEST) (NRF-2021M3E5E308120711). is paper is also supported by the Korea National Research Foun-
dation (NRF) Grant funded by the Korean government (MEST) (NRF-2021R1A2C100448711). is research
was supported by a fund (2022-03-008) by Research of Korea Disease Control and Prevention Agency (KDCA).
Author contributions
Conceptualization: J.L., R.M., V.M.P.M., J.L., Y.S., E.J.; Interpretation of data: J.L.; Formal analysis: J.L.; Funding
acquisition: E.J.; Methodology: Jongmin L., R.M., V.M.P.M., E.J.; Project administration: J.L., Renier M., V.M.P.M.,
E.J.; Visualization: J.L., R.M., V.M.P.M., E.J.; Writing—original dra: J.L., Renier M., V.M.P.M., J.L., Y.S., E.J.;
Writing—review & editing: J.L., R.M., V.M.P.M., E.J.
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at https:// doi. org/
10. 1038/ s41598- 023- 34121-y.
Correspondence and requests for materials should be addressed to E.J.
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