Managing bed capacity and timing of interventions: a COVID-19 model considering behavior and underreporting

Abstract

We develop a mathematical model considering behavioral changes and underreporting to describe the first major COVID-19 wave in Metro Manila, Philippines. Key parameters are fitted to the cumulative cases in the capital from March to September 2020. A bi-objective optimization problem is formulated that allows for the easing of restrictions at an earlier time and minimizes the number of additional beds ensuring sufficient capacity in healthcare facilities. The well-posedness of the model and stability of the disease-free equilibria are established. Simulations show that if the behavior was changed one to four weeks earlier before the easing of restrictions, cumulative cases can be reduced by up to 55% and the peak delayed by up to four weeks. If reporting is increased threefold in the first three months of the estimation period, cumulative cases can be reduced by 61% by September 2020. Among the Pareto optimal solutions, the peak of cases is lowest if strict restrictions were eased on May 20, 2020 and with at least 56 additional beds per day.

Full text

http://www.aimspress.com/journal/Math
AIMS Mathematics, 8(1): 2201–2225.
DOI: 10.3934/math.2023114
Received: 06 August 2022
Revised: 06 October 2022
Accepted: 20 October 2022
Published: 31 October 2022
Research article
Managing bed capacity and timing of interventions: a COVID-19 model
considering behavior and underreporting
Victoria May P. Mendoza1,2, Renier Mendoza1,2, Youngsuk Ko1, Jongmin Lee1and Eunok
Jung1,*
1Department of Mathematics, Konkuk University, Seoul 05029, Korea
2Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines
*Correspondence: Email: junge@konkuk.ac.kr.
Abstract: We develop a mathematical model considering behavioral changes and underreporting to
describe the first major COVID-19 wave in Metro Manila, Philippines. Key parameters are fitted to the
cumulative cases in the capital from March to September 2020. A bi-objective optimization problem
is formulated that allows for the easing of restrictions at an earlier time and minimizes the number of
additional beds ensuring sufficient capacity in healthcare facilities. The well-posedness of the model
and stability of the disease-free equilibria are established. Simulations show that if the behavior was
changed one to four weeks earlier before the easing of restrictions, cumulative cases can be reduced
by up to 55% and the peak delayed by up to four weeks. If reporting is increased threefold in the first
three months of the estimation period, cumulative cases can be reduced by 61% by September 2020.
Among the Pareto optimal solutions, the peak of cases is lowest if strict restrictions were eased on May
20, 2020 and with at least 56 additional beds per day.
Keywords: COVID-19; mathematical model; behavior change; underreporting; Metro Manila;
Philippines; community quarantine; bi-objective optimization
Mathematics Subject Classification: 65K05, 90C26, 92-10, 92D30
1. Introduction
Shortly after the first local transmissions of the coronavirus disease 2019 (COVID-19) in the
Philippines were confirmed, the government imposed social distancing policies, termed community
quarantines, which were largely implemented by the police and military [1, 2]. By March 30, 2020,
the country only had six laboratories that accommodated up to 1000 tests daily [3]. Contact tracing
began slowly due to an insufficient number of contact tracers [4]. Testing capacity was increased to
about 35000 tests daily by the end of September 2020 [5]. From April to September 2020, the weekly

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positivity rate ranged between 4.5 and 28%, higher than the 5% threshold set by the World Health
Organization during this time [6,7].
The Philippines’ capital region, Metro Manila, comprised around 12% of the country’s population
in 2020 [8]. Metro Manila was placed under Enhanced Community Quarantine (ECQ) on March 16,
2020 [9]. Under ECQ, movement was restricted to essential goods and services. Public transportation
and mass gatherings were suspended [2]. People were encouraged to work from home and businesses
were advised to do transactions online [1]. After two months, ECQ was replaced by the Modified
Enhanced Community Quarantine (MECQ), a transition phase before easing further to General
Community Quarantine (GCQ). On June 1, 2020, Metro Manila was placed under GCQ, wherein
public transportation and other establishments, except those for leisure, were allowed to operate [2].
A surge in the number of cases occurred from July to August 2020 and consequently, Metro Manila
was again placed under MECQ. During this time, the utilization of ICU, isolation, and ward beds in
Metro Manila reached 77%, 74%, and 84%, respectively, placing most facilities on critical or
high-risk, and prompting 80 medical societies, representing 80000 doctors and a million nurses, to
demand a ‘timeout’ [10, 11]. On August 19, 2020, Metro Manila returned to GCQ [12]. By the end of
September 2020, 53% of the 309303 total confirmed cases in the Philippines belonged to Metro
Manila [5].
Because of the lack of vaccines and limited antiviral therapies during the early phase of the COVID-
19 pandemic, NPIs such as wearing masks, school and workplace closures, and travel restrictions were
crucial disease control measures. In the Philippines, compliance with policies was not only prompted
by public health campaigns, but also driven by uncertainty and anxiety about the disease, and fear
of getting reprimanded by the authorities [13–15]. Some of those who got infected suffered stigma
and were blamed for not following the protocols [16, 17]. A study among low-income households in
the Philippines done in the early phase of the pandemic reported that 66% of respondents who might
experience symptoms considered staying at home instead of seeking medical attention [13].
