Multi-Faceted Analysis of COVID-19 Epidemic in Korea Considering Omicron Variant: Mathematical Modeling-Based Study

Abstract

Background:
The most recent variant of concern, omicron (B.1.1.529), has caused numerous cases worldwide including the Republic of Korea due to its fast transmission and reduced vaccine effectiveness.

Methods:
A mathematical model considering age-structure, vaccine, antiviral drugs, and influx of the omicron variant was developed. We estimated transmission rates among age groups using maximum likelihood estimation for the age-structured model. The impact of non-pharmaceutical interventions (NPIs; in community and border), quantified by a parameter μ in the force of infection, and vaccination were examined through a multi-faceted analysis. A theory-based endemic equilibrium study was performed to find the manageable number of cases according to omicron- and healthcare-related factors.

Results:
By fitting the model to the available data, the estimated values of μ ranged from 0.31 to 0.73, representing the intensity of NPIs such as social distancing level. If μ < 0.55 and 300,000 booster shots were administered daily from February 3, 2022, the number of severe cases was forecasted to exceed the severe bed capacity. Moreover, the number of daily cases is reduced as the timing of screening measures is delayed. If screening measure was intensified as early as November 24, 2021 and the number of overseas entrant cases was contained to 1 case per 10 days, simulations showed that the daily incidence by February 3, 2022 could have been reduced by 87%. Furthermore, we found that the incidence number in mid-December 2021 exceeded the theory-driven manageable number of daily cases.

Conclusion:
NPIs, vaccination, and antiviral drugs influence the spread of omicron and number of severe cases in the Republic of Korea. Intensive and early screening measures during the emergence of a new variant is key in controlling the epidemic size. Using the endemic equilibrium of the model, a formula for the manageable daily cases depending on the severity rate and average length of hospital stay was derived so that the number of severe cases does not surpass the severe bed capacity.

Full text

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ABSTRACT
Background: The most recent variant of concern, omicron (B.1.1.529), has caused numerous
cases worldwide including the Republic of Korea due to its fast transmission and reduced
vaccine eectiveness.
Methods: A mathematical model considering age-structure, vaccine, antiviral drugs, and
inux of the omicron variant was developed. We estimated transmission rates among age
groups using maximum likelihood estimation for the age-structured model. The impact
of non-pharmaceutical interventions (NPIs; in community and border), quantied by a
parameter μ in the force of infection, and vaccination were examined through a multi-faceted
analysis. A theory-based endemic equilibrium study was performed to nd the manageable
number of cases according to omicron- and healthcare-related factors.
Results: By tting the model to the available data, the estimated values of μ ranged from
0.31 to 0.73, representing the intensity of NPIs such as social distancing level. If μ < 0.55 and
300,000 booster shots were administered daily from February 3, 2022, the number of severe
cases was forecasted to exceed the severe bed capacity. Moreover, the number of daily cases is
reduced as the timing of screening measures is delayed. If screening measure was intensied
as early as November 24, 2021 and the number of overseas entrant cases was contained to 1
case per 10 days, simulations showed that the daily incidence by February 3, 2022 could have
been reduced by 87%. Furthermore, we found that the incidence number in mid-December
2021 exceeded the theory-driven manageable number of daily cases.
Conclusion: NPIs, vaccination, and antiviral drugs inuence the spread of omicron and
number of severe cases in the Republic of Korea. Intensive and early screening measures
during the emergence of a new variant is key in controlling the epidemic size. Using the
endemic equilibrium of the model, a formula for the manageable daily cases depending on
the severity rate and average length of hospital stay was derived so that the number of severe
cases does not surpass the severe bed capacity.
Keywords: COVID-19; Mathematical Modeling; Omicron Variant, Nonpharmaceutical
Interventions; Endemic; Vaccination
J Korean Med Sci. 2022 Jul 4;37(26):e209
https://doi.org/10.3346/jkms.2022.37.e209
eISSN 1598-6357·pISSN 1011-8934
Original Article
Preventive & Social Medicine Multi-Faceted Analysis of COVID-19
Epidemic in Korea Considering
Omicron Variant: Mathematical
Modeling-Based Study
Received: Apr 7, 2022
Accepted: Jun 7, 2022
Published online: Jun 28, 2022
Address for Correspondence:
Eunok Jung, PhD
Department of Mathematics, Konkuk
University, 120 Neungdong-ro, Gwangjin-gu,
Seoul 05029, Korea.
Email: junge@konkuk.ac.kr
© 2022 The Korean Academy of Medical
Sciences.
This is an Open Access article distributed
under the terms of the Creative Commons
Attribution Non-Commercial License (https://
creativecommons.org/licenses/by-nc/4.0/)
which permits unrestricted non-commercial
use, distribution, and reproduction in any
medium, provided the original work is properly
cited.
ORCID iDs
Youngsuk Ko
https://orcid.org/0000-0001-9063-176X
Victoria May Mendoza
https://orcid.org/0000-0003-0953-9822
Renier Mendoza
https://orcid.org/0000-0003-3507-0327
Yubin Seo
https://orcid.org/0000-0001-5183-1996
Jacob Lee
https://orcid.org/0000-0002-7041-065X
Jonggul Lee
https://orcid.org/0000-0002-5771-3015
Donghyok Kwon
https://orcid.org/0000-0003-4756-6477
Eunok Jung
https://orcid.org/0000-0002-7411-3134
Youngsuk Ko ,1 Victoria May Mendoza ,1,2 Renier Mendoza ,1,2 Yubin Seo ,3
Jacob Lee ,3 Jonggul Lee ,4 Donghyok Kwon ,4 and Eunok Jung 1
1Department of Mathematics, Konkuk University, Seoul, Korea
2Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines
3
Division of Infectious Disease, Department of Internal Medicine, Kangnam Sacred Heart Hospital, Hallym
University College of Medicine, Seoul, Korea
4
Division of Public Health Emergency Response Research, Korea Disease Control and Prevention Agency,
Cheongju, Korea

