(2022) 22:656
Ejima et al. BMC Infectious Diseases
https://doi.org/10.1186/s12879-022-07646-2

Open Access

RESEARCH

Estimation of timing of infection
from longitudinal SARS‑CoV‑2 viral load data:
mathematical modelling study
Keisuke Ejima1,2*†, Kwang Su Kim3,4†, Ana I. Bento1†, Shoya Iwanami3, Yasuhisa Fujita3, Kazuyuki Aihara5,
Kenji Shibuya2 and Shingo Iwami3,6,7,8,9,10*

Abstract
Background: Multiple waves of the COVID-19 epidemic have hit most countries by the end of 2021. Most of thosewaves are caused by emergence and importation of new variants. To prevent importation of new variants, combination of border control and contact tracing is essential. However, the timing of infection inferred by interview is influenced by recall bias and hinders the contact tracing process.
Methods: We propose a novel approach to infer the timing of infection, by employing a within-host model to capture viral load dynamics after the onset of symptoms. We applied this approach to ascertain secondary transmission
which can trigger outbreaks. As a demonstration, the 12 initial reported cases in Singapore, which were considered asimported because of their recent travel history to Wuhan, were analyzed to assess whether they are truly imported.
Results: Our approach suggested that 6 cases were infected prior to the arrival in Singapore, whereas other 6 cases
might have been secondary local infection. Three among the 6 potential secondary transmission cases revealed thatthey had contact history to previously confirmed cases.
Conclusions: Contact trace combined with our approach using viral load data could be the key to mitigate the riskof importation of new variants by identifying cases as early as possible and inferring the timing of infection with highaccuracy.
Keywords: SARS-CoV-2, COVID-19, Mathematical model, Viral load, Contact trace
Background
Most countries have experienced multiple waves of
COVID-19 epidemic by the end of 2021. Those different waves are due to emergence and importation of newvariants of SARS-CoV-2. Further, given that waning†

Keisuke Ejima, Kwang Su Kim and Ana I. Bento contributed equally to thiswork.
*Correspondence: kejima@iu.edu; iwami.iblab@bio.nagoya-u.ac.jp

1

Department of Epidemiology and Biostatistics, Indiana University Schoolof Public Health-Bloomington, Bloomington, IN, USA
3Interdisciplinary Biology Laboratory, Division of Natural Science,
Graduate School of Science, Nagoya University, Nagoya, Japan
Full list of author information is available at the end of the article

immunity is inevitable even after infection or (multiple
doses of ) vaccination [1–5], more waves of the epidemic
are anticipated to happen in the next few years [6].
There are three major control knobs to prevent a resurgence: border control; test, trace contacts and isolate;and social distancing measures [7]. To avoid future outbreaks, governments may need to implement various
border control programs: quarantine of suspicious cases
and isolation of confirmed cases, and travel restriction
to and from countries with ongoing outbreaks. Bordercontrol, which is generally based on symptoms, cannot
capture all cases given approximately a 5-day incubation
period and non-negligible number of asymptomatic cases© The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/. The Creative Commons Public Domain Dedication waiver (http://​creat​iveco​
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Ejima et al. BMC Infectious Diseases(2022) 22:656