Mathematical modeling has been extensively used to understand the dynamics of COVID-19
transmission [18–24]. Non-pharmaceutical interventions, behavior change, and underreporting of
cases have been incorporated into mathematical models of COVID-19 [25–32]. In this study, we aim
to investigate the effects of reporting and behavior on the timing and magnitude of the peak of
COVID-19 infections in Metro Manila from March to September 2020. We extend the SEIQR model
by Kim et al., which includes a compartment for behavior-changed susceptible individuals [28], by
adding an unreported compartment to account for individuals who were undetected due to inadequate
testing and tracing, or unwillingness to be detected. We provide a detailed mathematical analysis of
the model showing the existence, nonnegativity, boundedness of solutions, computation of the
threshold parameter, and two forms of disease-free equilibria (DFEs). We show the global stability of
one of the DFEs by utilizing a suitable Lyapunov function and analyzing the asymptotic behavior of
the states. Furthermore, a bi-objective optimization problem is formulated that allows the easing from
ECQ to GCQ at an earlier time and minimizes the number of additional beds necessary to ensure
sufficient capacity in healthcare facilities. The multiple optimal solutions (Pareto optimal solutions) of
the bi-objective problem offer trade-offsolutions in which the user, a policymaker or healthcare
authority, can use in decision-making.
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2. Methods
2.1. Data
The number of cumulative confirmed cases from March 8 to September 30, 2020, was obtained
from the Philippine Department of Health (DOH) data drop [6]. These data were used to estimate
the rates of transmission, behavior change, and reporting. The data on COVID-19 bed capacity and
occupancy rate in Metro Manila were gathered from the number of occupied and available isolation,
ward, and ICU beds from the weekly DOH bulletin from June 20 to September 26, 2020 [33]. The
total population of Metro Manila was set to 13484462, based on the 2020 census data of the Philippine
Statistics Authority [8].
2.2. Mathematical model
The model we present is an extension of the model in [28] wherein an unreported compartment is
added to represent the undetected or unreported COVID-19 cases in the early phase of the pandemic in
the Philippines. The constant total population Nis divided into eight mutually exclusive compartments:
susceptible (S), behavior-changed susceptible (SF), exposed (E), reported infectious (I), unreported
infectious (Iu), isolated (Q), recovered (R), and deaths (D). A schematic diagram of the model is
shown in Figure 1.
Figure 1. COVID-19 transmission model that incorporates behavior change and unreported
cases. Susceptible (S) may change behavior (SF) and vice versa at rates βFor µ. These
classes can be exposed (E) to the virus and become infectious (I,Iu) in 1/κ days on average.
Transmission rate βis reduced by a factor δfor a behavior-changed SF. Reporting ratio is
denoted by ρ. Confirmed cases are isolated (Q) in 1/α days and recover (R) 1/γ days on
average or die (D). The average fatality rate is denoted by f. Those in Iurecover 1/η days
on average.
Assuming a local, prevalence-based spread of fear of the disease and following the study of Perra
et al. [34], the transition rate of a susceptible to a behavior-changed susceptible is given by
βFQ/(N−Q−D). This means that a susceptible individual is more likely to change behavior as the
number of confirmed cases among one’s contacts increases. The movement back to Sis assumed to
be influenced by the number of recoveries and susceptible individuals without behavior change [34].
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As the recoveries and susceptible individuals increase among the contacts of a behavior-changed
susceptible, the more likely the individual to exit the SFclass and resume regular social behavior. The
parameter µrepresents the rate of the easing of behavior and its value is assumed to be 1/14 [28].
Susceptible individuals (Sand SF) move to the exposed class upon contact with infectious
individuals (Iand Iu) at a rate β. The transmission rate for the behavior-changed susceptible class is
assumed to be reduced by a factor δ. The reporting ratio ρpartitions the exposed class to reported I
and unreported Iuclasses. Assuming that an individual becomes infectious 2 days before symptom
onset [35], the mean incubation period of the original virus strain is 6 days [36], and the mean
duration between symptom onset and the first medical consultation in the Philippines during this time
was 6.75 days [37], we set the mean latent period (1/κ) to 4 days and the mean infectious period of
the reported cases (1/α) to 8.75 days. From isolation, individuals recover 1/γ days on average or die.
The average fatality rate, which is the ratio of daily deaths to daily active cases and denoted by f, is
set to 1.9% [38]. Those in the unreported class are assumed to have less severe symptoms and move
to the recovered class 1/η days on average. The following system of differential equations describe
the model used in the study:
dS
dt =−βSI+Iu
˜
N+µSF
S+R
˜
N−βFSQ
˜
N,
dS F
dt =−δβSF
I+Iu
˜
N−µSF
S+R
˜
N+βFSQ
˜
N,
dE
dt =δβSF
I+Iu
˜
N+βSI+Iu
˜
N−κE,
dI
dt =ρκE−αI,
dIu
dt =(1 −ρ)κE−ηIu,
dQ
dt =αI−γQ,
dR
dt =(1 −f)γQ+ηIu,
dD
dt =fγQ,
˜
N=N−Q−D,
N=S+SF+E+I+Iu+Q+R+D,
(2.1)
where S(0) ≥0,SF(0) ≥0,E(0) ≥0,I(0) ≥0,Iu(0) ≥0,Q(0) ≥0,R(0) ≥0,and D(0) ≥0. We
introduce the term ˜
Nbecause those in Qand Dare assumed to be isolated and have no contact with the
rest of the population. All the parameters are assumed to be positive and ρ, f∈(0,1]. In the numerical
simulations, we set the initial population of the infectious I(0), exposed E(0), and unreported Iu(0)
classes equal to the number of cases 1/α, 1/α +1/κ, and 10 ×I0days from March 8, 2020, respectively.
The initial number of isolated individuals Q(0) was the number of cases at the start of the estimation
period. The initial susceptible population S(0) was computed by getting the difference between the
total population and E(0), Iu(0), I(0), and Q(0). The rest of the state variables were initially set to zero.
The model parameters and initial values of the state variables are shown in Table 1.
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Table 1. List of model parameters, their values, and references. The subscript denotes the
estimation period; 1 if from March 8 to May 31,2020 or 2 if from June 1 to September 30,
2020.
Symbol Description (unit) Value Ref.
