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http://www.aimspress.com/journal/Math
AIMS Mathematics, 7(11): 19922–19953.
DOI: 10.3934/math.20221091
Received: 06 July 2022
Revised: 28 August 2022
Accepted: 01 September 2022
Published: 09 September 2022
Research article
Adjusting non-pharmaceutical interventions based on hospital bed capacity
using a multi-operator differential evolution
Victoria May P. Mendoza1,2, Renier Mendoza1,2, Jongmin Lee1and Eunok Jung1,*
1Department of Mathematics, Konkuk University, Seoul, 05029, Republic of Korea
2Institute of Mathematics, University of the Philippines Diliman, Quezon City, 1101, Philippines
*Correspondence: Email: junge@konkuk.ac.kr
Abstract: Without vaccines and medicine, non-pharmaceutical interventions (NPIs) such as social
distancing, have been the main strategy in controlling the spread of COVID-19. Strict social distancing
policies may lead to heavy economic losses, while relaxed social distancing policies can threaten public
health systems. We formulate optimization problems that minimize the stringency of NPIs during
the prevaccination and vaccination phases and guarantee that cases requiring hospitalization will not
exceed the number of available hospital beds. The approach utilizes an SEIQR model that separates
mild from severe cases and includes a parameter µthat quantifies NPIs. Payoffconstraints ensure
that daily cases are decreasing at the end of the prevaccination phase and cases are minimal at the
end of the vaccination phase. Using a penalty method, the constrained minimization is transformed
into a non-convex, multi-modal unconstrained optimization problem. We solve this problem using
the improved multi-operator differential evolution, which fared well when compared with other
optimization algorithms. We apply the framework to determine optimal social distancing strategies
in the Republic of Korea given different amounts and types of antiviral drugs. The model considers
variants, booster shots, and waning of immunity. The optimal µvalues show that fast administration
of vaccines is as important as using highly effective vaccines. The initial number of infections and
daily imported cases should be kept minimum especially if the bed capacity is low. In Korea, a gradual
easing of NPIs without exceeding the bed capacity is possible if there are at least seven million antiviral
drugs and the effectiveness of the drug in reducing severity is at least 86%. Model parameters can be
adapted to a specific region or country, or other infectious diseases. The framework can be used as
a decision support tool in planning economic policies, especially in countries with limited healthcare
resources.
Keywords: COVID-19; social distancing; mathematical model; metaheuristic algorithm; improved
multi-operator differential evolution; optimal control
Mathematics Subject Classification: 34A55, 34H05, 90C26, 92-10
19923
1. Introduction
In the early outbreaks of COVID-19, many countries have banned public gatherings, closed down
schools, restaurants, land, and sea borders, and forced people to stay at home in an attempt to curb
the spread of this disease [1–3]. Few countries such as the Republic of Korea and Singapore focused
their control measures on intensive contact tracing and testing [2,4]. Without vaccines and medicine,
controlling the spread of COVID-19 relied mainly on non-pharmaceutical interventions (NPIs). These
control measures were essential to public health but have caused an immense burden on the social and
economic aspects of life [5, 6].
During the early stages of COVID-19, most countries have monitored incidence cases and deaths
and relied on this information in crafting policies [2, 3]. The development of vaccines and oral
antiviral drugs greatly impacted public health by reducing the severity of infections [7]. The fast
rollout of vaccines, use of highly effective vaccines and medicine, and implementation of strict social
distancing policies are expected to greatly minimize infections [8–10]. However, the protection given
by pharmaceutical interventions alone did not guarantee the suppression of COVID-19 as variants of
the virus continued to emerge. During the highly-transmissible omicron wave, the focus of control
shifted to managing severe cases and minimizing deaths. The timing of lifting and intensity of easing
NPIs have been important policy questions during the course of the COVID-19 pandemic. To this
end, mathematical models that incorporate various aspects of COVID-19 and NPIs are proving to be
important decision support tools [11–14].
Motivated by the events of the COVID-19 pandemic, we aim to provide a framework for planning
social distancing policies based on the number of infections requiring hospitalization. The approach
utilizes a Susceptible-Exposed-Infected-Isolated-Recovered (SEIQR) model that distinguishes mild
from severe cases and includes a parameter that quantifies NPIs. The policy period is divided into two
phases: a prevaccination phase, where only NPIs are implemented, followed by a vaccination phase,
where vaccines are assumed to be available as additional control measures. We formulate optimization
problems that minimize the stringency of NPIs during the prevaccination and vaccination phases and
guarantee that cases requiring hospitalization will not exceed the number of available hospital beds. A
measure of cost-effectiveness is presented as a basis for the choice of the frequency of policy change.
To solve the optimization problems, we transform the constrained problems into unconstrained ones
and implement a metaheuristic algorithm called the improved multi-operator differential evolution
(IMODE) [15]. We compare IMODE with 20 other global optimization algorithms to demonstrate
its effectiveness.
The presented framework can address key policy issues on the timing and level of adjusting NPIs.
It is most relevant during an early epidemic phase, wherein pharmaceutical interventions are not yet
available but managing the number of infections is a top priority. The timing of easing of NPIs is crucial
at the start of vaccination in order not to compromise the benefits of vaccines [16–18]. Since a time-
dependent epidemiological model is embedded in the method, data can be readily updated to fit the
country or region where the framework is applied. As an application, we forecast the optimal intensity
and timing of social distancing policies in Korea. An extended, more elaborate epidemiological model
is adopted that includes variants, vaccination with primary and booster shots, waning of immunity,
and administration of oral antiviral drugs [19]. Optimal solutions during the vaccination phase are
determined under different scenarios.
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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2. Materials and methods
In this section, we present the epidemiological models utilized in the optimization problems. We
formulate the optimization framework and transform the constrained into unconstrained problems via
the penalty method. We also present a measure of the cost-effectiveness of the strategies and discuss
IMODE in more detail.
2.1. Epidemiological model
The mathematical model, illustrated in Figure 1, follows an SEIQRD structure. Susceptible class
(S) can be effectively (V) or ineffectively (U) vaccinated with vaccine effectiveness e. The subscript
vdenotes vaccination and θis the number of people vaccinated per day. After an average of 1/ω
days, individuals in Vdevelop full immunity to the disease (P). Without full immunity, susceptible
individuals (S,U,V) can become exposed (E) with a force of infection λ(t), infectious (I) after 1/κ
days, confirmed and isolated in 1/α days on average. Infection can be classified as mild (Qm) or severe
(Qs). The parameter prepresents the severe rate or the proportion of infected individuals that becomes
severe. Isolated individuals can recover (R) at a rate γmor γs. A proportion fof severe cases is assumed
to die (D). We assume that the severity of symptoms is reduced by a factor of 1 −esfor the vaccinated
classes. The number of daily imported cases (ξ) is depicted by the red dashed arrow going into E.