Funding
This paper is supported by the Korea
National Research Foundation (NRF) grant
funded by the Korean government (MEST)
(NRF-2021M3E5E308120711). This paper
is also supported by the Korea National
Research Foundation (NRF) grant funded
by the Korean government (MEST) (NRF-
2021R1A2C100448711).
Disclosure
The authors have no potential conflicts of
interest to disclose.
Author Contributions
Conceptualization: Ko Y. Data curation: Ko
Y, Lee J,2 Kwon D. Formal analysis: Ko Y,
Mendoza R, Mendoza VM, Seo YB, Lee J,1 Lee
J,2 Kwon D, Jung E. Funding acquisition: Jung
E. Investigation: Ko Y, Mendoza R, Mendoza
VM, Jung E. Methodology: Ko Y. Software: Ko
Y. Validation: Ko Y, Mendoza R, Mendoza VM,
Seo YB. Visualization: Ko Y. Writing - original
draft: Ko Y, Mendoza R, Mendoza VM, Jung E.
Writing - review & editing: Ko Y, Mendoza R,
Mendoza VM, Seo YB, Lee J,1 Lee J,2 Kwon D,
Jung E.
Lee J,1 Jacob Lee; Lee J,2 Jonggul Lee.
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INTRODUCTION
The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) coronavirus disease 2019
(COVID-19), which originated in China at the end of 2019, has become a global public health
issue and was declared a pandemic by the World Health Organization (WHO) on March 11,
2020.1,2 In mid-November 2021, a new variant, called omicron, was detected in Gauteng
province, South Africa.3 On November 26, 2021, omicron variant was designated by the
Technical Advisory Group on SARS-CoV-2 Virus Evolution of WHO as a variant of concern
(VOC).4 In sample serums from vaccinated individuals, the neutralization of omicron variant
was much less compared to the previous variants.5 Moreover, the vaccine eectiveness of
primary dose was shown to be reduced against symptomatic omicron infections.6,7 Vaccine-
breakthrough omicron infections are higher when compared to delta.8 On the other hand,
reduced hospitalization rates and fewer severe cases are observed.9,10 Vaccination remains
a key intervention strategy as it oers protection against hospitalization.6,11-13 Furthermore,
booster shots can provide a substantial increase in protection against symptomatic
infection.6,7,14,15 The development of safe and eective oral antiviral drugs can signicantly
impact control measures for COVID-19.16 In particular, Pzer’s Paxlovid has been shown to
be 89% eective in reducing the risk of hospitalization.17
In Korea, omicron variant cases have been detected since November 2021 and later, omicron
variant has become the dominant strain, reaching over 50% in mid-January 2022 and more
than 90% among conrmed cases since February 2022.18 Omicron infections were shown to
have caused signicant local community transmission in Korea.19 Aer the omicron variant
became dominant, the number of cases increased signicantly. Average daily conrmed
cases in December 2021 (delta-dominant) and March 2022 (omicron-dominant) were
approximately 6,000 and 300,000, respectively. Since February 10, 2022 and February 21,
2022, the antiviral drug Paxlovid has been given to infected individuals over 60 years and over
40 years, respectively.20,21
Mathematical modelling has been extensively used throughout the dierent phases of the
pandemic. During the early stage of COVID-19, mathematical models were used to forecast
the number of cases in various countries.22-25 Non-pharmaceutical interventions (NPIs) such
as massive testing, contact tracing, social distancing, mobility restrictions, school closure,
mask mandate, etc., have been incorporated in models to come up with eective policies
in curbing the rise of infections.26-30 Strategies for vaccine rollout were also proposed
using mathematical models.31-34 Because variants may have dierent epidemiological