β1, β2Transmission rate of COVID-19 in 0.199, 0.361 Estimated
Periods 1 and 2 (1/day)
βF,1, βF,2Transmission rate of awareness or fear of 471.057, 68.783 Estimated
the disease in Periods 1 and 2 (1/day)
µBehavior change ease rate (1/day) 1/14 Assumed
δTransmission reduction factor for 0.202 Estimated
behavior-changed individuals
1/κ Mean latent period (day) 4 [35, 36]
1/α Mean infectious period of reported cases (day) 8.75 [35, 37]
1/γ Mean recovery period (day) 16 [37]
fMean fatality rate 1.9% [38]
ρ1, ρ2Reporting ratio in Periods 1 and 2 0.289, 0.866 Estimated
1/η Mean infectious period of unreported cases (day) 10 [39]
S(0) Initial susceptible population 13483232 Calculated
SF(0) Initial behavior-changed susceptible population 0 Assumed
E(0) Initial exposed population 139 Assumed
I(0) Initial reported infectious population 99 Calculated
Iu(0) Initial unreported infectious population 990 Assumed
Q(0) Initial isolated population 2 [6]
R(0) Initial recovered population 0 Assumed
D(0) Initial deaths population 0 Assumed
2.3. Least-squares fitting of parameters
The values of the transmission rate, reporting ratio, and reduction factor were estimated from the
cumulative cases data in Metro Manila from March 8 to September 30, 2020. We divide the estimation
period into two: Period 1 is from March 8 to May 31, and period 2 is from June 1 to September 30,
2020. Metro Manila was mostly under ECQ during period 1, while it was mostly under GCQ during
period 2. It was during period 2 that the first major epidemic wave in the Philippines occurred. Since
the intensity of NPIs and the behavior of the population during ECQ and GCQ vary, the values for the
transmission rates (βand βF) and reporting ratio (ρ) in periods 1 and 2 are assumed to be different. The
reduction in transmission (δ) for the behavior-changed susceptible class is assumed to have the same
value in the two periods. We denote the transmission rates and reporting ratio for periods 1 or 2 by the
subscripts 1 or 2, respectively.
Denote by p=β1, β2, βF,1, βF,2, δ, ρ1, ρ2the vector of parameters to be estimated and set the domain
as Γ = [0,1] ×[0,1] ×[0,103]×[0,103]×[0,1] ×[0,1] ×[0,1]. Define the objective functional
J:Γ⊂R7→Ras
J(p)=X
iαI(ti;p)−˜
C(ti)2,(2.2)
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where ˜
C(ti) is the cumulative reported cases on day tibased on the data and I(ti;p) is the solution
Iof (1) at t=tiusing the parameter vector p. To estimate the unknown parameters, we calculate
the minimizer p⋆of the objective function (2.2) using a trust-region method based on the interior-
reflective Newton method [40, 41]. Each iteration of this approach solves a large linear system using
preconditioned conjugate gradient method [42]. We utilize the Matlab built-in function lsqcurvefit
to implement this numerical optimization technique. The program requires the bounds, which we set
based on the domain Γ, and initial guess, which we set to pinit =[0.3,0.3,285,285,0.2,0.25,0.25].
2.4. LHS-PRCC and parameter bootstrapping
Sensitivity analysis is a numerical technique that is widely used in identifying and ranking critical
parameters for a given model output [43]. A parameter is said to be influential to an output if small
perturbations of its value lead to significant changes in the model output. In this work, we use the
Partial Rank Correlation Coefficient (PRCC) method paired with the Latin Hypercube Sampling (LHS)
technique. LHS-PRCC is a global sensitivity analysis technique and is widely used in infectious disease
modeling [44–48]. We investigate the sensitivity of the ten parameters χ:=β, ρ, α, δ, βF, η, µ, κ, γ, f∈
[0.01,1]×[0.01,1]×[0.01,0.5]×[0.01,1]×[0.01,103]×[1/40,1/7]×[0.05,1]×[0.1,0.5]×[1/40,1/7] ×
[0.01,0.1]. To consider every infection, we use the cumulative number of infected individuals,
F(t;χ)=Zt
0
κE(σ;χ)dσ, (2.3)
as the model output. For each parameter χj,where j∈ {1,2,...,10}, we sample 10000 values {Xi j}10000
i=1,
all following a uniform distribution, using LHS. Hence, X∈R10000×10 with each row representing a
sampled combination of χ. For brevity, we denote Xias the ith row of Xand Xjas the jth column of
X. Denote by Y∈R10000 the output vector whose ith component is calculated by evaluating F(t;Xi).
The correlation coefficient (CC) is a metric to quantify the linear association between input and output.
If the data is ranked-transformed first before calculating the CC, the result is a Spearman or rank
correlation coefficient (RCC). Ranking is done by sorting both Xjand Yin descending order, and
the integer rank values XR
jand YR, respectively, are stored. Partial correlation coefficient (PCC) is
used to characterize the linear relationship between Xjand Yafter the linear effects on Yof the other
inputs are discounted [43]. The partial rank correlation coefficient (PRCC) between Xjand Yis the
PCC between XR
jand YR. The PRCC is computed numerically using the Matlab built-in function
partialcorr, with the ‘type’ set to ‘Spearman’. Note that the output Yis time-dependent as seen
in (2.3). In our simulations, the PRCC values are calculated at five time points: April 19, May 31,
August 2, October 4, and November 1, 2020.
Parameter bootstrapping is a statistical technique to quantify uncertainty and construct confidence
intervals of estimated parameters. In this study, we utilize the algorithm introduced in [49], where large
samples of synthetic data sets using the estimated model parameters are generated assuming a certain
probability distribution structure. In our simulations, parameters are re-estimated from 10000 (number
of realizations) synthetic data sets each generated by assuming a Poisson error structure. The mean,
standard deviation, and 95% confidence intervals of the re-estimated parameters are determined. This
technique is summarized in Algorithm 1.