Figure 1. The SEIQRD model used in the optimization problem. Vaccinated compartments
are denoted by the subscript v. The compartments V,U,and Prepresent the effectively
vaccinated, ineffectively vaccinated, and fully protected groups, respectively, while the
isolated mild and severe groups are denoted Qmand Qs, respectively. The other subclasses
are susceptible (S), exposed (E), infectious (I), recovered (R), and death (D).
The non-infected compartments are described by the following differential equations
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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dS
dt =−λ(t)S−θ,
dU
dt =(1 −e)θ−λ(t)U,
dV
dt =eθ−λ(t)V−ωV,
dP
dt =ωV.
(2.1)
The infected, recovered, and death compartments are given by
dE
dt =λ(t)(S+V)−κE+ξ,
dI
dt =κE−αI,
dQm
dt =(1 −p)αI−γmQm,
dQs
dt =pαI−γsQs,
dR
dt =γmQm+(1 −f)γsQs,
dD
dt =fγsQs,
dEv
dt =λ(t)U−κEv,
dIv
dt =κEv−αIv,
dQm
v
dt =(1 −(1 −es)p)αIv−γmQm
v,
dQs
v
dt =(1 −es)pαIv−γsQs
v,
dRv
dt =γmQm
v+(1 −f)γsQs
v,
dDv
dt =fγsQs
v.
(2.2)
The force of infection λ(t) is defined as
λ(t)=(1 −µ(t))R0αI+Iv
N,(2.3)
where N=S+V+U+P+E+Ev+I+Iv+R+Rv,µ(t)∈(0,1) is a time-dependent parameter representing
the intensity of NPIs, and R0is the basic reproductive number. A µvalue close to 1 translates to a high
reduction in transmission, which is assumed to be a result of strict implementation of NPIs. On the
other hand, a µvalue close to 0 implies more relaxed NPIs. The model parameters are summarized in
Table 1.
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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Table 1. Description and values of the epidemiological model parameters.
Parameter Description (unit) Value Ref.
R0Basic reproductive number 3.17 [20–22]
1/ω Mean period to develop full immunity (day) 40 [19, 23, 24]
1/κ Mean latent period (day) 4 [25, 26]
pProportion of infections that becomes severe 2.28% [27]
fMean fatality rate in severe cases 0.439 [19, 27]
eVaccine effectiveness against infection 0.91 (high) [28]
0.468 (low) [29]
esVaccine effectiveness against severe disease 0.96 (high) [30]
0.555 (low) [29]
1/γmMean duration of hospitalization for mild cases (day) 11.7 [31]
1/γsMean duration of hospitalization for severe cases (day) 14 [19, 32]
1/α Mean infectious period (day) 4 [25, 33]
The number of administered vaccines per day (θ), imported cases (ξ), periods from confirmation to
isolation (1/α), and hospitalization (γm, γs) are specific to a country’s policies and resources. In the
prevaccination phase, we simply set θto zero. If ξis zero, then screening measures are assumed to
detect all non-locally transmitted cases. Other parameters in Table 1 can be obtained from the literature.
Because vaccines have different effectiveness, we considered low and high values for eand es. The
initial susceptible, exposed E0, and infectious I0population are set to 999945, 50, and 5, respectively.
The rest of the state variables are initially set to zero. The initial total population N0is 1000000.
2.2. Formulation of the optimization problem
The stringency of NPIs is incorporated into the model as a factor in (2.3) that reduces transmission
by 1 −µ(t). In the optimization problem, we aim to determine the least value of µ(t) such that the
number of severe patients Qs(t)+Qs
v(t) throughout the policy period does not exceed the maximum
severe bed capacity Hmax. Payoffconstraints to ensure that daily cases are decreasing at the end of the
prevaccination period and there are minimal cases at the end of the vaccination period are incorporated.
A gradual easing of policies during the vaccination phase is also added as a constraint.
Suppose that the prevaccination phase is divided into n1equal periods P1(t),P2(t), . . . , Pn1(t) and
the vaccination phase into n2equal periods Pn1+1(t),Pn1+2(t), . . . , Pn1+n2(t). We assume that µ(t) is a
piecewise constant function over disjoint periods given by
µ(t)=
n1+n2
X
i=1
µiχi(t),
where χi(t) is the characteristic function of Pi(t),i=1,2,...,n1+n2. Note that each µiassumes a
constant value between 0.05 and 0.95. The goal in the prevaccination phase is to
min
[0,1]n1
1
n1
n1
X
i=1
µi(2.4)
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19927
such that
max
tQs(t;µ)<Hmax.(2.5)
The objective in (2.4) means that we want to find the set of µi(i=1,...,n1) that gives the minimum
average value. Here, Qs(t;µ) is solved from (2.1) to (2.3). Constraint (2.5) ensures that the number
of severe cases at any time during the prevaccination period does not exceed the threshold value Hmax.
Furthermore, we include a payoffconstraint which guarantees that the number of cases at the end of
the prevaccination phase is decreasing. That is,
Rt(Pend
n1)<ρ<1,(2.6)
where Pend
n1is the last time point on period Pn1,ρis a constant less than 1, and Rt(t) is the effective
reproduction number given by
Rt(t)=R0(1 −µ(t))S+U+V
N.
Hence, the optimization problem for the prevaccination phase consists of (2.1) to (2.6).
For the vaccination phase, we consider the following objective function,
min
[0,1]n2
1
n2
n1+n2
X
i=n1+1
µi(2.7)
such that
max
tQs(t;µ)+Qs
v(t;µ)<Hmax (2.8)
and
Qs(Pend
n2)+Qs
v(Pend
n2)<˜
H,(2.9)
where Pend
n2denotes the last time point on period Pn2. The payoffconstraint (2.9) ensures that the
number of severe patients by the end of the vaccination phase is less than a constant ˜
H, with ˜
H<Hmax.
To have a gradual easing of policies during the vaccination phase, we further impose that
µn1≥µn1+1≥µn1+2≥. . . ≥µn1+n2.(2.10)
This means that successive values of µiin the vaccination phase are decreasing. The full optimization
problem for the vaccination phase is given by (2.7)–(2.10) and the model (2.1)–(2.3).
To solve the constrained optimization problems, we use a penalty method and convert them into
unconstrained problems. For the prevaccination phase, we minimize over [0,1]n1the objective function
1
n1
n1
X
i=1
µi+η1max max
tQs(t;µ)−Hmax,0
| {z }
constraint (2.5)
+η2max Rt(Pend
n1)−ρ, 0
| {z }
constraint (2.6)
,(2.11)
where the penalty terms η1, η2>> 1. For the vaccination phase, the unconstrained problem is to
minimize
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19928
1
n2
n1+n2
X
i=n1+1
µi+η1max max
tQs(t;µ)+Qs
v(t;µ)−Hmax,0
| {z }
constraint (2.8)
+η3max Qs(Pend
n2)+Qs
v(Pend
n2)−˜
H,0
| {z }
constraint (2.9)
+
η4max
max
µn1+1−µn1
.