of COVID-19 (i.e., cases can pass the control program if
they are not symptomatic) [8–16]. Matteo and colleagues
examined that travel border control only delayed epidemic progression by 3–5 days within China, suggesting border control solely cannot prevent outbreaks [17].
Therefore, “test, contact trace and isolate” measure plays
an essential role in controlling local transmission andreducing the risk of subsequent outbreaks.
Contact trace starts once cases are confirmed. Confirmed cases are followed by further investigation–
through interviews, contact tracing, and genomic
analysis in some cases–to assess when and where they
were infected [18, 19]. This tells us a lot about the transmission mode and benefits contacts. First, contact trace
benefits cases with better clinical outcomes. Bi and colleagues reported that the cases identified by contact trace
were identified and treated earlier than cases identified
by symptom-based survey in Shenzhen, China [20]. Second, it is used to identify highly transmissible situation.
For example, Nishiura and colleague found SARS-CoV-2
is highly transmissible in closed environment throughcontact trace data [21]. The third, and what we will focus
on in this study, is that contact trace is used to ascertain
whether the cases are locally transmitted or imported.
Epidemics are triggered by importation of new cases.
Indeed, Kathy and colleagues reported that the first wave
was initiated by imported cases from Hubei province tomega cities such as Beijing, and the further local transmissions created the first wave [22]. Interestingly, newcases reported afterwards were imported from overseas,suggested potential risk of further outbreak in China.
Thus many studies separately reported imported casesand cases of local transmission [22, 23]. Ascertaining
whether new COVID-19 cases are imported or due tolocal secondary transmission is essential for a government to develop public health strategies. If they are thelatter and the number of local transmissions is substantial, outbreaks are inevitable, and the goal of intervention programs should be shifted from containment to
mitigation.
Traditionally, the ascertainment of local transmission
requires interview-based assessments, which would
be time-consuming and potentially biased because thetiming of infection is not directly observable in manycases (extremely hard for SARS-CoV-2 because it is anairborne disease [24–26]) except some special caseswith specific known infection route such as HIV [27,28]. As such, most of the studies inferring the timingof infection for COVID-19 were based on active surveillance or contact trace [8–11]. However, if cases
cannot report correct day(s) of exposure, the investigation could be biased and the source of infection

Page 2 of 9

will not be identified. Especially for COVID-19, thebias could be worsen given relatively long incubation
period of COVID-19 compared with influenza [17].
For example, if ones travel frequently during the longincubation period, they may not remember when andwhere they were exposed. Indeed, Lauer et al. and Bi
et al. demonstrated large uncertainly on the estimation of the timing of infection for COVID-19 in China[8, 20]. Even accounting for uncertainty for the timingof infection with exposure information using Bayesian
approach when estimating incubation period [9, 29],bias is not perfectly removed. Further, given that there
is substantial portion of pre-symptomatic and asymptomatic infection [12–16], specification of the exposure
events is challenging even with stringent contact tracing protocol.
Mathematical models have been used to study infectious diseases including COVID-19 at any scale fromthe population level to the within-host level [30]. Especially, between-hosts models are widely adopted todescribe the transmission process (between hosts) of an
infection, study the epidemiology of infectious diseases,
and assess the effectiveness of different intervention
measures at the population level [31–34]. These models found ample adoption by public health officials andgovernments in a number of ways that range from situational awareness to intervention planning and projections [35] as well as for filling data gaps, estimating key
epidemiological parameters (such as the reproduction
number), and explaining the mechanisms behind theobserved patterns [20]. On the other hand, the modelwith another scale, within-host models have beendeveloped to describe the process of viral replication
and removal as well as immune response within a single person [36]. Overall, these models capture both theviral dynamics within a host while it is infected as well
as changes in immune level after infection (e.g., waning of immunity, boosting). Furthermore, these models
can be used to describe and test personalized interventions through in-silico experiments, providing insights,
for instance, on the development of treatments andtreatment regimens, timing of booster vaccination, andclinical trial design [37, 38]. Within-host models allow
considering the intrinsic biological difference between
patients, which may alter the epidemiology and transmission patterns of an infectious disease outbreak (asobserved for SARS-CoV-2 infection [16, 39–41]).
In this study, we propose to use the viral dynamics
model to differentiate secondary local transmission
from imported cases, which complements contact traceand reduces its inherent bias. For the purpose of illustration, we analyzed cases in the early phase of the firstwave in Singapore.