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Algorithm 1 Uncertainty analysis.
1: Input: The estimated parameter vector p⋆, the number of realizations M
2: Calculate αI(ti;p⋆)) at each time point tiby solving (1).
3: for j=1 : Mdo
4: Generate the noisy data ˆ
Cj(ti) by adding Poisson noise to αI(ti;p⋆)).
5: Using the Matlab built-in function lsqcurvefit, calculate the minimizer ˆ
pjof
ˆ
J(p)=X
iαI(ti;p)−ˆ
Cj(ti)2.
6: end for
7: Perform statistical analysis on the re-estimated parameter vectors {ˆ
pj}M
j=1.
8: Output: Histograms to display the empirical distributions of the estimates and the corresponding
mean, standard deviation, and 95% confidence interval.
2.5. Optimization problem
Using the bed occupancy data from the DOH [33], we calculated that an average of 16% of the
active cases Q(t) occupied COVID-19 beds from June to September 2020. Fitting the weekly data
on available beds, a linear function representing 75% of the bed capacity was obtained. The DOH
categorizes a facility as high risk if the bed occupancy is 70% to 85%, and critical if bed occupancy is
greater than 85%. From July 18 to August 8, 2020, most facilities in Metro Manila were on critical or
high-risk, with a combined COVID-19 bed occupancy (isolation, ward, and ICU beds) exceeding 75%
of the capacity. Here, we propose an optimization approach to determine the number of additional beds
per day so that the number of cases requiring beds does not exceed 75% of the total bed capacity and
if it is possible to transition from ECQ to GCQ earlier than June 1, 2020.
We denote by QH(t) the number of active cases requiring beds and H(t) the linear, time-dependent,
data-fitted bed capacity. We consider two objectives:
(1) Minimize the number of additional hospital beds needed per day (ω) so that the 75% capacity is
not reached;
(2) Determine an earlier timing (τ) of easing from ECQ to GCQ.
The problem can be formulated as a bi-objective constrained optimization problem expressed as
follows:
min "ω
τ#(2.4)
such that
QH(t)≤H(t;ω, τ) :=0.75[ω(t−τ)+H0] for all t,(2.5)
where QH(t)=0.16 ·Q(t), H0is the baseline number of beds at time τbased on the data, and Q(t) is
solved from (2.1) by setting τas the day when GCQ started. Since this is a bi-objective optimization
problem, the solution is not unique but a pareto optimal set. We solve (2.4) and (2.5) using Genetic
Algorithm, which has found a growing number of applications in various fields of science and
engineering [50–52]. In particular, we implement the Matlab built-in function gamultiobj, which is
based on a variant of Non-dominated Sorting Genetic Algorithm II (NSGA-II) [53,54].
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3. Results
3.1. Model analysis
In the following theorems, we establish the existence, nonnegativity, and boundedness of the
solutions to model (2.1) for t≥0.Moreover, we derive the disease-free equilibria and a threshold
parameter for the given model.
Theorem 1 (Existence and uniqueness of solutions).There exists a unique solution to (2.1) for t ≥0.
Proof. Model (2.1) can be written as
˙
x=F(t,x(t)),x(0)=x0,(3.1)
where the column vector x(t)=(S(t),SF(t),E(t),I(t),Iu(t),Q(t),R(t),D(t))T∈R8defines a mapping
from [0,+∞] to R8. Moreover, let F(t,x(t)) =(F1(t,x(t)),...,F8(t,x(t)))Tbe a vector-valued function
from R8to R8equal to the right-hand side of the equations in (2.1). It can be verified that Fis
continuous and each of the first-order partial derivatives of Fwith respect to xis continuous in its
domain. By the existence and uniqueness theorem in [55], system (2.1) has a unique solution for
t≥0. □
Theorem 2 (Nonnegativity and boundedness of solutions).Solutions to system (2.1) with nonnegative
initial conditions will remain nonnegative for all t >0. Moreover, the solutions are bounded.
Proof. Using the notation in (3.1), assume that x0≥0. First, we show by contradiction that S(t)≥0
for all t>0. Following the approach in [56, 57], suppose that there exists t1>0 such that S(t1)=
0,S′(t1)<0,SF(t)>0,E(t)>0,I(t)>0,Iu(t)>0,Q(t)>0,R(t)>0, and D(t)>0 for t∈(0,t1).
Then, from the first equation of system (2.1),
S′(t1)=µSF(t1)R(t1)
˜
N(t1)>0,
which is a contradiction to the assumption S′(t1)<0. Here we choose t1such that SF(t1)>0 and
R(t1)>0. Hence, S(t)≥0 for all t≥0. Similarly, we can show that SF,I, and Iuremain nonnegative
for t≥0.
Next, since I(t)≥0 for all t≥0 then Q′(t)=αI−γQ≥ −γQand hence, Q(t)≥Q(0)e−γt≥0. It
also follows that D′(t)=fγQ≥0 and thus, D(t)≥0 for all t≥0. In a similar approach, it can be
shown that E(t) and R(t) are nonnegative for all t≥0.
Since Nis constant and S,SF,E,I,Iu,Q,R,D≥0, then from the last equation of (2.1),
S,SF,E,I,Iu,Q,R,D≤N.□
By equating the right-hand side of (2.1) to zero, the model has DFEs of the following forms:
E1=[θN,0,0,0,0,0, ϕN,(1 −θ−ϕ)N] and
E2=[0,N,0,0,0,0,0,0] (3.2)
for any 0 ≤θ≤1 and 0 ≤ϕ≤1 such that θ+ϕ≤1. Using the next-generation method and
following the notations in [58], the matrix Fof new infections and the matrix Vrepresenting the flow
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of individuals between compartments are given by
F=









0β β
0 0 0
0 0 0










and V=









κ0 0
−κρ α 0
−κ(1 −ρ) 0 η










.(3.3)
The basic reproduction number R0is calculated by finding the spectral radius of the matrix FV−1and
is given by
R0=ρβ
α+(1 −ρ)β
η.(3.4)
Moreover, the reproductive number R(t), which is the average number of secondary infections from an
individual during one’s infectious period, is expressed as
R(t)=βρ
α δSF(t)+S(t)
N!+β(1 −ρ)
η δSF(t)+S(t)
N!.