.
.
µn1+n2−µn1+n2−1
,0
| {z }
constraint (2.10)
(2.12)
over [0,1]n2, where η3, η4>> 1. Note that the penalty terms chosen are exact [34], which means that
the solution of the unconstrained problem is the same as the constrained one.
The lengths of the prevaccination and vaccination periods and frequency of policy change are
decided by the user. In the simulations, we assume that the entire policy period is 18 months (540
days), consisting of 9-month prevaccination and 9-month vaccination phases. In the prevaccination
phase, we vary the frequency of policy change, that is, we set n1=1 (uniform), n1=3 (quarterly),
n1=9 (monthly), n1=18 (biweekly), or n1=36 (weekly). We assume that the severe bed capacity
Hmax is 100, which is 0.01% of the initial total population N0, and the number of severe cases by
the end of the vaccination phase ˜
His 1% of Hmax. Moreover, vaccination is assumed to proceed at
a constant rate of θ=0.8N0/30σ, where σis the number of months it takes to vaccinate 80% of
N0. Different speeds of vaccination is considered: σ=6,9,12,or 24 months. To investigate the
effect of importation, we also perform simulations on varying ξ. Because the constraints have different
magnitudes relative to the objective functions, we chose appropriate weight constants ηi. The parameter
settings used in the simulations are summarized in Table 2. To solve (2.11) and (2.12), we implement
the metaheuristic algorithm IMODE.
Table 2. Description and values of the parameters in the optimization problem. The
parameter N0denotes the initial total population which is set to 1000000.
Parameter Description (unit) Value
Hmax Severe bed capacity 0.0001N0
˜
HThreshold on the number of severe cases by 0.01Hmax
the end of vaccination phase
θNumber of vaccines administered per day (1/day) 0.8N0
30σ,σ∈ {6,9,12,24}
ξNumber of imported cases per day (1/day) 0,10,15,20
ηi,i=1,2,3,4 Penalty weight constants 106,103,103,1010
2.3. Mathematical model of COVID-19 in Korea
As variants, booster vaccines, antiviral drugs, and waning of immunity have emerged as critical
drivers of infection, a model that captures these factors is considered. We adopt the model in [19],
which is an extension of the previously presented model to describe COVID-19 transmission in Korea.
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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The model illustrated in Figure 2 is divided into two parts: infection flow (top) and vaccination
flow (bottom). In the vaccination flow, Sdenotes the susceptible group with no vaccine- or infection-
induced immunity. The number of people administered with primary and booster vaccines per day
are denoted by θvand θb, respectively. The protected class for the pre-delta and delta variants is P2,
and against all variants is P3. Note that e1is the primary vaccine effectiveness and e∗
3=e3/e1is
a conditional probability depending on the vaccine effectiveness against the different variants. We
assume that vaccine-induced immunity of the protected groups wanes after 1/τvdays on average.
Those in the waned group Wwill go to Vbafter getting a booster shot three or four months after
completing the primary vaccines. After an average of 1/ωbdays, those in Vbmove back to the protected
groups depending on the probabilities e∗
b,j, which are computed similarly as before.
Figure 2. Flowchart of the COVID-19 model considering variants, waning of immunity,
booster vaccines, and antiviral drugs.
In Figure 2, the variable Xdenotes the class where the infection was from and it can be sus (from
Sor V), U(from U), or P(from P1,W, or Vb). The subscript jdenotes the variant and it can be 1
(infected by pre-delta), 2 (infected by delta), or 3 (infected by omicron). The reason for dividing the
infectious group is to distinguish the different characteristics of infection, e.g. effect of vaccines for
those infected, and the transmissibility and latent period of the variants. The E,I,Qm,Qs,R, and D
classes are the same as in the simple model, each having nine variations depending on the subscripts.
The probability of getting severe symptoms is assumed to be reduced by a factor (1 −es
X,j)(1 −epill)
due to the antiviral drugs. To include the waning of natural immunity depicted by the green arrow in
Figure 2, individuals in Rare assumed to have immunity only for 1/τX,ndays on average. Note that the
natural waning rate and vaccine effectiveness against severe disease are different for sus,V,and P. The
system of differential equations describing the model is in Appendix A.
Before applying the optimization framework using model (A.1), we first establish the baseline value
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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for µin Korea by fitting the model to the cumulative confirmed cases data from February 26, 2021,
when vaccination began, until February 3, 2022, when the testing method was changed. We then
forecast the number of infections given a fixed µvalue and different amounts of antiviral drugs and
compare the results to the forecasts using the optimal µvalues.
The estimation period covered four distinct social distancing phases in Korea: Social Distancing
level 2 (SD2) from February 26 until July 11, 2021, Social Distancing level 4 (SD4) from July 12
until October 31, 2021, Gradual Recovery (GR) from November 1 until December 17, 2021, and
Suspended Gradual Recovery (SGR) from December 18, 2021 to February 3, 2022. These phases do
not directly reflect the stringency of NPIs. For example, the number of people in a private gathering,
which is an important policy in controlling transmission, changed multiple times during SGR. We
divide the estimation period every two weeks and determine µon each interval by fitting the model
to the cumulative confirmed data (Y(t)) using IMODE. That is, we look for the best-fitted µvalue by
minimizing the objective functional
N
X
t=1
Y(t)−
3
X
j=1αjIsus,j(t)+αjIU,j(t)+αjIP,j(t)
2
.(2.13)
As of April 29, 2022, Korea had used 2504630 drugs (Paxlovid: 2328610, Lagevrio: 176020) and
has a remaining supply of 5775670 drugs (Paxlovid: 4942710, Lagevrio: 832960) [35]. This means
that the proportion of Paxlovid among the used antiviral drugs is 93%, among the reserve antiviral
drugs is 86%, and the total supply is 88%. We investigate future scenarios by first considering a fixed
µ=0.1, 0.3, 0.5, 0.7 and setting the number of antiviral drugs to five or seven million. Then we apply
the optimization framework to determine the optimal µwith varying amounts of antiviral drugs (five
million, six million, and seven million) and proportions of Paxlovid φ(80%, 84%, 88%, 92%, 96%,
and 100%) among the antiviral drugs used. We assume that the reduction in severity by Paxlovid and
Lagevrio are 89% [36] and 30% [37], respectively. Simulation for the forecast starts on February 3,
2022, which is the final time of estimation, and ends on December 31, 2022. The threshold value on
the number of severe cases at the end of the simulation period ˜
His set to 500.
2.4. Measure of cost-effectiveness
To determine the frequency of policy change, we adopt a measure of cost-effectiveness based on the
cost of implementation of NPIs and the number of cases averted by the intervention strategy [38]. The
cost-effectiveness ratio (CER) is the basis for choosing n1,n2=9 (monthly policy change) and is used
to compare the optimal strategies under different amounts of antiviral drugs in Korea.