characteristics, they have been incorporated into models to capture their dynamics and
propose strategies to mitigate their spread.35,36
In our proposed mathematical model, we considered the delta and omicron variants. We
incorporated age structure, foreign entrant cases, vaccination, and antiviral drugs in the
model. The aim of this study is to quantify and analyze the impacts of NPIs, such as social
distancing and screening measures at the border, in controlling the spread of the disease.
Furthermore, we forecasted the number of daily incidence and severe infections caused by
the omicron variant in the Republic of Korea in 2022. By analyzing the endemic equilibrium
of the model, we determined the number of manageable daily cases so that the number of
severe cases will not surpass the severe bed capacity in Korea.
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METHODS
Data
Both public and non-public data that were used in this study were from the Korea Disease
Control and Prevention Agency (KDCA). Publicly available daily number of conrmed
cases and vaccine administration were aggregated from daily presses and were used in the
mathematical model simulation.37 Two types of information, symptom onset date and
diagnosis date, were aggregated from the non-publicly available, individual based data and
were used in the maximum likelihood estimation (MLE) process.
Mathematical modeling of COVID-19 considering delta and omicron variants
In this study, a deterministic mathematical model that includes age, vaccines, antiviral
drugs, and inux of the omicron variant was developed. We consider eight age groups and
two strains of COVID-19 virus, delta (
δ
) and omicron (
o
). These variants have dierent basic
reproductive numbers, transmission rates, and severe rates. Fig. 1 illustrates the ow diagram
of the mathematical model. Note that
X
indicates vaccine- or waning-related status of the
host and
i
refers to age group. There are seven vaccine- or waning-related compartments (
X
);
u
(unvaccinated),
w
(unvaccinated, previously infected, but natural immunity has waned),
v1
(two weeks before nishing primary doses),
v2
(two weeks aer nishing primary doses),
wv
(waned aer primary doses),
b
(boostered),
wb
(waned aer booster).
An unvaccinated host (
u
i
) moves to the
v
1i compartment aer administration of the rst
dose and has partial vaccine eectiveness. Two weeks aer receiving the second dose, the
https://doi.org/10.3346/jkms.2022.37.e209
Multi-Faceted Analysis of COVID-19 Using Mathematical Model
,
,
,
, ,
,
,
,
,
1−,
,
1−,
,
(1 −,)
(1 −,)
,
,
,
,
,
,
2
A
BInfection-recovery waning
(boostered;b, wb)
Infection-recovery waning
(primary dosed; v1, v2, wv)
Infection-recovery waning
(unvaccinated; u, w)
min ,
−min ,
,
−,
,
Unvaccinated Vaccinated (primary) Boostered
Vaccine effectiveness against infection
Vaccine effectiveness against severity
Fig. 1. Flow diagrams of the mathematical model of coronavirus disease 2019 in Korea. (A) Epidemiological flow diagram, where Xi represents a vaccine- or
waning-related status of a host in compartment X and age group i. Note that X can be u, w, v1, v2, wv, b, or wb and each follows this epidemiological flow. (B)
Flow diagram describing vaccination, including booster, and waning of immunity after vaccination or infection, which constitute the IN flow to and OUT flow from
each Xi in (A). The time-dependent parameters νi(t) and νib(t) are the number of primary and booster doses administered per day and are obtained from data.
The blue line in the bottom graph shows that the values used for vaccine effectiveness against severe infection are the same across all vaccinated individuals but
the vaccine effectiveness against infection (red curve) peaks after completing the primary dose and after getting a booster shot.