In the following theorems, we analyze the stability of the DFEs of the model.
Theorem 3 (Local asymptotic stability).E1is locally asymptotically stable if R0<1and E2is
unstable.
Proof. We calculate the eigenvalues λof the Jacobian matrix derived from system (2.1) at the DFEs.
For E1, the corresponding characteristic polynomial is given by
C1(λ)=λ3(λ+γ)(λ+µ)P1(λ),
where
P1(λ)=λ3+a1λ2+a2λ+a3,
a1=α+η+κ,
a2=ϵθ(αη +ακ −βκ +ηκ)+ϵϕ(αη +ακ +ηκ),
a3=ϵϕαηκ +ϵ θ(αηκ −αβκ +ραβκ −ρβηκ),
and ϵ=1/(ϕ+θ). It is clear from C1(λ) that five of its roots are nonpositive. Thus, it is left for us
to show that all of the roots of the polynomial P1(λ) are negative or have negative real parts. By the
Routh-Hurwitz criteria, it is sufficient to show that a1>0, a3>0, and a1a2−a3>0 [59].
Suppose R0<1. Since all of the model parameters are positive, a1>0. Next, we can rewrite a3as
a3=ϵϕαηκ +ϵ θαηκ(1 − R0).(3.5)
Since R0<1, then a3>0. Note that
a1a2−a3=A0+ϵθ(A1+A2+A3+A4),(3.6)
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where
A0=ϵϕ(α+η+κ)(αη +ακ +ηκ)−ϵϕαηκ,
A1=α2η+αηκ +αη2,
A2=ακ(α−ρβ),
A3=ηκ(η−(1 −ρ)β),
A4=κ2(α−β+η).
It suffices to show that A0,A1,A2,A3,A4>0. By expanding A0, its last term will be cancelled out and
so A0>0. Clearly, A1>0. Define
R0,I:=ρβ
αand R0,Iu:=(1 −ρ)β
η
so that R0=R0,I+R0,Iu. Since 0 <R0<1, then R0,I<1 and R0,Iu<1. Hence,
α−ρβ > 0 and η−(1 −ρ)β > 0.
Using the above-mentioned inequalities, it follows that A2and A3are both positive. Finally,
A4=κ2(α−β+η)=κ2 A2
ακ +A3
ηκ !>0.(3.7)
Therefore, E1is stable.
For E2, the characteristic polynomial is given by
C2(λ)=λ3(λ+γ)(λ−µ)P2(λ)
for some third-degree polynomial P2(λ). Observe that one of the eigenvalues is µ > 0. Thus, E2is
unstable. □
Remark. If θ=0, then a3in (3.5) and a1a2−a3in (3.6) are both positive regardless of the value of
R0, which means that [0,0,0,0,0,0, ϕN,(1 −ϕ)N)] is stable. Meanwhile if ϕ=0, then
[θN,0,0,0,0,0,0,(1 −θ)N)] is unstable if R0>1since a3<0. By some algebraic manipulation, E1is
unstable if ϕ<θ(R0−1).
In the next theorem, the global stability of the DFE E1was studied. Lyapunov stability was
established using Lyapunov’s direct method and asymptotic stability was shown following the
definition in [60].
Theorem 4 (Global asymptotic stability of E1).The DFE E1of the model (2.1) is globally
asymptotically stable in the domain
Ω:={(S,SF,E,I,Iu,Q,R,D)∈R8: 0 ≤S,SF,E,I,Iu,Q,R,D≤N},
whenever R0<1.
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Proof. Consider the Lyapunov function
V(x)=R0E+β
αI+β
ηIu,(3.8)
where xis the vector of state variables as in (3.1) and R0=βρ
α+β(1−ρ)
ηis the basic reproduction number.
Note that Vis continuous and differentiable in Ω,V(E1)=0, and V(x)>0 for x∈Ω,x,E1.
Next, we obtain ˙
V(x)=R0E′(t)+β
αI′(t)+β
ηI′
u(t). Note that ˙
V(E1)=0. Substituting the expressions
for E′(t),I′(t),and I′
u(t) from the model and simplifying, we get
˙
V(x)=R0δβSF
I+Iu
˜
N+R0βSI+Iu
˜
N−β(I+Iu)
=β(I+Iu) R0δSF
˜
N+R0S
˜
N−1!
≤β(I+Iu)(R0−1),
since δSF+S
˜
N≤1. It follows that ˙
V(x)≤0 for all x∈Ωif R0<1. Therefore, E1is Lyapunov stable [60].
To show that the DFE E1is globally asymptotically stable, we need to show that x→ E1as t→ ∞ [60].
First, we show that E,I,Iu→0 by showing that V→0 as t→ ∞. Since Vis decreasing and
nonnegative whenever R0<1, then by the monotone convergence theorem, Vconverges to, say, ξas
t→ ∞. We claim that ξ=0. Suppose otherwise, that is, ξ > 0. Then ξ≤V(x(t)) ≤V(x(0)). Define
the set U:={x∈Ω: 0 < ξ ≤V(x(t)) ≤V(x(0))}. Clearly, Uis compact and E1<U. Since ˙
Vis
continuous, then by the extreme value theorem, there exists ζ > 0 such that
sup
x∈U
˙
V=−ζ < 0.