The total cost of implementation of NPIs is given by C · 1
n1
n1
P
i=1
ui=C · µave, where Cis the average
cost of implementation of NPIs for the whole population for n1periods and µave is the average of all
the µiin the prevaccination phase. The CER is computed as
CER =Cµave
Ak
,(2.14)
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19931
where k=1 or 2,
A1=Ztf
t0
pαI(t;µ=0) dt −Ztf
t0
pαI(t;µ=µ?)dt
Ztf
t0
pαI(t;µ=0) dt
,
A2=
Ztf
t0X
X,j
(1 −es
X,j)αjIX(t;µ=0; ϕ=0) dt − A?
2
Ztf
t0X
X,j
(1 −es
X,j)αjIX(t;µ=0; ϕ=0) dt
,
and
A?
2=Ztf
t0X
X,j
(1 −es
X,j)(1 −epill)αjIX(t;µ=µ?, ϕ =ϕ?)dt.
The number of averted severe cases Akis computed as the relative difference between the number of
severe cases when there are no NPIs and when there are NPIs with intensity µ?. In A2,ϕdenotes
the number of antiviral drugs and ϕ?can be five, six, or seven million. We use A2in comparing the
optimal strategies in Korea depending on the supply of antiviral drugs (Figure 2 model). Meanwhile,
A1is used in comparing the strategies resulting from different frequencies of policy change (Figure 1
model). For simplicity of calculations, we set C=1 since it only appears as a factor in the CER in
(2.14) and will not affect the ranking of the different policies.
2.5. Improved multi-operator differential evolution
In Figure 3, we observe a non-convex and multi-modal surface plot of the objective function (2.11)
with n1=2. Since the surface has many local minima near the global minimum, a local optimizer is
not suitable for this problem. Evolutionary algorithms have become popular because of their capability
of obtaining global minimum and ease of use. They only use function evaluations and do not require
the derivative of the function. Applications of these algorithms have been explored in many areas of
science and engineering [39–45].
Several algorithms are continuously being developed that can obtain the global minimum with
high accuracy and low computational time [46–51]. In 2020, the Competition on Single Objective
Bound Constrained Numerical Optimization was held and the Improved multi-operator differential
evolution (IMODE) [15] ranked first in this competition. Hence, we use IMODE in solving the
minimization problems (2.11) and (2.12). Since evolutionary algorithms have been shown to be
effective in estimating parameters of biological systems [52–56], we also use IMODE in estimating
the parameters for Korea using (A.1) by minimizing (2.13).
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19932
Figure 3. Surface plot in log-scale of the objective function in (2.11) for n1=2. The
objective function is non-convex, multi-modal, and has many local minima near the global
minimum.
The inputs of IMODE are the objective function, dimension, and bound constraints. At the
beginning of the IMODE process, an initial population from the search space is generated. The
objective function using each population member is calculated and then sorted in ascending order.
IMODE divides the population into several sub-populations, which are all evolved using three mutation
operators. To preserve population diversity, archiving is done. The three operators are given as follows:
Operator 1 : vi,j=xi,j+Fi×(xφ, j−xi,j+xr1,j−xr2,j),
Operator 2 : vi,j=xi,j+Fi×(xφ, j−xi,j+xr1,j−xr3,j),
Operator 3 : vi,j=Fi×(xr1,j+(xφ, j−xr3,j)),
where r1,r2,r3,iare randomly generated integers, ~
xr1,~
xr3are randomly chosen from the population,
~
xφis selected from the top 10% members of the population, and ~
xr2is randomly selected from the
union of the archive and the population. The scaling factor Fiis assigned based on success-historical
memory [57]. Operators 1 and 2 move the member in the current population to the best points
with and without archiving, respectively [58]. Operator 3 is a weighted random-member-to-best
operator [15]. The size of each sub-population (N Po p,op =1,2,3) is iteratively adjusted based on
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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the diversity of the sub-populations and the quality of the solutions. The size of the population per
generation (NPG) is also reduced linearly [57]. After the mutation operators are carried out, crossover
is implemented randomly to create a new set of solutions. To speed up the convergence of IMODE, a
local search is implemented when the number of function evaluations exceeds 85% of the maximum
function evaluations (MAXFE S ). Figure 4 shows the flowchart of the IMODE algorithm. For a detailed
discussion of the IMODE, we refer the readers to [15].
Begin
Input:
objective function
f, dimension
D, search space
bounds Xmin,Xmax
Set G=1, ls =0.1,
maximum function
evaluations
MAXFES, initial
population
size NPG
Evaluate
f(X) and sort
G←G+1
FES ≤MAXFES ?
Update NPGand
assign the number
of solutions NPop
to each operator
Operator 2:
current-to-φbest
without archive
Operator 1:
current-to-φbest
with archive
Operator 3:
weighted-
rand-to-φbest
Generate the
new population
based on the
operators and
update the archive
Calculate
NPop based on
solutions’ quality
and diversity
Sort new
population
FES >
0.85 MAXFES
and rand <ls?
Local search from
the best point in
current population
Did the local
search improve
the solution?
Update the
best solution
ls ←0.1
ls ←0.01
Output:
best solution
End
Yes
Yes
Yes
No
No
No
Figure 4. Flowchart of IMODE algorithm.
We compare IMODE to 20 other global optimization techniques in Appendix B by solving (2.11)
for n1=3. The best, worst, mean, median, and standard deviation of the cost function values obtained
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
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from the 20 independent runs of each algorithm are computed. The hyperparameters of the algorithms
are set to their default values except for the maximum function evaluations (MAXFES ), which is set
to 10000. If the stopping criterion of the algorithm is based on the number of iterations, we fix the
population (or swarm) size (NP) to the default value and set the maximum number of iterations to
dMAXFE S /N Pe. For (2.11), (2.12), and (2.13), we use IMODE and set MAXF ES to 20000 times the
dimension of the problem. The other hyperparameters are fixed to their default values.
3. Results
3.1. Comparative analysis of performance of IMODE with other metaheuristic algorithms
Appendix B summarizes the results using various metaheuristic algorithms. The top five with the
least cost function values for each metric are written in boldface. Notably, IMODE ranked first in the
mean and median. Eight other algorithms, RACS [59], LSHADE-cnEpSin [60], EBOwithCMAR [61],
LSHADE-SPACMA [62], WFS [63], ABC [64], GBO [65], and AMO [66], have obtained 100%
feasibility rate and have metric scores of less than 1. These algorithms may also perform well in solving
the optimization problems given enough function evaluations or tuning of some hyperparameters.
3.2. Optimal solutions and cost-effectiveness analysis during the prevaccination phase
Figure 5 shows the results of the optimization problem (2.11) during the prevaccination phase for
different frequencies of policy change. Here we considered uniform (n1=1), quarterly (n1=3),
monthly (n1=9), biweekly (n1=18), or weekly (n1=36) policy changes in a span of nine months.
The black curves are the plots of the severe patients Qs(t) and the colored lines are the optimal values
of µper period.