Ejima et al. BMC Infectious Diseases(2022) 22:656

Methods
In Singapore, the first case was identified on 23rd January 2020 (Fig. 1A). The first 18 cases had travel history
to Wuhan, China, thus were considered to be imported
cases[42]. Two days after the 18th case was confirmed (3rdFebruary 2020), a new case was identified which had no
history of traveling to China. To investigate the possibilityof some of the original 18 being evidence of ongoing local
transmission, we leveraged viral load data collected[42] for
multiple time points after the onset of symptoms using a
within-host viral dynamics model for SARS-CoV-2. This
enables us to infer time of infection (i.e., before or after
arrival to Singapore).
Viral load data

We extracted data from two published papers [42, 43] (wehave not collected original data in this study). Nasopharyngeal swabs were collected from the 18 cases reported for
up to 30 days from the onset of symptoms. Viral loads
were measured by RT-PCR[42]. We excluded 5 cases who
received antivirals (i.e., lopinavir, ritonavir) because theantiviral effect could not be quantified with the limited
data, and a case whose viral load was detected only twice(i.e., not enough to hindcast the viral load trajectory). In
total, we analyzed the first 12 cases. In order to find “infection establishment boundary” (see “Viral load boundary
for infection establishment” section) and derive robust
parameter estimation, we obtained an additional dataset of viral loads measured in nasal swab collected fromthe 8 cases reported in Zhuhai, China[43]. Three of thesecases were confirmed as secondary infections and theirdays of infection were reported, thus we used the information to compute the viral load threshold for the infection
establishment.
Cycle threshold (Ct) values reported in Zou et al. [43] andYoung et al. [42], which are the cycle numbers when the
fluorescent signal crosses the threshold, were converted toviral RNA copies (copies/mL); these quantities were shown
to be inversely proportional to each other from a previous SARS-CoV study [44]. The values under the detection limit were assumed to be at the detection limit forthe purposes of fitting the model (see later for detail). We
used the program datathief III (version 1.5, Bas Tummers,
www.​datat​hief.​org) to extract the data from images in thosepublications. Waiver of informed consent was granted bypublic health authorities or written informed consent wasPage 3 of 9

obtained from study participants as described in the original studies.
Viral load modelling to estimate the day of infection
establishment

Based on a standard viral dynamics model, to describe
SARS-CoV-2 dissemination among target cells, we used
the following simple mathematical model previously proposed [45–47]:

df (t)dV (t)= −βf (t)V (t),= γ f (t)V (t) − δV (t),
dt
dt
(1)
where f (t) and V (t) are the ratio of uninfected target
cells and the amount of virus, respectively. The parameters β, γ , and δ represent the rate constant for virus
infection, the maximum rate constant for viral replication, and the death rate of infected cells, respectively. All
viral load data including Singapore and Zhuhai patientswere simultaneously fitted using a nonlinear mixed-effectmodelling approach, which uses the whole samples toestimate population parameters while accounting forinter-patient variation.
Further, sampled parameter sets were used to predict
the estimated day of SARS-CoV-2 infection establishment, that is, the start of the exponential growth phaseof viral loads [47]. The infection establishment time, Tinf ,was estimated by hindcasting the mathematical model,
when the viral load reaches a boundary. The viral loadboundary for infection establishment was computed
using the three secondary infection cases reported in
Zhuhai, whose initial days of exposure to the primarycases are known [43]. We assumed that the initial day of
exposure is equal to the day of infection establishment.
Once the day of infection establishment is obtained foreach case, it was compared against the date of arrival in
Singapore. If the estimated day of infection is before thearrival in Singapore, it suggests that the infection wasestablished outside Singapore, otherwise, the case is theresult of secondary local transmission within Singapore.
Viral load boundary for infection establishment

To define “viral load boundary for infection establishment”, we used the information of the three secondary
cases with known primary cases in Zhuhai (i.e., Patients
D, H and L) [43]: a primary infected patient (Patient E)(See figure on next page.)
Fig. 1 Epidemic curve of COVID-19 and clinical course of patients in Singapore. A Epidemic curves of COVID-19 as of March 10, 2020 in Singapore
are shown. The green and red solid bars correspond to the newly reported cases by date of symptom onset and by date of laboratory confirmation,
respectively. B Each panel presents timeline of infection for each case. Expected SARS-CoV-2 viral dynamics and observed viral load for the first 13
cases are depicted by grey (or black) solid lines and grey open circles, respectively. The timing of arrival to Singapore (red dashed lines), the timingof symptom onset (black dashed lines), the estimated timing of infection establishment (blue shaded areas), and the detection limit of viral load
(grey dashed lines) are also shownEjima et al. BMC Infectious Diseases(2022) 22:656

Fig. 1 (See legend on previous page.)