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host has full vaccine eectiveness (
v
2i). Later, vaccine-induced immunity wanes and so
the host moves to the
wv
i
compartment. The host goes to
b
i
compartment aer receiving
a booster shot and later to
wb
i
, considering the waning of booster shots. In this study, we
assume that the immunity against symptomatic infection wanes but the immunity against
severe infection does not. This assumption is supported by studies, where a population-
wide decline in eectiveness against infection was observed but eectiveness against
hospitalization remained high and with no signicant change over time.38,39 For both
variants, an exposed host
E
X,i
becomes infectious (
I
X,i
) and spreads the disease until case
conrmation, and so the host moves to the
Q
X,i
compartment. Aer conrmation, the
isolated host either develops mild symptoms
M
X,i
, including asymptomatic case, or severe/
critical symptoms
C
X,i
. An isolated host with mild symptoms recovers (
R
X,i
), while those with
severe symptoms may either recover or die. We assume that recovered individuals, whether
vaccinated or not, develop natural immunity which wanes over time. Since recovered
individuals retain protection against severe infection, they move to a dierent compartment
w
i
for unvaccinated,
wv
i
for primary-dosed, or
wb
i
for boostered, aer the natural immunity
has waned.39 It was demonstrated that in unvaccinated participants, the infection-acquired
immunity waned aer about 1 year but remained consistently high in previously vaccinated-
participants, even for individuals who were infected 18 months prior.40 Hence, we use a
dierent natural immunity waning rate for those who were unvaccinated (
ζ
) and vaccinated
(
ζ
v
), with
ζ
v
<
ζ
. The parameter Γi represents the number of overseas entrant cases from age
group
i
who are not screened and entered the local community. Its value is calculated using
data on average daily number of overseas entrant cases across all ages from November 24
to December 31, 2021. The detailed description of the mathematical model, including the
governing equations, can be found in the Supplementary Data 1.
Parameter estimation
Transmission rates among age groups were estimated using MLE. For MLE, we considered
two events for a host at one unit time: not being infected and being infected. Individual
based data provided by the KDCA were used for the MLE procedure to capture every infection
event (also uninfected events) of age groups. Detailed formulation is described in the
Supplementary Data 1.
To quantify the impact of NPIs, a time-dependent parameter μ is introduced to indicate
the reduction in transmission caused by NPIs. For example, ignoring other factors, if the
basic reproductive number is 2 and μ is 0.7, then the eective reproductive number becomes
(1 − 0.7) × 2 = 0.6. We estimated the value of μ every week using least squares curve tting
method, by minimizing the dierence between the cumulative incidence calculated using the
model (∫∑X∑iα(IX,iδ+IX,io))dt) and the available data. The model simulation time was done
from August 1, 2021 to February 2, 2022, because the testing policy has been changed since
February 3, 2022.41 Furthermore, we proceeded with parameter bootstrapping to examine the
reliability of the estimation and data. Detailed description of the bootstrapping method and
results are in the Supplementary Data 1. During the parameter estimation period, antiviral
drugs were provided for certain age groups.20,21 To apply impact of antiviral drugs, we simply
set that severity rate of age over 60 and 40 has reduced since January 14 and February 21,
2022, by 80%, respectively. Note that the 80% severity reduction is between the lowest and
highest values that were considered in a recent study.42 For example, delta variant’s severity
rate for the unvaccinated individuals aged 60 to 69 reduces from 7.49% to 1.50% since
January 14, 2022.
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In this research, we performed a multifaceted approach to examine critical factors which
aected the COVID-19 epidemic. First, we did a forecast considering dierent NPIs-related
factors and vaccine hesitancy. Second, we examined the time-dependent sensitivity of
screening measures to the disease spread since the omicron variant has arrived in Korea.
Finally, we derived a manageable daily incidence number from the endemic equilibrium state
of the mathematical model.
Forecast of omicron variant epidemic in 2022
For the forecast, we extended the simulation time until the end of 2022 and varied the
factors related to NPIs and booster shots. We set the range for the NPIs-related reduction
factor (µ) from 0.4 to 0.65 in 0.05 increments (six scenarios), and the maximum number
of daily booster shots as 300,000 or 100,000, to observe the inuence of vaccine hesitancy.
Furthermore, to consider the implementation of the antiviral drugs, we set that groups of age
over 60 (age over 40) have reduced severity aer February 10, 2022 (February 21, 2022).
Examination of the time-dependent impact of screening measure
We examined the impact of screening measures by varying the value of Γi by factors of 0.1 to
10, and the date of entry of omicron to the local community from November 24 to December
1, December 8, and December 22. The rest of the parameters are xed to their values on
Supplementary Tables 1 and 2.
Endemic equilibrium study
As the number of cases rapidly increases, endemicity of COVID-19 becomes an issue. We
performed an endemic equilibrium analysis to investigate how COVID-19 cases can be
maintained on a manageable level. Ignoring age structure (
i
), vaccination-related history (
X
),
and strains, and considering endemic equilibrium (assuming that there is natural death and
birth in susceptible groups, therefore endemic equilibrium can exist), ordinary dierential
equations of conrmed (
Q
) and severe patients (
C
) are:

 =  −  = 0, ∴ = ,

 = ∗ − ∗ = 0, ∗ = ∗,∴ = 1
∗∗,
where
p
* and
γ
* are average severe rate and recovery rate, respectively. Combining the two
results above, we get:
 = 1
∗∗
Because the number of severe patients should be below the severe bed capacity,
C
max
, the
following inequality is formulated:
 = 1
∗∗ < 1
∗∗
Considering that
αI
is the daily incidence and 1/
γ
* is the average length of hospital stay, then
the threshold value of the inequality, referred to as the manageable daily incidence, is given as
follows:
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   = 1
∗ × 1

× 
Equation 1
The manageable daily incidence is a function of three input parameters, average length of
hospital stay (
t
H
), average severe rate (
p
*), and severe patient capacity (
C
max
).
Ethics statement
The study was conducted according to the guidelines of the Declaration of Helsinki, and
approved by the Institutional Review Board of Konkuk University (7001355-202101-E-130).
Informed consent was submitted by all subjects when they were enrolled.
RESULTS
Estimation of transmission rates among age groups
Fig. 2 shows the transmission rate matrix,
M
1, represented as a heatmap. The maximum
value is 6.14, which is the value among age group 8. Estimated reproductive number
from
M
1 is 6.16, which is aected by NPIs but not by vaccine because reduced probability
of being infected was considered in the MLE process. To exclude the eect of NPIs, the
adjusted matrix
M
2 was introduced using the basic reproductive number of the variant
and the estimated eective reproductive number from the transmission rate matrix,
M
1
(Supplementary Data 1). The adjusted matrix
M
2 was applied into the mathematical model.
Qualification of NPIs in Korea
Fig. 3 shows that the daily and cumulative incidences from the model simulation t the data
well (Fig. 3A and B, respectively). Also, the number of administered severe patients captures
the trend well (Fig. 3C), even if the parameters related to severe patients were not tted but
aggregated from references.
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Multi-Faceted Analysis of COVID-19 Using Mathematical Model
Transmission rate (βXY)
≥ 80 (VIII)
70–79 (VII)
60–69 (VI)
50–59 (V)
40–49 (IV)
30–39 (III)
18–29 (II)
0–17 (I)
≥ 80 (VIII)
70–79 (VII)
60–69 (VI)
50–59 (V)
40–49 (IV)
30–39 (III)
18–29 (II)
0–17 (I)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
> 4.0
Fig. 2. The transmission rate matrix among age groups using maximum likelihood estimation.

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During social distancing level 4 (August to October 2021), the range of estimated µ was from
0.61 to 0.73, except near the national holiday season (Chuseok, September 20 to September
22, 2021) when µ dropped to 0.4. Since November, as the gradual recovery policy began,
µ decreased and ranged from 0.31 to 0.41, and later becomes 0.52 as suspended gradual
recovery was announced because the number of severe patients reached more than 1,100.
The obtained µ estimates and the corresponding values of the eective reproductive number
R
t
are illustrated as horizontal lines and dashed curves, respectively, in Fig. 3A. Note that
all the estimated values of µ were within the 95% condence intervals of the parameter
bootstrapping results and the details are displayed in Supplementary Fig. 1 and listed in
the Supplementary Table 3. The proportion of omicron among new cases (magenta curve)
increased from 7% to 71% from December 16, 2021 to January 16, 2022 and reached 97% by
the end of the simulation period.
https://doi.org/10.3346/jkms.2022.37.e209
Multi-Faceted Analysis of COVID-19 Using Mathematical Model
01-Aug-2021
15-Aug-2021
29-Aug-2021
12-Sep-2021
26-Sep-2021
10-Oct-2021
24-Oct-2021
07-Nov-2021
21-Nov-2021
05-Dec-2021
19-Dec-2021
02-Jan-2022
16-Jan-2022
30-Jan-2022
0
1
2
3
4
5
6
7
8
Cumulative incidence, ×105
ModelData
A
01-Aug-2021
15-Aug-2021
29-Aug-2021
12-Sep-2021
26-Sep-2021
10-Oct-2021
24-Oct-2021
07-Nov-2021
21-Nov-2021
05-Dec-2021
19-Dec-2021
02-Jan-2022
16-Jan-2022
30-Jan-2022
0
0.5
1.0
1.5
2.0
0
0.5
1.0
1.5
2.0
2.5
Daily incidence, ×104
NPIs related reduction rate, µ
Effective reproductive number
B
01-Aug-2021
15-Aug-2021
29-Aug-2021
12-Sep-2021
26-Sep-2021
10-Oct-2021
24-Oct-2021
07-Nov-2021
21-Nov-2021
05-Dec-2021
19-Dec-2021
02-Jan-2022
16-Jan-2022
30-Jan-2022
0
500
1,000
1,500
2,000
2,500
Administered severe patients
Severe patient capacity
C
0
10
5
15
20
25
30
Proportion of incidence by age, %
D
≥ 80
(VIII)
70–79
(VII)
60–69
(VI)
50–59
(V)
40–49
(IV)
30–39
(III)
18–29
(II)
0–17
(I)
Model (from Aug 1st, 2021 to Dec 31st, 2021)
Model (from Jan 1st, 2022 to Feb 3nd, 2022)
Data (same period as model)
Fig. 3. Estimation results of the qualification of NPIs. (A) Daily incidence, NPIs-related reduction factor, and reproductive number. Dark solid curve is the model
simulation and red boxes are data. Pale blue solid lines indicate NPIs-related reduction factor and dashed curve is the effective reproductive number. The
magenta curve is the proportion of omicron variant among new cases. (B) Cumulative incidence. (C) Administered severe patients. Red dashed curve indicates
the severe patient capacity of Korea. (D) Proportion of incidence by age on two different periods. The blue bar is from August 1 to December 31, 2021, and the red
bar is from January 1 to February 3, 2022. The green boxes indicate data.
NPI = non-pharmaceutical intervention.