Because ˙
V(x(t)) ≤ −ζfor all t, we get
V(x(T)) =V(x(0)) +ZT
0
˙
V(x(t))dt ≤V(x(0)) −ζT.
If we take T>V(x(0))/ζ, then V(x(T)) <0, which is a contradiction. Therefore, V→0 as t→ ∞,
which implies that E(t),I(t),Iu(t)→0.
Next, since R′(t),D′(t)≥0 then R(t) and D(t) are both increasing. Moreover, since R(t) and D(t) are
bounded, then by the monotone convergence theorem, there exist R⋆,D⋆∈[0,N] such that R(t)→R⋆
and D(t)→D⋆.
Let ε1>0. Since I(t)→0, then there exists T1>0 such that Q′(t)+γQ(t)=αI< ε1for all
t>T1. Letting ε1→0, we have Q′(t)≤ −γQ(t). Equivalently, Q(t)≤Q(T1)e−γt. Therefore, as t→ ∞,
Q(t)→0.
By removing some negative terms and noting that S≤˜
N, it follows from the second equation
in (2.1) that
S′
F(t)≤ −µSF
R
˜
N+βFQ.(3.9)
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Let ε2>0. Because Q(t)→0, then there exists T2>0 such that
Q(t)<ε2
2βF
(3.10)
for all t>T2. Also, since R(t)→R⋆and D(t)→D⋆, then there exists T3>0 so that

µR(t)
N−D(t)−ψ
<ε2
2N
for all t>T3and ψ=µR⋆
N−D⋆. It follows that
ψ−ε2
2N< µ R(t)
N−D(t).(3.11)
Taking T⋆=max{T2,T3}, (3.9)–(3.11) imply that
S′
F(t)+ψ−ε2
2NSF(t)<ε2
2(3.12)
for all t>T⋆. Because SF(t)≤N, (3.12) becomes
S′
F(t)+ψSF(t)<ε2
2+ε2
2NSF(t)≤ε2
2+ε2
2NN=ε2.
Letting ε2→0, we get S′
F(t)+ψSF(t)≤0 as t→ ∞. Thus, SF(t)≤SF(T⋆)e−ψtand SF(t)→0.
Finally, S=N−(SF+E+I+Iu+Q+R+D) implies S(t)→N−R⋆−D⋆=:S⋆.
Because S⋆+R⋆+D⋆=N, then there exist θ, ϕ ∈[0,1] with θ+ϕ≤1 so that
E1=[θN,0,0,0,0,0, ϕN,(1 −θ−ϕ)N]. Thus, if R0<1, all solutions of (2.1) converge to the DFE of
the form E1.□
3.2. Parameter estimation
The best model fit for the cumulative and daily cases is shown as the black curves in Figure 2. The
red circles represent the data points. The estimated transmission rates β1and βF,1in period 1 were 0.199
and 471.057, respectively. In period 2, the transmission rate of the disease β2increased to 0.361, while
the transmission rate of awareness or fear of the disease βF,2dropped to 67.783. The reporting ratio ρ1
in period 1 was estimated at 28%, which increased to 86% in period 2. The reduction in transmission
induced by behavior change δwas estimated at 0.202. The parameter estimates are given in Table 1.
The reproductive number R(t) is shown as the blue curve in Figure 2. Initially, R(t) was at 1.9 then
decreased to 0.9 by the end of period 1. During period 2, R(t) remained above 1 from early June until
mid-August, with a peak of up to 1.5 in mid-July 2020.
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Figure 2. The best model fit to the cumulative cases from March 8 to September 30, 2020.
The black curves are the plots of the cumulative cases (top) and daily cases (bottom) using
the model and parameter estimates. The vertical dashed lines mark the end of periods 1 and
2. Metro Manila was under ECQ from March 8 to May 15, 2020 (dark gray), MECQ from
May 16 to May 31 and August 4 to 18, 2020 (gray), and GCQ from June 1 to August 3 and
August 19 to September 30, 2020 (light gray). The red circles represent the data points and
the blue curve is the reproductive number R(t).
3.3. Sensitivity and uncertainty analyses
Using the cumulative number of infected individuals as the model output, the results of the
sensitivity analysis showed that β(range: 0.82 to 0.92) and δ(range: 0.52 to 0.68) have the highest
PRCC values, followed by ρ(range: −0.49 to −0.44) and the parameters for the infectious periods, α
(range: −0.48 to −0.46) and η(range: −0.45 to −0.28). PRCC values of κdeclined over time, with its
highest value at 0.436. The rest of the parameters have small magnitudes of PRCC. Figure 3
illustrates the results. Moreover, results of the parameter bootstrapping showed that the re-estimated
values of β, βF, δ and ρfollow a normal distribution and the mean values of the estimates all fall
within their respective 95% confidence intervals. Figure 4 shows the distributions of the parameter
re-estimates. The mean, standard deviation, and confidence interval are also indicated in the figure.
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Figure 3. The PRCC of the model parameters with respect to the cumulative number of
infected individuals. The bars represent the PRCC values on April 19, May 31, August 2,
October 4, and November 1, 2020.
Figure 4. The distribution of the re-estimates of β1, β2, βF,1, βF,2, δ, ρ1, and ρ2using the
parameter bootstrapping method. The mean, standard deviation (SD), and 95% confidence
interval (CI) are also shown.
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3.4. Effects of behavior change and reporting
The solid curves in the upper panel of Figure 5 are the plots of the susceptible class S,while the
dashed lines are the behavior-changed susceptible class SF. Here we investigate what happens if people
changed their behavior one (orange), two (yellow), three (purple), or four (green) weeks earlier. At the
beginning of period 2, we calculated that the proportion of SFwith respect to the total susceptible
population was 88% (SF: 11849000; S: 1592900). To incorporate early behavior change, we scale the
value of βFto yield the same proportion of SFone to four weeks before the start of GCQ. The black
curves in Figure 5 represent the plots of the model using the parameters in Table 1.