Page 4 of 9

Ejima et al. BMC Infectious Diseases(2022) 22:656

worked in Wuhan and visited Patient D and Patient L
on January 17th, then Patients D and L developed symptoms on January 23rd and 20th, respectively. Other primary infected patients (Patients I and P) visited Patient
H on January 11th, and fever developed in Patient H
on January 17th. This implies that exposure started on
the day when the primary cases visited those secondary
cases. Assuming that infection established on the initial
day of exposure in the secondary cases, we hindcasted
the mathematical model, and obtained the viral load on
the initial day of exposure, which is defined as the infection establishment boundary: 1­0–6.67 to 1
­0–5.18, ­10–5.20
–3.88
–1.14
0.03
to ­10
and 1
­0
to 1
­0
for Patients D, H and L,
respectively. We used the lowest ­(10–6.67) and highest
0.03­(10 ) values as the boundary (Fig. 2).
Nonlinear mixed‑effects model

The nonlinear mixed-effects modelling approach incorporates fixed effects as well as random effects which
describe the inter-patient variability in parameters.
Including random effects amounts to a partial pooling of
the data of all patients to improve estimates of the fixed
effect parameters applicable across the cases. The parameter of patient k , θk (= θ × eπk ) is a product of θ (the fixed
effect) and πk (the random effect), where πk is assumed
to follow a normal distribution: N (0, �). The fixed effectparameters and random effect parameters were estimatedusing the stochastic approximation Expectation/Maximization (SAEM) algorithm and empirical Bayes method,respectively. A right-truncated normal distribution wasused in the likelihood function to account for the left
censoring of the viral load data (i.e., when the viral load
is not detectable) [48]. MONOLIX 2019R2 (www.​lixoft.​com), a program for maximum likelihood estimation fora nonlinear mixed-effects model, was employed to fit themodel to the viral load data. We changed the initial values multiple times to avoid local minimum of AIC andconfirmed the robustness of parameter estimation.

Page 5 of 9

Results
Before describing the viral load data, here is the epidemiological situation when the data were collected. Figure 1A depicts the weekly epidemic curve in Singapore
from January 21st to March 15th based on the onset ofsymptoms and laboratory confirmation. The epidemic
curve based on laboratory confirmation follows the curve
based on the onset of symptoms because of the reporting
delay. For the first few weeks, the epidemic in Singapore
was not in the phase of exponential growth, which suggests secondary transmissions were limited and any successive transmission did not take place yet. The first 18
cases discussed here are observed in the first two weeksof the epidemic.
For the first 12 ‘imported’ cases with travel history to
Wuhan (6 were removed from the analysis due to insufficient viral load data or antiviral treatment. See the “Viralload data” section), their viral load data are plotted alongwith the curve of viral load estimated from the model(Fig. 1B) (note that time since the onset of symptoms wasused as a time scale). Typically, the viral load exponentially increases since infection because the viruses reproduce themselves successively in the target cells. The viralload hits peak when uninfected target cells run out. Thenthe viral load starts to decrease over time as the viruses
and infected cells are removed from the host body.
Although the viral dynamics curves of different patients
share the above characteristics, there was also huge heterogeneity in the dynamics. Especially, the virus persistence (i.e., the length of time the viral load is above thedetection limit) and the hight of peak viral load were different between the patients. Such variability is due to thedifference in biological (immunological) characteristics
of each patient. For example, lower death rate of infected
cells, δ, is translated into longer virus persistence (slower
decay in viral load) (Additional file 1: Table S1). However, interestingly, the viral load consistently hit the peak
about 2–3 days after the onset of symptoms, suggesting viral shedding is high even before the clinical onset.