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Age groups 1 and 2 (age under 30) showed the maximum and second maximum incidence
among age groups in both phases, August 1 to December 31, 2021 and January 1 to February
3, 2022, respectively. Age group 6 (60 to 69) had the third highest incidence number in 2021
but third lowest in 2022. Age groups 8 and 7 (age over 70) had the minimum and second
minimum incidence during the simulation period.
Forecast results of omicron epidemic in 2022
Forecast from February 3 to the end of 2022 considering various NPIs-related reduction
factor (µ) and maximum number of daily booster shot administration, showing the range of
conrmed cases and administered severe patients, are displayed in Fig. 4. If the maximum
number of booster per day is 300,000 (100,000), the peak size of incidence and administered
severe patients will range from 320,300 (420,900) to 1,409,200 (1,518,900) and 1,210 (1,530) to
5,120 (5,410) according to the value of µ which varies from 0.4 to 0.65, respectively. A secondary
wave towards the end of 2022 is observed in each scenario, and the size of the secondary peak
(incidence: less than 300,000, severe patient: less than 1,000) is smaller than the rst peak. We
display the data (red boxes) until March 13, 2022, before testing policy was changed to include
positive rapid antigen test done in an accredited facility as a conrmed case.43
Time dependent impact of screening measure
Fig. 5 shows the log-scaled simulation results using lled curves with dierent colors. Red,
green, blue, and cyan areas indicate the ranges of daily incidence for the various numbers
of overseas entrant cases (0.1Γi to 10Γi), initiated on November 24, December 1, December
8, and December 22, respectively. As the date is delayed, the ranges of incidence become
narrow. The black curve indicates the incidence when Γi is set to its baseline value and
initiated on November 24. The ratio of the maximum (minimum) to the baseline incidence
value when Γi is initiated on November 24, December 1, December 8, and December 22 are
8.64 (0.13), 2.61 (0.84), 1.36 (0.96), and 1.04 (0.99), respectively. In particular, if the impact
of screening (Γi) is varied since November 24, 2021, the range of values of the number of daily
cases by February 3, 2022, is (2,730–190,420). On the contrary, if the screening is varied since
December 22, 2021, the range of daily cases is (21,800–22,780).
Endemic equilibrium study
Using Equation 1, if the severe rate, length of hospital stay, and severe patient capacity are
5%, 28 days, and 500, respectively, which might be similar to the early stage of COVID-19 in
Korea, then the manageable number of incidence is approximately 360. Fig. 6 illustrates the
manageable daily incidence if the severe patient capacity is xed to 2,800 (Fig. 6A) or when
the average length of hospital stay is set to 7 days (Fig. 6B). Fig. 6C shows the actual incidence
data and theory-driven manageable incidence using aggregated data of each day, interpolated
data of hospital stay, severe rate, and severe patient capacity.44,45 Blue square indicates
that the manageable daily incidence of Korea in mid-February 2022 is around 600,000
assuming that severe patient capacity is 2,800, average severe rate is 0.16%, and average
length of hospital stay is 7 days. Moreover, it is visible that actual daily incidence exceeded the
theory-driven manageable incidence in December 2021, when the government declared the
suspended gradual recovery.
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DISCUSSION
Age-structured models are useful to analyze the heterogeneity of transmission patterns
according to dierent age groups and suggest age-specic policies, such as vaccine
prioritization or protocols related to school closures. To solve an age-structured model,
transmission rate matrix (or contact matrix) is required. However, obtaining a contact
matrix through survey during epidemic would be challenging. In this work, we construct the
transmission matrix using MLE. The maximum value of the estimated transmission rate for
age over 60 and under 60 were 6.22 and 4.06, respectively. Considering that approximately
300,000 of seniors are using elderly facilities, the transmission matrix shows the importance
of disease control in elderly facilities during an epidemic.46
We quantied the impact of NPIs by using µ, whose value was varied to indicate the dierent
levels of social distancing policies. The range of the value of µ is a useful guide for the
https://doi.org/10.3346/jkms.2022.37.e209
Multi-Faceted Analysis of COVID-19 Using Mathematical Model
03-Feb-2022
17-Feb-2022
03-Mar-2022
17-Mar-2022
31-Mar-2022
14-Apr-2022
28-Apr-2022
12-May-2022
26-May-2022
09-Jun-2022
07-Jul-2022
04-Aug-2022
01-Sep-2022
29-Sep-2022
23-Jun-2022
21-Jul-2022
18-Aug-2022
15-Sep-2022
13-Oct-2022
27-Oct-2022
10-Nov-2022
24-Nov-2022
08-Dec-2022
22-Dec-2022
0
5
10
15
Daily incidence, ×105
0.40, 300,000 per day
0.45, 300,000 per day
0.50, 300,000 per day
0.55, 300,000 per day
0.60, 300,000 per day
0.65, 300,000 per day
0.40, 100,000 per day
0.45, 100,000 per day
0.50, 100,000 per day
0.55, 100,000 per day
0.60, 100,000 per day
0.65, 100,000 per day
A
NPIs related reduction rate (µ), maximum booster dose administration per day
03-Feb-2022
17-Feb-2022
03-Mar-2022
17-Mar-2022
31-Mar-2022
14-Apr-2022
28-Apr-2022
12-May-2022
26-May-2022
09-Jun-2022
07-Jul-2022
04-Aug-2022
01-Sep-2022
29-Sep-2022
23-Jun-2022
21-Jul-2022
18-Aug-2022
15-Sep-2022
13-Oct-2022
27-Oct-2022
10-Nov-2022
24-Nov-2022
08-Dec-2022
22-Dec-2022
0
8,000
10,000
6,000
2,000
4,000
12,000
Administered severe patients
B
Fig. 4. Forecast results considering different intensity of NPIs and vaccine hesitancy. (A) Daily incidence. (B)
Administered severe patient. Colors of model simulation curves indicate the value of NPIs related factor (µ). The
solid and dashed curves correspond to maximum daily booster shot administration set to 300,000 and 100,000,
respectively. Blue dotted line in (B) is the expected severe patient capacity assuming that the increasing trend
continues (25.42 per day, based on historical data), while the red dotted line indicates a constant trend. Red
boxes are the data points.
NPI = non-pharmaceutical intervention.