In the uniform policy, the optimal value of µthat should be maintained throughout the
prevaccination phase is 0.630. The number of severe patients grew exponentially, almost reaching
Hmax =100 by the end of this phase. Having two constraints on a one-dimensional optimization
problem may result in an infeasible solution. For this reason, although the severe patients did not
exceed Hmax, the payoffconstraint, which ensures that incidence cases are decreasing by the end of the
period, is not satisfied.
If the policy is changed every three months, the resulting optimal values of µon the three
consecutive periods are 0.873, 0.214, and 0.708. If the policy is changed monthly, the optimal values of
µon the consecutive months are 0.363, 0.812, 0.355, 0.944, 0.778, 0.917, 0.207, 0.085, and 0.787. The
optimal solutions for the biweekly and weekly policies are illustrated in Figure 5. The average values
of µand the corresponding CER for the different policies are calculated using (2.14). The resulting
CER are 0.016, 0.017, 0.012, 0.014, and 0.031 and are shown at the bottom right panel in Figure 5. In
the succeeding simulations, we set n1,n2=9, which translates to a monthly policy change.
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Figure 5. Severe patients, optimal µvalues, and cost-effectiveness ratio (CER) for different
frequencies of policy change. The policy changes once (uniform) if n1=1, quarterly if
n1=3, monthly if n1=9, biweekly if n1=18, or weekly if n1=36. The black curves are
the severe patients Qs(t), while the colored lines correspond to the optimal values of µ. The
bottom right panel shows the CER for the different policies.
3.3. Optimal policy strategies during the vaccination phase
Figure 6 shows the optimal solutions and number of severe patients (Qs(t)+Qs
v(t)) considering high
(solid) or low (dashed) vaccine effectiveness (eand es) and speeds of vaccination depending on the
value of σ.
If vaccination proceeds at a speed equivalent to vaccinating 80% of the initial total population in
six months (red solid) using a highly effective vaccine, the optimal strategy is to keep the value of µ
at around 0.785 from the start of the vaccination phase until day 359, when the first small noticeable
reduction in µto 0.733 is observed. This value of µis reduced further to 0.05 on day 390. If the speed
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of vaccination is slowed down and vaccinating 80% of the initial total population is achieved in 9
(black solid), 12 (blue solid), or 24 months (green solid), the first observable easing of NPIs is delayed
to day 419, 449, and 479, respectively. Despite the delay, the corresponding reduction in the value of
µis greater. The value of µdecreased to 0.530 (black solid), 0.303 (blue solid), or 0.146 (green solid).
In all scenarios, the number of severe patients by day 540 is around ˜
H=1. Notably, a small peak in
severe patients is observed on day 480 when the speed of vaccination is the fastest (red solid).
Figure 6. Optimal strategies considering different values for vaccine effectiveness and speed
of vaccination. The optimal values of µper month for highly (solid) and low (dashed)
effective vaccines are shown on the top panel, while the corresponding number of severe
patients is shown on the bottom panel. The red, black, blue, and green curves correspond to
vaccinating 80% of the initial total population in 6, 9, 12, and 24 months, respectively.
When a low effective vaccine is used, the first significant reductions in NPIs are further delayed.
From around µ=0.785 at the start of the vaccination phase, the value of µnoticeably reduced to 0.145
(red dashed) or 0.687 (black dashed) on day 450. Meanwhile, if the speed of vaccination is slower,
the first noticeable reduction in µis on day 480 to 0.075 (blue dashed) or 0.403 (green dashed). In all
cases, the number of severe patients declined.
3.4. Effects of importation and initial number of infection on NPIs
On the left panel in Figure 7, we set the severe bed capacity Hmax to 50, 100, 150, and 200, and
vary the initial number of infected individuals I0. We assume that the daily imported cases ξis zero
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and E0=10 ·I0. For simplicity, we assume that there is a uniform policy (n1=1) throughout the
prevaccination phase. Results show that for Hmax =150 or 200, the optimal values of µrange from
0.61 to 0.69 as the initial number of infections varies from 0 to 500. If Hmax =100, the optimal values
of µcan climb to 0.7 or higher if I0is at least 350. If Hmax =50 and the initial number of infections is
at least 420, the maximum value for µis reached.
Figure 7. Optimal values of µconsidering imported cases, initial number of infections, and
severe bed capacity. The left panel shows the optimal value of µfor varying initial number
of infectious individuals I0and Hmax, while the right panel shows the optimal value of µfor
varying severe bed capacity Hmax and daily imported cases ξ.
On the right panel in Figure 7, we fix ξto 0, 5, 10, or 20, and vary Hmax . If Hmax is only 0.001%
of the total population, then the optimal values of µare 0.818, 0.750, 0.714, and 0.657 if ξ=20,10,5,
or 0, respectively. Meanwhile, if Hmax is increased to 0.025% of the total population, then the optimal
values of µrange from 0.689 to 0.704 as ξvaries from 0 to 20. As Hmax goes to 0.1% of the total
population, all four cases converge to an optimal µvalue of about 0.6.
3.5. Application of the framework using COVID-19 data of Korea
The estimation results for µfrom February 26, 2021, when vaccination was begun, until February 3,
2022, when the testing method changed are shown in Figure 8 and summarized in Table 3. The lowest
µvalue was 0.590 during GR, and the highest value was 0.795 during SD4. Considering the average
values in each phase, GR (0.623) was the most relaxed, followed by SD2 (0.665), SGR (0.682), and
SD4 (0.744). If we compare the length of time for the variants to reach 90% of the infections, delta
took 16 weeks while omicron took 10 weeks. In the final two weeks of the estimation period (January
19 to February 3, 2022), daily cases exceeded 30000 and µwas estimated at 0.653.
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Figure 8. Fitting results. Model simulation (black curve) and data points (circles) for the
daily (top panel) and cumulative (middle panel) cases. The bottom panel shows the fitted
values of µ(t) (blue lines) and the proportion of delta (green) and omicron (brown) variants.
Table 3. Estimated µ(t) values on each SD phase from February 26, 2021 to February 3,
2022.
Period SD Policy Average µ(t) Range of µ(t)
Feb 26 to Jul 11, 2021 SD2 (Level 2) 0.665 (0.596,0.737)
Jul 12 to Oct 31, 2021 SD4 (Level 4) 0.744 (0.653,0.795)
Nov 1 to Dec 18, 2021 GR (Gradual Recovery) 0.623 (0.590,0.653)
Dec 19, 2021 to Feb 3, 2022 SGR (Suspended GR) 0.682 (0.653,0.791)
Figure 9 shows the forecasts for the daily confirmed cases and severe patients for different values
of µand amounts of antiviral drugs from February 3 to December 31, 2022. In this simulation, the
amount of antiviral drugs is set to seven million (solid) or five million (dashed), and µis kept constant
until the end of the year. The horizontal line on the right panel depicts the severe bed capacity on
February 3, 2022 equal to 2825. Note that the drugs only affect the severe patients and not the daily
confirmed cases since the drugs are assumed to reduce the severity of infections. If µis fixed at 0.10 or
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0.30, the number of severe patients surpasses the threshold. If the amount of antiviral drugs is limited
to five million, a surge in severe infections starting from around July 2022 may occur if µ=0.10,0.30,
or 0.50. Moreover, if µ=0.10 and with five million antiviral drugs, the number of severe patients may
once more exceed the threshold. In the next simulations, we apply the framework and investigate when
and by how much should NPIs be eased given different amounts of antiviral drugs.