Fig. 2 Viral load dynamics of the three patients in China. The three panel presents timeline of infection for the three cases in Zhuhai, China used
to compute the viral load boundary for infection establishment. Expected SARS-CoV-2 viral dynamics and observed viral load are depicted by grey(or black) solid lines and grey open circles, respectively. The timing of symptom onset (black dashed lines), the timing of infection establishment(known; blue shaded areas), and the estimated viral load boundaries for infection establishment (red dashed lines) are also shownEjima et al. BMC Infectious Diseases(2022) 22:656

Although viral shedding does not necessarily reflect themagnitude of infectiousness, this finding is concordance with the previous studies suggesting non-negligible
amount of pre-symptomatic infection [12–16]. Notably
He and colleagues demonstrated that the infectiousness
profile peaks around the onset of symptoms [16], which
is similar to the viral load dynamics we estimated in thisstudy.
Hindcasting the estimated viral load dynamics, the dayof infection establishment was estimated, where viralload reaches the boundary (indicated by shaded blue areain Fig. 1B). Note that the estimation of the day of infection establishment has some uncertainty (about 6 days)
because of uncertainty on the boundary of viral load
threshold. By comparing the day of infection establishment and the reported day of arrival in Singapore (whichshould be available from the immigration record in general), suggested by red dotted line in Fig. 1B, whether thecase was infected in or out of Singapore was assessed.
Additional file 1: Table S2 summarized the day of infection establishment and arrival to Singapore in calendar
time. We found that 6 of the 12 cases (Case 2, 3, 4, 14, 16,17) were clearly concluded as imported cases, whereas
the remainder 6 cases (Case 6, 8, 9, 11, 12, 18) could haveresulted from ongoing transmission locally in Singapore.
For those suspicious secondary cases, contact tracing
could provide further confirmation as to the timing of
infection.
In fact, although all of the 12 were considered to beimported cases due to their travel history, the detailed
investigation [42] revealed 3 cases (Case 8, 11, 18) among
the 6 cases (Case 6, 8, 9, 11, 12, 18) for whom we could
not exclude the possibility of secondary transmission had
close contact with previously confirmed cases: Case 8 isa spouse of Case 9 travelled together to Singapore; Case
11 and 18 were identified as close contacts of Case 4 and12, respectively. These findings suggest that our approachcomplements the traditional interview-based approach,and together with it, we can differentiate imported casesand cases of local transmission.

Discussion
Many countries in Europe, Asia, and North America
experienced several waves of COVID-19 epidemic, whichwere slowed down due to stringent lockdown measurements. The lockdown is not a sustainable measure
because it downregulated world’s economy and impacted
citizens’ health both physically and mentally [49, 50].
Therefore, ways to control epidemics without the lockdown measures have been warranted.
In order to avoid further waves of the epidemic without
stringent lockdown measures, the following three measures should be implemented together: border control;