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healthcare authorities in deciding the intensity of the intervention. Using a parameter
bootstrapping approach, we showed that the estimated impact of NPIs was within the
condence interval. Furthermore, sensitivity analysis of the
μ
i
’s and the other parameters
(latent period, infectious period, vaccine eectiveness, waning rates, impact of omicron
variant, impact of antiviral drugs) was performed. Results of the sensitivity analysis
(displayed in Supplementary Fig. 2) emphasized that the factors with most impact to the
epidemic are the NPIs (μ) and the infectious period (1/
α
). The arrival time (To) of the omicron
variant has an increasing correlation on the number of cases as time goes by. Antiviral
drugs do not aect the number of cumulative conrmed cases but over time, the eect on
cumulative severe cases becomes more apparent. We could also observe the risk of spreading
during the holiday season, with an estimated lower µ value, which is a considerable factor
for the policymakers. Strict social distancing, associated with high µ value, remains a good
control measure to minimize the size of epidemic. However, there is serious economic
burden if a strict policy is continued. Therefore, our model can be used as a guide in
determining a more relaxed policy considering changes in the number of severe bed capacity.
In Fig. 4, we observe the impact of booster shots on the number of administered severe
patients under dierent values of NPIs-related reduction factor µ. In particular, it is possible
for the initial peak of the green curve (µ = 0.6) to reach the assumed value of severe patient
capacity. If the maximum number of booster shots per day is small (dashed), indirectly
expressing vaccine hesitancy, then the green curve reached the red-dotted line, which is a
pessimistic assumption that the number of severe beds has not increased. On the other hand,
the number of administered severe patients is manageable if the number of booster shots is
large (solid). This result highlights the importance of booster shots in reducing the number
of mild and severe infections. Finally, we observe that incidence data (red boxes) follow the
green dashed curve while the administered severe patients follow the green solid curve. In
the ocial national data, critically ill patients are dened as individuals who have SpO2 < 94%
https://doi.org/10.3346/jkms.2022.37.e209
Multi-Faceted Analysis of COVID-19 Using Mathematical Model
28-Nov-2021
05-Dec-2021
12-Dec-2021
19-Dec-2021
26-Dec-2021
02-Jan-2022
09-Jan-2022
16-Jan-2022
23-Jan-2022
30-Jan-2022
104
105
Daily incidence in log-scaled
Initial time of Γ changing
24-Nov-2021
01-Dec-2021
08-Dec-2021
22-Dec-2021
Fig. 5. Examination of screening measure. Colored area indicates the range of daily incidence in log-scaled
simulation as the number and date of daily overseas entrant case is varied.

11/16https://jkms.org
on room air at sea level, a ratio of arterial partial pressure of oxygen to fraction of inspired
oxygen (PaO2/FiO2) < 300 mm Hg, a respiratory rate > 30 breaths/min, or lung inltrates >
50%. Therefore, a patient who is infected with SARS-CoV-2 and needs intensive care unit care
for a disease other than a respiratory system problem is not dened as a critically ill patient.
For this reason, data on bed use may be underestimated.
Screening measures are the primary NPIs in blocking the arrival of a new strain in the local
community. However, we found that the impact of screening measures is reduced as the
incoming strain becomes more dominant in the local community. Since strict screening
policies incur serious socio-economic costs, strengthening screening measures might have less
eect on the current situation (March 2022). Nevertheless, strengthening screening measures
would be important if there is an emerging VOC outside of the country because our results
suggest that strong screening measures can delay the new peak if they are applied early.
The derived formula (Equation 1) calculates the manageable number of daily incidence cases
using the data on severe rate, the average duration of hospital stay, and severe patient capacity.
Factors such as emergence of relatively mild variants, vaccines, and enhanced medical support
https://doi.org/10.3346/jkms.2022.37.e209
Multi-Faceted Analysis of COVID-19 Using Mathematical Model
1,000 1,500 2,000 2,500 3,000
Severe patient capacity
0.5
1.0
1.5
2.0
Average severe rate, %
B
 K
 K
 K
 K
 K
1
2
3
4
5
6
Manageable number of daily incidence, ×