Figure 9. Daily cases and severe patients for different amounts of antiviral drugs and levels
of NPIs. Red, black, blue, and green curves correspond to the constant values µ=0.1,
µ=0.3, µ=0.5, and µ=0.7, respectively. The supply of antiviral drugs is set to seven
million (solid) or five million (dash-dot).
Because Korea is in the vaccination phase, the optimal µis solved by minimizing (2.12), where the
number of severe patients is obtained by solving system (A.1). We set the upper bound for µto 0.653,
which was the last estimated value of µ, and set different amounts of antiviral drugs (five, six, or seven
million). Figure 10 shows that if the amount of drugs is five million (red), the optimal values of µare
around 0.6 until July 2022, then around 0.48 until the end of the year. A rise in severe patients may
occur towards the last quarter of the year when the supplies are all used up. Meanwhile, if the amount
of drugs is six million (blue), the optimal values of µrange from 0.487 to 0.438 until around October
2022 before it is reduced to 0.365. If the supply of antiviral drugs is seven million (green), the optimal
values of µare 0.534 at the beginning, eased to 0.318 from March 2022, then reduced significantly to
0.05 from April 2022. At the end of the forecast period, the cumulative severe cases are 18494, 11152,
and 13233 for the five, six, and seven million supply of antiviral drugs, respectively. The corresponding
CER values are 0.652, 0.493, and 0.124.
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Figure 10. Forecast results using the optimal values of µgiven different amounts of antiviral
drugs. Panels on the left show the number of severe patients and optimal values of µfrom
February 3 to December 31, 2022, if the supply of antiviral drugs is five million (red), six
million (blue), or seven million (green). The bar plots show the cumulative severe cases by
the end of the forecast period and the corresponding cost-effectiveness ratio (CER) for five,
six, or seven million antiviral drugs.
Figure 11 shows the optimal policy plans and the resulting number of severe and cumulative severe
patients from February 3 until December 31, 2022, assuming different proportions φof administered
antiviral drug Paxlovid. Since the effectiveness of Paxlovid (89%) is assumed to be much higher than
Lagevrio (30%), the net effectiveness of using both antiviral drugs would vary depending on how much
of each drug is administered. For example, φ=0.8 means out of all the administered antiviral drugs,
80% is Paxlovid and 20% is Lagevrio. This results in a net effectiveness of 77%. In the forecast,
the optimal values of µon the first period, if φ=1 (green) or φ=0.96 (teal), are 0.534 or 0.617,
respectively. On the other hand, if φ=0.92,0.88,0.84 and 0.8, the optimal values of µare 0.674,
0.706, 0.728, and 0.743, respectively, which are higher than the estimated value of µ=0.653 before
February 3, 2022. At the end of the forecast period, the optimal value of µreached the minimum (0.05)
in all scenarios and the cumulative severe cases are 23861, 21815, 20100, 17922, 15656, and 13231
for φ=0.8,0.84,0.88,0.92,0.96,and 1, respectively. The corresponding CER values are 0.290, 0.272,
213, 0.181, 0.166, and 0.124.
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Figure 11. Forecast results using the optimal values of µgiven different proportions of the
antiviral drugs. Panels on the left show the number of severe patients and optimal values of
µfrom February 3 to December 31, 2022 if the proportion of Paxlovid φused among the
antiviral drugs is 80% (red), 84% (purple), 88% (black), 92% (blue), 96% (teal), or 100%
(green). The bar plots show the cumulative severe cases by the end of the forecast period and
the corresponding cost-effectiveness ratio (CER) for different values of φ.
4. Discussion
In Figure 5, the oscillating values of µ, which results in the shifts in the number of severe patients,
are observed on the quarterly, monthly, biweekly, and weekly policy changes. In the quarterly
policy change (n1=3), relaxation of NPIs is suggested to be implemented from months 3 to 6.
Correspondingly, we observe a very small number of severe cases until towards the end of the sixth
month. Under this strategy, NPIs should be intensified in the last period as the number of severe cases
increased and peaked at almost the capacity Hmax before it declined. As the frequency of policy change
is increased, there are more frequent adjustments in the intensity of NPIs, depicted by the jumps in the
values of µ. Consequently, more peaks in the number of severe patients occur. Naturally, a rise or drop
in cases follows when the policy is eased or relaxed. In all cases, Hmax is always almost reached to
allow the most relaxed level of NPIs possible, as long as the number of severe cases is kept below the
capacity. Results of the cost-effectiveness analysis show that the monthly policy change has the least
CER (0.012) and hence, the most cost-effective strategy. On the other hand, the weekly policy change
is the least cost-effective (0.031).
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In Figure 6, we see that a gradual easing of NPIs (or decreasing values of µ) is possible in all
scenarios. We observe that NPIs can be noticeably eased much earlier if the vaccines used are highly
effective. Moreover, if vaccination is slow (green curves), then strict NPIs (µ≈0.78) are maintained
longer, even if the vaccine has high or low effectiveness (easing on day 479 for high and day 480 for
low vaccine effectiveness). These emphasize the importance of not only using highly effective vaccines
but also fast administration of the vaccines to the population.
In Figure 7, results show that with higher Hmax, a lower value for the optimal µis possible. For
low values of initial infections (I0less than 150), the level of NPIs is relatively stable (µfrom 0.61 to
0.69). As I0increases and Hmax reduces, the importance of maintaining strict NPIs is more evident. For
example, if I0=500 (or 0.05% of N0) and Hmax =50 (or 0.005% of N0), the healthcare capacity can be
overwhelmed with severe cases even with the most strict level of NPIs (µ≈0.95). This approach can
serve as a guide in assessing the adequacy of severe beds given that the initial number of infections is
known. Furthermore, the impact of daily imported cases on the optimal values of µis more apparent
for smaller values of Hmax. Here, the severe bed capacity Hmax is presented as a percentage of the
total population. For example, in Germany and the USA in 2020, the number of ICU beds is more
than 0.025% of the population, while in most African and Southeast Asian countries, this percentage
is less than 0.001% [67]. In Figure 7, if the severe bed capacity is 0.001% of the initial population,
the intensity of NPIs is less if there are fewer imported cases (µ=0.657 if ξ=0, while µ=0.818 if
ξ=20). Therefore, for countries with a low number of severe beds, NPIs such as screening measures
at the border are crucial in keeping a minimal number of imported cases and preventing strict NPIs.