Page 6 of 9

test, trace contacts and isolate; and social distancing.
Along with them, closely monitoring transmission modeby contact trace is essential to identify cases and avoid
further transmission. However, the contact tracing is
generally interview-based and the timing of infection
inferred by this method is influenced by recall bias. We
proposed viral load-based approach to help contact trace
identify the timing of infection establishment.
Indeed, contact trace is widely conducted in many
countries, however, the process is time-consuming. Using
digital technology has been proposed to identify contacts
faster [51], although there are issues of privacy and dataprotection [52–54]. The Singapore government releasedsuch application as a step toward lifting lockdown [55],however, only limited population have installed the application so far [56]. If many people do not install the app,all contacts cannot be identified. Further, infection events
can occur indirectly (transmission can occur from contaminated surfaces [57, 58]), whether such apps can identify when and where infection occurred and eventually
prevent outbreak is still uncertain.
In this study, we assessed whether the 12 initial cases
which were classified as ‘imported’ were in fact imported
or the result of secondary local transmission within Singapore. We found that 50% of the cases (6 out of 12) wereclearly infected before arrival to Singapore (i.e., imported
cases), the remaining half of the cases, however, could
have been infected after the arrival in Singapore (i.e., secondary local transmission). This implies the possibility ofwithin-country transmission prior to the 19th case (whois the result of local transmission) being reported on
February 3rd. Combined with interview-based contact
tracing, this approach can identify the trace of local secondary transmission.
Our method is useful to infer the timing of infection,
discerning between cases imported or autochthonous(i.e., before or after arrival to the country) complementing contact trace. The advantage of using this method
is that computation is solely based on viral load data.
Collecting viral load could be a part of clinical practice
as viral load has been collected for clinical purposessuch as to understand the pathophysiology and aetiology of new diseases especially in the early phase of outbreak [59–62]. Further, viral load is measurable withpatients’ saliva (not nasopharyngeal swabs, which has
been widely used) in COVID-19 cases. It will reduce
the effort of measuring viral load and the risk of infection for health practitioners [63, 64]. Given that recall
bias is an issue of contact tracing, our method can
assist inferring the timing of infection which is usually
done by interview. In other words, estimation using ourapproach will be further enhanced if combined with thecomplementary information (e.g., travel and contact

Ejima et al. BMC Infectious Diseases(2022) 22:656

history and genetic information) thus reducing uncertainty in our predictions. Therefore, we empathize that
we are not undermining the value of contact tracing.
There are a few limitations in our approach. First,
our approach requires viral load data over multiple
time points (only the patients with more than threeviral load data points were included in the analysis);
therefore, we may not be able to accurately estimate
the timing of infection if the number of data points arelimited. In addition to the number of data points, thetiming of data collection (i.e., immediately after or longafter symptom onset) will be the key that determines
whether the dynamics can be accurately estimated.
Future studies should argue data collection process todefine the best practices for the use of the viral dynamics model in the context of inferring the timing of infection. Second, we need to note that both the boundaries
and the day of infection establishment estimated using
our approach could be underestimated, because infection is established after exposure starts. Third, we
estimated the viral load boundaries of infection establishment with limited available data. Further studies
need to investigate what factors are associated withthe boundaries, such as countries, strains, and viruses.
Fourth, we did not include the cases received antiviraltherapy because of lack of information about its effectfor SARS-CoV-2. However, once enough data about
those antivirals are available, we can include thosecases (we have already proposed a model accounting
for antiviral effect [38, 65, 66]).

Conclusions
Now is the time to think about the balance between
risk of further waves and the social and economic damage accompanied by lockdown. To secure social andeconomic activities whilst controlling the risk of nextwave, we need to closely monitor infection events aswell as implement various preventive measurements,
which were effective to hammer the waves of the epidemic. Contact trace plays an essential role. Collecting
and analysing viral load data would help calibrate thetiming of infection establishment estimated by contact
tracing. As such, the method we used may be critical tohelp shape a country’s early response to the next wave.
Supplementary Information
The online version contains supplementary material available at https://​doi.​
org/​10.​1186/​s12879-​022-​07646-2.
Additional file 1: Table S1. Estimated parameters for each case. Table S2.
Date of arrival, symptom onset, estimated day of infection for Singapore
cases infected with SARS-CoV-2.