Figure 5. The dynamics of the susceptible population (top panel; Ssolid and SFdashed
curves), daily (lower left panel), and cumulative cases (lower right panel) if the population
changed their behavior one (orange), two (yellow), three (purple), or four (green) weeks
earlier than the start of period 2.
During period 1, we observe a switch in the populations of SFand S. By the end of period 1, SF
comprises the majority (88%) of the susceptible classes. The impact of early behavior change is seen
in the daily and cumulative cases in period 2, shown in the bottom panels of Figure 5. As behavior
changed one, two, three, or four weeks earlier, the cumulative cases decreased to 140468, 115573,
93041, or 73328 from 163191 (model, black), respectively. These translate to reductions of 14%, 29%,
43%, or 55% in cumulative cases by September 30, 2020. The peak of the daily cases also reduced to
2148, 1870, 1618, and 1392 from 2408 (model, black), and the timing was delayed from one to four
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weeks.
When the community quarantine was relaxed to GCQ, we observe that the size of SFdeclined,
while Sincreased. Around mid-July, when the number of daily cases were increasing (R(t)>1),
the behavior of the susceptible classes switched again. From that point on until the peak of the first
big wave in Metro Manila (∼August 14 according to the model), the proportion of SFamong the
susceptible classes increased from 59% to 89%.
The upper panels in Figure 6 show the effect of varying the values of µand βFon the cumulative
cases and timing of the peak of infections. Higher values of βFand lower factors of the behavior change
ease rate µin period 2 result in notable reductions in the number of cases and delay in the occurrence
of the peak. For instance, if in period 2 we set βF,2as 3 times βF,1and µreduced by 90%, then the
cumulative cases by the end of September 2020 would have been approximately 30000 and the peak
would have occurred around July 25 (140 days from March 8). The blue area on the heatmap for peak
timing indicates that the big wave in period 2 did not occur until September 2020.
Figure 6. The effect of varying the behavior parameters (µand βF) in period 2 and reporting
ratio in period 1 (ρ1) to the timing of the peak and cumulative cases by September 30, 2020.
Finally, the bottom panel in Figure 6 shows the effect of increasing the reporting ratio ρ1in period 1
on the cumulative cases by September 30, 2020. Only slight differences in the reported cumulative
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cases (blue bars; range: 11817 to 13421) were observed if ρ1was increased by factors of 1.5, 2, 2.5, or
3. On the other hand, cumulative cases, including the unreported (red bars), can be reduced by 29%,
45%, 54%, or 61%, if ρ1was increased by factors of 1.5, 2, 2.5, or 3, respectively.
3.5. Optimal bed capacity and timing of policy change
Panel (A) in Figure 7 shows the fitted bed capacity H(t) (dashed curve) from the data (red circles)
and the number of cases requiring beds QH(t) from the model. Note that the red circles depict 75% of
the total COVID-19 bed capacity during this time and QH(t) is 16% of Q(t), representing the average
number of active cases that occupy beds. By calculating the slope of H(t), denoted by ωdata , an
estimated 33 beds were added per day from June 21 to September 30, 2022, in Metro Manila.
Notably, QH(t)>H(t) during the second MECQ, when healthcare workers demanded a ‘timeout’.
The peak of QH(t) was calculated at 5307 cases.
Figure 7. The Pareto optimal solutions of the bi-objective optimization problem. (A) The
black curve QH(t) is the number of cases requiring beds, calculated as 16% of the reported
active cases Q(t) and the black dashed line H(t) is the bed capacity obtained by fitting the data
(red circles) using linear regression. (B) Pareto optimal set of (2.4). (C) Plots of QH(tω⋆, τ⋆)
(cases requiring beds) and the optimal hospital bed capacity H(t;ω⋆, τ⋆) corresponding to the
three Pareto optimal solutions colored blue, purple, and yellow. (D) Peaks of QH(t;ω⋆, τ⋆)
corresponding to the Pareto optimal solutions, compared to the model.
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The optimal solutions of the bi-objective optimization problem in (2.4) form a Pareto optimal set
illustrated in Figure 7 Panel (B). The circles are all optimal solutions depicting different policies. The
blue-colored optimal solution corresponds to the earliest easing to GCQ on May 2, 2020, and requires
131 additional beds per day to ensure that the bed capacity is adequate (up to 75% occupancy) during
the surge in cases following the lifting of restrictions (blue curves in Panel (C)). On the other hand,
the yellow-colored optimal solution has the least number of additional beds per day (47 beds) and
the latest start of GCQ (on May 28, 2020). This solution has a delayed and lower peak of infections
compared to the blue Pareto optimal solution (see Panel (C)). Compared to the curves in Panel (A),
the number of cases requiring beds QH(t;ω⋆, τ⋆) shown in Panel (C) is below the optimal bed capacity
H(t;ω⋆, τ⋆). Hence, constraint (2.5) of the optimization problem is satisfied. Panel (D) shows the peak
sizes of the epidemic waves corresponding to the various Pareto optimal solutions. The smallest peak
size (4807 cases, purple) is the optimal solution with GCQ starting on May 20, 2020, and with at least
56 additional beds.