From February 26, 2021 to February 3, 2022, the estimated µvalues in Korea and the number of
daily confirmed cases during SD2 and SD4 were relatively stable except around July 12, 2021, when
the proportion of the delta variant among the infections increased rapidly. On November 1, 2021,
GR was implemented and the maximum number of people allowed in a private gathering increased to
eight. As a result, daily confirmed cases soared, even though the proportion of omicron among the
cases was still below 10%. As daily confirmed cases reached 7000, the government suspended the GR
policy. Since SGR initially had a stricter private gathering policy, the daily confirmed cases decreased.
However, the daily confirmed cases increased again as the proportion of the omicron variant rose to
over 50%. Despite the increasing number of cases towards the end of the estimation period, policies in
Korea kept easing.
Furthermore, we investigate the effects of antiviral drugs and the easing of NPIs in the number of
severe patients. We see in Figure 9 that for non-optimal µvalues of 0.1 or 0.3, even with seven million
antiviral drugs, the number of severe patients may surpass the severe bed capacity. Moreover, with
µ=0.5 or lower and five million antiviral drugs, a surge in severe patients is likely to happen. Results
in Figure 10 with the optimal µvalues show that in all three cases, a gradual easing of NPIs is possible
without exceeding the severe bed capacity, even during the second surge of cases when all the five
million supply of antiviral drugs are used up. We note that all the obtained optimal µvalues are less
than the estimated µvalues during SD2, SD4, GR, and GR (see Table 3). Although the cumulative
cases by the end of the year when the antiviral drugs are six million (11152) are less compared to when
the supply is seven million (13233), the CER for the seven million supply (0.124) is considerably less
than the six million supply (0.493). Finally, we see on the bottom left panel in Figure 11 that if the
proportion of administered Paxlovid is 92% or less (blue, black, purple, and red), then more strict NPIs,
greater than the previously estimated µ=0.653, should be implemented at the beginning of the forecast
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period so that the number of severe patients will not exceed the set threshold of 2825 beds. In all cases,
a relaxed level of NPIs (µ=0.05) is possible starting June 2022. A lower proportion φmay delay the
peak of severe infections but the wave is wider resulting in higher cumulative cases. For instance, if
φ=0.80, the predicted cumulative severe cases are almost double compared to when φ=1. This is
interesting because social distancing policies are more relaxed if φ=1 compared to when φ=0.8.
Thus, the use of highly effective antiviral drugs not only reduces the number of severe cases but may
also lead to earlier easing and less strict intensity of NPIs. In Korea, COVID-19 restrictions were eased
in April 2022, while the relaxation of mask mandates began in May 2022. Since 93% of antiviral drugs
used in Korea were Paxlovid, the results of the simulations support the Korean policies on the easing
of restrictions.
5. Conclusions
In this work, we have formulated nonlinear optimization problems that minimize the intensity
of NPIs (such as social distancing and mask-wearing) and ensure that the number of severe cases
will not surpass the severe bed capacity during the prevaccination and vaccination phases. An
epidemiological model that considers vaccination, reduction in transmission due to NPIs, and severe
cases is embedded in the optimization problem. The constrained optimization problem is transformed
into an unconstrained one using the penalty method. Because we use exact penalty functions,
the resulting objective function is non-differentiable, non-convex, and highly multi-modal. The
metaheuristic optimization algorithm IMODE, which is capable of obtaining the global minimum,
is used to solve the problems.
In the simulations, we observed in the prevaccination phase that more frequent policy changes result
in more oscillations in the number of severe cases and a lower average µ. However, results of the cost-
effectiveness analysis show that a monthly policy change is the most cost-effective. In the vaccination
phase, faster administration of highly effective vaccines results in the earlier easing of NPIs. If vaccine
rollout is slow (80% of the population is vaccinated in 24 months), vaccine effectiveness does not
impact the timing of easing of NPIs. Moreover, we have demonstrated that the initial number of
infected individuals and daily imported cases should be kept at a minimum value especially when the
severe bed capacity is low.
As an application, we determine optimal social distancing policy plans in Korea considering
different amounts and types of antiviral drugs. A mathematical model that includes variants, booster
vaccines, antiviral drugs, and waning of immunity was used. To establish the relationship between
social distancing policies and the values of µ, the model is first fitted to the data on confirmed cases by
minimizing a least-squares formulation using IMODE. Forecast results for the optimal µvalues show
that gradual easing of policies is possible and the number of severe patients can be maintained below
capacity even with at most five million antiviral drugs, compared to when µis kept fixed to 0.3 or
below. The easing of NPIs may occur earlier and with less intensity, if the supply of antiviral drugs is
enough and the antiviral drugs are highly effective in reducing the severity of the disease.
Results in Appendix B show that recent algorithms performed better compared to classical
algorithms when solving the proposed optimization problem. Hence, this study contributes to the
growing importance of optimization algorithms that converge faster with more accurate solutions.
Since the study of evolutionary algorithms is an active research area, we expect that new algorithms
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will be developed which can be used to solve the constrained problems. In our simulations, a constant
value for the severe bed capacity Hmax was assumed. However, the optimization problem can be
reformulated to allow Hmax to be time-dependent, as in the case when there is a surge of infections
and hospital bed capacity needs to be increased to accommodate more patients. A study that considers
a time-dependent Hmax can be done in future work. One can also consider a more general optimization
problem that allows for a non-constant duration of policies (e.g. P1is 30 days, P2is 15 days, and
so on). Furthermore, we assumed that both the prevaccination and vaccination phases span 9 months.
Depending on the capability to implement and availability of resources of a region or country, the
length of these phases can be modified, and consequently, the optimal strategies will change. In the
application using COVID-19 data of Korea, second booster shots and possible underreporting of cases
were not considered. Nevertheless, the epidemiological model can be modified to incorporate these and
other factors, and the framework is still applicable. The proposed scheme is general enough that it can
be applied to any model, regardless of complexity, as long as NPIs are quantified as a reduction in the
force of infection, and the severe case has a separate compartment. Finally, the presented framework
may be applied to other variants of COVID-19 or other infectious diseases.
Acknowledgments
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).
Conflict of interest
All authors declare no conflicts of interest in this paper.