Average length of hospital stay:  days
5 10 15 20
Average length of hospital stay
0.5
1.0
1.5
2.0
Average severe rate, %
A
 K
 K
 K
 K
 K
 K
1
2
3
4
5
6
Manageable number of daily incidence, ×
Severe patient capacity: ,
Jun
Jul
Jul
Aug
Aug
Aug
Sep
Sep
Oct
Oct
Nov
Nov
Dec
Dec
Jan
Jan
Feb
102
103
104
105
Daily incidence
C
Data
Theory-driven manageable number
Fig. 6. Theory-driven manageable number of daily incidence considering endemic equilibrium. (A) Color-scaled result considering varying length of hospital
stay and severe rate, with fixed severe patient capacity to 2,800. Blue square indicates the severe rate of Korea in mid-February 2022. (B) Color-scaled result
considering varying severe patient capacity and severe rate, with fixed length of hospital stay as 7 days. (C) Real daily incidence data and theory-driven
manageable number of daily incidence using real data.

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https://jkms.org
have decreased the severe rate of COVID-19 infections. Furthermore, the average duration
of hospital stay reduced signicantly, from 28 days to 7 days, since February 2020 to March
2022.45,46 The endemic equilibrium study can be useful in craing policies that ensure the
number of incidence and severe cases are within safe levels. For example, our theory-driven
model indicates that the declaration of suspended gradual recovery on mid-December 2021
might have been inevitable to control the surge in daily incidence. Data on the severe patient
capacity also showed a steep rise during this period (dashed curve in Fig. 3C).
Our mathematical-modeling-based approach is not only valid on the delta or omicron
variants of COVID-19 but can be adopted for other emerging infectious disease in the future,
or new variant of COVID-19. Because NPIs are incorporated using the parameter µ, our model
would be useful as a guide in policymaking. Analysis considering various important factors,
such as waning eects of vaccine and natural recovery, or variants, may give insights for the
disease control.
A limitation of the study is that breakthrough infection during MLE process was not
considered due to the lack of available data. On March 14, 2022, conrmation of cases was
expanded to include positive rapid antigen tests, which has a lower accuracy compared
to polymerase chain reaction test. This may lead to under-reporting, which is also not
considered in this study. In this study, we assumed that the waning of immunity decays
exponentially aer vaccination (or natural recovery). Waning rates were estimated using
vaccine eectiveness of primary and third doses, and a single value for the waning rate of
booster shot for all age groups is applied. Furthermore, we did not include the administration
of a second booster shot in the model and assumed that the protection against severe
infections does not wane. Because of these model uncertainties, we added an appendix in
the supplementary le to analyze the sensitivity of the relevant parameters with respect
to the cumulative conrmed and severe cases. The model was formulated to reect the
COVID-19 policy of the Korean government. If a policy is changed, for example, when the
self-isolation policy for conrmed individuals is lied, our model may need to be modied.
In our simulations, antiviral drugs were incorporated during the last 20 days of the parameter
estimation. A more comprehensive analysis of administering antiviral drugs to all age groups,
the eect of varying eectiveness on future scenarios, and cost-eectiveness analysis are
interesting research topics but will demand a separate study. These and the above-mentioned
limitations can be pursued in future works.
SUPPLEMENTARY MATERIALS
Supplementary Data 1
Click here to view
Supplementary Table 1
Model parameters, non-age dependent
Click here to view
https://doi.org/10.3346/jkms.2022.37.e209
Multi-Faceted Analysis of COVID-19 Using Mathematical Model

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Supplementary Table 2
Model parameters, age-and-strain dependent. Note that subscript and superscript indicate
strain and age group, respectively
Click here to view
Supplementary Table 3
Bootstrapping results showing the mean values, standard deviations, and 95% CIs of the re-
estimates of μ
Click here to view
Supplementary Fig. 1
Distribution of the bootstrapping results. Titles of the panels indicate the central date of each
estimation period. Red triangles are the estimated values.
Click here to view
Supplementary Fig. 2
PRCC values of the parameters for model outputs (A) cumulative conrmed cases and (B)
cumulative severe cases.
Click here to view
Supplementary References
Click here to view
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