4. Discussion
Using the estimated parameter values, we observe that the model captures the trend of the daily
and cumulative data from March until November 2020. The model shows a small peak in the number
of daily cases (204 cases) and a slow increase in cumulative cases during period 1. A much higher
peak (2408 cases) around mid-August 2020 and a sharp rise in cumulative cases are seen from the
model during period 2. A delay of about one month between the drop in Rtand the decline in daily
cases was also observed. Results of the parameter bootstrapping suggest good reliability of the
estimated parameters. Moreover, sensitivity analysis showed that the transmission rate βwas the most
sensitive parameter with respect to the number of cumulative infections. A higher reporting ratio ρor
shorter mean infectious period of reported cases 1/α reduces the cumulative infections. These results
suggest that intensifying testing and tracing efforts can effectively reduce new infections. The average
latent period (1/κ), which has the effect of delaying infection, becomes less sensitive to the
cumulative number of infections as the epidemic progresses, while the mean infectious period of
unreported cases (1/η) becomes more sensitive as the epidemic progressed.
Reporting was low in the early pandemic phase, possibly resulting from low testing capacity, slow
contact tracing, uncertainty, and lack of knowledge about the disease and protocols, or fear of social
stigma. This changed during period 2, where the estimated reporting ratio went up three times. These
results are consistent with a study on time-varying under-reporting estimates in various countries,
including the Philippines, during the same period [61]. The impact of the community quarantines
imposed by the government is reflected in the reduction parameter δand transmission rates βand βF.
The estimated 20% reduction in transmission for the behavior-changed class compares with a
mathematical model of COVID-19 transmission in the Philippines which showed that the minimum
health standards reduced the probability of transmission per contact by 13 −27% [62]. As the
community quarantine was relaxed from period 1 to 2, the transmission rate of the disease (β)
increased from period 1 to 2, while the rate of behavior change or transmission rate of awareness or
fear of COVID-19 (βF) decreased from period 1 to 2.
Results in Figures 5 and 6 emphasize the importance of early public health campaigns and positive
behavior changes (e.g., mask-wearing, improved hygiene practices, and social distancing) on reducing
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and delaying the peak of infections. Although fear and stigma can influence behavior changes [63,
64], these can also affect reporting and negatively impact disease control. We see in Figure 6 that as
reporting increased, total cumulative cases including the unreported decreased significantly.
Without vaccines and antiviral therapy, control of epidemic diseases relies on effective NPIs and
the management of healthcare systems. The bi-objective optimization approach can be used as a
decision support tool because of the multiple optimal solutions provided by the method, wherein
policymakers can choose depending on how much priority is given to minimizing the number of
additional beds or earlier easing of restrictions. Although the method is applied to the Philippines, the
optimization approach can also be used by other cities or countries by adapting location-specific
epidemiological parameters. For countries with limited resources, the solutions corresponding to later
easing of restrictions and a smaller number of additional beds may be better options. On the other
hand, for those that can provide sufficient additional beds, the approach can be used to identify the
optimal timing of adjusting NPIs. Results in Figure 7 suggest that if Metro Manila eased to GCQ on
June 1, 2020, at least 47 beds per day should have been prepared so that the bed occupancy in the
capital did not reach critical or high-risk, and MECQ was not needed to be reimposed. The blue
solutions in Figure 7 prioritizes the earlier timing (τ) of easing protocols over the number of
additional beds (ω). With this policy, GCQ could have been started 30 days earlier. However, this
requires 131 additional beds per day, which is 4 times ωdata. On the other hand, the yellow solutions
correspond to implementing GCQ on May 29, 2020. This would require 47 additional beds per day,
which is still more than ωdata. In Panel (D), the policy in purple has the lowest peak among all the
Pareto solutions. For this policy, even though GCQ starts on May 20 (12 days earlier than what
happened), the peak of cases (purple curve in Panel (C)) was 500 less than the peak from the model
(black curve in (A)). This policy would have required 56 beds per day, which is almost double than
ωdata.
5. Conclusions
In this work, we used an SEIQR model that considers behavior change and underreporting to study
the spread of COVID-19 during the early phase of the pandemic in Metro Manila, Philippines.
Behavior change can be influenced by awareness or fear of the disease, and willingness to observe
NPIs such as social distancing and mask-wearing. It was incorporated into the model by introducing a
two-compartment susceptible population: one for the behavior-changed population and the other for
those with regular behavior. The probability of getting infected is reduced for the behavior-changed
susceptible class. Due to limited testing and tracing, or negative attitudes of people towards seeking
healthcare, a compartment for the unreported cases was also added.
The results of this study highlight the importance of early behavior change and a high reporting rate
in reducing the number of cases and delaying the peak of infections. These can be done by intensifying
case surveillance and public health campaigns promoting compliance with NPIs, seeking healthcare,
and discouraging social stigma. Moreover, this study provides an optimization approach that quantifies
the additional bed requirement when policies are eased. The approach can be helpful in planning
strategies that address strengthening or easing of policies, especially during the early phase when NPIs
were the only control measures. Although the study focused on the transmission of COVID-19 in the
Philippines, the proposed model is general enough that it can be applied to any city or country. The
AIMS Mathematics Volume 8, Issue 1, 2201–2225.

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optimization problem can also be applied to other disease outbreaks by adjusting key epidemiological
parameters.
The model is best suited to represent the early phase of an epidemic. So, for a future work, the
model can be extended to describe succeeding epidemic waves and incorporate vaccination, variants,
and NPIs. Since the simulations did not consider the economic impact of NPIs, a model that maximizes
the economic output of a city or country while minimizing the number of infections is another work
that can be pursued. Moreover, one can also modify the model using fractional derivatives to account
for memory effects [65, 66]. Recently, numerous works on the theoretical analysis and applications of
fractional differential equations have been done [67–72]. To the best of our knowledge, this approach
has not yet been applied to a behavior change model similar to the one utilized in this study. Although
a compelling research direction, this generalization requires an intensive investigation and demands a
separate study.
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). The authors acknowledge Dr. Peter Julian Cayton of the UP
Resilience Institute for his assistance with the data collection.
Conflict of interest
All authors declare no conflicts of interest in this paper.
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