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Supplementary
A. Equations of the COVID-19 model in the application using data of Korea
The system of differential equations describing the model are given by
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19951
dEX,j
dt =λjSX−κjEX,j,
dIX,j
dt =κjE−αjI,
dQm
X,j
dt =1−1−es
X,j1−epill αjI−γmQm,
dQs
X,j
dt =1−es
X,j1−epill αjI−γsQs,
dRX,j
dt =γmQm
X,j+(1 −f)γsQs,
dDX,j
dt =fγsQs
X,j,
dS
dt =−θv−X
j∈{1,2,3}
λjS+X
j∈{1,2,3}
τnRsus,j
dU
dt =e1θv−X
j∈{1,2,3}
λjU+X
j∈{1,2,3}
τnRU,j,
dV
dt =(1 −e1)θv−ωV−X
j∈{1,2,3}
λjV,
P2
dt =ω(1 −e∗
3)V+ωb(1 −e∗
b,3)Vb−τv,2P2−λ3P2,
P3
dt =ωe∗
3V+ωbe∗
b,3Vb−τv,3P3,
W
dt =τv,2P2+τv,3P3−qθbW+X
j∈{1,2,3}
τnRP,j,
Vb
dt =qθbW−ωbVb,
(A.1)
where
X∈ {sus,U,P},j∈ {1,2,3},
Ssus =S+V,SU=U,SP=P2+W+Vb,
λj=(1 −µ)R0,jαj
Isus,j+ηIU,j+ηIP,j
N
N=S+U+V+P2+P3+W+Vb+X
j∈{1,2,3}
X
X∈{sus,U,P}EX,j+IX,j+RX,j
.
The model parameters and their values considering the pre-delta, delta, and omicron variants are
shown in Table S1. The parameters ω,f,p, and vaccine effectiveness (e,eb,es) are from the literature
and adjusted to fit Korea (e.g. vaccines administered, age structure, etc.).
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19952
Table S1. List of parameters in the COVID-19 model for Korea.
Symbol Description (unit) pre-δ δ ø Ref.
R0Basic reproductive number 3.17 6.24 8.11 [20–22, 68, 69]
1/α Mean infectious period (day) 4 6 6 [25, 33]
1/ω Mean period to develop 40 40 40 [19,23,24]
full immunity (day)
1/κjMean latent period (day) 4 2 2 [25, 26]
pjProportion of infections 2.28% 2.28% 0.57% [27, 70]
that becomes severe
1/γmMean duration of hospitalization 11.7 11.7 11.7 [31]
for mild cases (day)
1/γsMean duration of hospitalization 14 14 14 [19,32]
for severe cases (day)
fMean fatality rate among 0.439 0.439 0.439 [19, 27]
severe cases
ejAdjusted effectiveness of 0.85 0.85 0.53 [23, 24]
primary vaccines
eb,jAdjusted effectiveness of 0.96 0.96 0.67 [23, 24]
booster vaccines
esVaccine effectiveness against 0.73 0.73 0.73 [30]
severe disease
epill Effectiveness of antiviral pills [0.30,0.89] [0.30,0.89] [0.30,0.89][36, 37]
against severe infections
τnWaning rate of infection-induced 1/120 1/120 1/120 [71–74]
immunity (1/day)
τv,jWaning rate of vaccine-induced 1/480 1/480 1/480 [75]
immunity (1/day)
ηVaccine-induced reduction in 23% 23% 4.2% [76]
transmission
E0Initial exposed population 0 1 50 Assumed
B. Application of different metaheuristic algorithms in solving the prevaccination phase
optimization problem
Table S2 summarizes the results of the 21 metaheuristic algorithms in solving (2.11) for n1=3. The
following abbreviations are used: Improved Multi-operator Differential Evolution (IMODE), Ranking-
based Adaptive Cuckoo Search (RACS), Ensemble sinusoidal differential covariance matrix adaptation
with Euclidean neighborhood (LSHADE-cnEpSin), Effective Butterfly Optimizer with Covariance
Matrix Adapted Retreat Phase (EBOwithCMAR), LSHADE with Semi-Parameter Adaptation Hybrid
with CMA-ES (LSHADE-SPACMA), Wingsuit Flying Search (WFS), Artificial Bee Colony(ABC),
Gradient-based Optimizer (GBO), Animal Migration Optimization (AMO), Ant Lion Optimizer
(ALO), Simulated Annealing (SA), Particle Swarm Optimization (PSO), Grey Wolf Optimizer (GWO),
AIMS Mathematics Volume 7, Issue 11, 19922–19953.
19953
Elephant Herding Optimization (EHO), Genetic Algorith, (GA), Mexican Axolotl Optimization
(MAO), Harris Hawks Optimizer (HHO), Generalized Pattern Search (GPS), Whale Optimization
Algorithm (WOA), Chimp Optimization Algorithm (ChOA), Bat Algorithm (BA).
Table S2. Mean, median, best, worst, and standard deviation (SD) of the cost function values
resulting from 20 independent runs of some known and recent metaheuristic algorithms. The
top five least values in each metric are written in boldface. The year when the algorithm was
published and references are also listed.
Algorithm Mean Median Best Worst SD Year Ref.
IMODE 6.09E-01 6.07E-01 5.60E-01 6.42E-01 2.13E-02 2020 [15]
RACS 6.14E-01 6.22E-01 5.59E-01 6.47E-01 2.58E-02 2022 [59]
LSHADE-cnEpSin 6.20E-01 6.34E-01 5.65E-01 6.34E-01 2.34E-02 2017 [60]
EBOwithCMAR 6.23E-01 6.29E-01 5.88E-01 6.37E-01 1.42E-02 2017 [61]
LSHADE-SPACMA 6.29E-01 6.34E-01 5.80E-01 6.38E-01 1.31E-02 2017 [62]
WFS 6.23E-01 6.30E-01 5.68E-01 6.77E-01 3.23E-02 2020 [63]
ABC 6.24E-01 6.32E-01 5.85E-01 6.44E-01 2.10E-02 2022 [64]
GBO 6.23E-01 6.35E-01 5.73E-01 6.39E-01 2.48E-02 2020 [65]
AMO 6.25E-01 6.35E-01 5.77E-01 6.50E-01 2.10E-02 2014 [66]
ALO 5.14E+01 6.60E-01 5.64E-01 5.05E+02 1.36E+02 2015 [77]
SA 5.54E+01 6.36E-01 5.64E-01 1.10E+03 2.45E+02 1996 [78]
PSO 2.65E+01 6.34E-01 5.65E-01 5.17E+02 1.16E+02 1995 [79]
GWO 6.59E+01 6.37E-01 5.68E-01 1.08E+03 2.44E+02 2014 [80]
EHO 2.20E+01 6.71E-01 6.29E-01 4.26E+02 9.52E+01 2015 [81]
GA 3.03E+02 6.73E-01 5.82E-01 1.62E+03 4.58E+02 1989 [82]
MAO 1.17E+02 7.04E-01 6.34E-01 7.34E+02 2.15E+02 2021 [83]
HHO 2.45E+02 7.10E-01 6.28E-01 2.05E+03 4.97E+02 2019 [84]
GPS 4.25E+02 1.37E+02 6.07E-01 1.81E+03 6.18E+02 2002 [85]
WOA 4.46E+02 1.70E+02 6.19E-01 1.58E+03 5.44E+02 2016 [86]
ChOA 1.14E+03 1.58E+03 5.88E-01 1.95E+03 8.14E+02 2020 [87]
BA 2.96E+03 1.07E+03 6.45E-01 1.25E+04 3.61E+03 2010 [88]
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AIMS Mathematics Volume 7, Issue 11, 19922–19953.