Page 7 of 9

Acknowledgements
Not applicable
Author contributions
Conceived and designed the study: KE AIB KS SIwami. Analyzed the data: KSK
KE SIwanami SIwami. Wrote the paper: KSK KE AIB YF KA SIwami. All authors
read and approved the final manuscript.
Funding
This study was supported in part by The Tokyo Foundation for Policy Research(to K.E. and K.S.); Grants-in-Aid for JSPS Scientific Research (KAKENHI) Scientific
Research B 18KT0018 (to S.IWAMI), 18H01139 (to S.IWAMI), 16H04845 (toS.IWAMI), Scientific Research S 15H05707 (to K.A.), Scientific Research in
Innovative Areas 20H05042 (to S.IWAMI), 19H04839 (to S.IWAMI), 18H05103(to S.IWAMI); AMED JP20dm0307009 (to K.A.); AMED CREST 19gm1310002 (toS.IWAMI); AMED Japan Program for Infectious Diseases Research and Infrastructure, 20wm0325007h0001, 20wm0325004s0201, 20wm0325012s0301,
20wm0325015s0301 (to S.IWAMI); AMED Research Program on HIV/AIDS
19fk0410023s0101 (to S.IWAMI); AMED Research Program on Emerging andRe-emerging Infectious Diseases 19fk0108156h0001, 20fk0108140s0801
and 20fk0108413s0301 (to S.IWAMI); AMED Program for Basic and ClinicalResearch on Hepatitis 19fk0210036h0502 (to S.IWAMI); AMED Program on the
Innovative Development and the Application of New Drugs for Hepatitis B
19fk0310114h0103 (to S.IWAMI); Moonshot R&D Grant Number JPMJMS2021(to K.A. and S.IWAMI) and JPMJMS2025 (to S.IWAMI); JST MIRAI (to S.IWAMI);
Mitsui Life Social Welfare Foundation (to S.IWAMI); Shin-Nihon of Advanced
Medical Research (to S.IWAMI); Suzuken Memorial Foundation (to S.IWAMI);Life Science Foundation of Japan (to S.IWAMI); SECOM Science and Technology Foundation (to S.IWAMI); The Japan Prize Foundation (to S.IWAMI);
Foundation for the Fusion of Science and Technology (to S.IWAMI); This work
was supported by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (2022R1C1C2003637)(to K.S.K.); the
MIDAS Coordination Center (MIDASSUGP2020-6) by a grant from the National
Institute of General Medical Science (3U24GM132013-02S2) (to K.E.). The study
does not necessarily represent the views of the funding agencies listed above.
Availability of data and materials
The data are publicly available from following two publications: Young BE,
Ong SWX, Kalimuddin S, Low JG, Tan SY, Loh J, Ng OT, Marimuthu K, Ang LW,
Mak TM et al.: Epidemiologic Features and Clinical Course of Patients InfectedWith SARS-CoV-2 in Singapore. JAMA 2020, 323(15):1488–1494. Zou L, Ruan
F, Huang M, Liang L, Huang H, Hong Z, Yu J, Kang M, Song Y, Xia J et al.: SARSCoV-2 Viral Load in Upper Respiratory Specimens of Infected Patients. N Engl J
Med 2020, 382(12):1177–1179.

Declarations
Ethics approval and consent to participate
We used the de-identified secondary data from published studies, and thus
ethics approval for this study was not necessary.
Consent for publication
Not applicable.
Competing interests
All authors have no conflicts of interest with respect to this paper. Indiana University and the Indiana University Foundation have received funding from Eli
Lilly and Company and from private philanthropists to support other research
projects pertaining to COVID-19 in which the authors participate.
Author details
1
Department of Epidemiology and Biostatistics, Indiana University Schoolof Public Health-Bloomington, Bloomington, IN, USA. 2 The Tokyo Foundation
for Policy Research, Tokyo, Japan. 3 Interdisciplinary Biology Laboratory, Divisionof Natural Science, Graduate School of Science, Nagoya University, Nagoya,Japan. 4 Department of Science system simulation, Pukyong National University, Busan, South Korea. 5 International Research Center for Neurointelligence,
The University of Tokyo, Tokyo, Japan. 6 Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan. 7 Institute for the Advanced Studyof Human Biology (ASHBi), Kyoto University, Kyoto, Japan. 8 NEXT-Ganken

Ejima et al. BMC Infectious Diseases(2022) 22:656

Program, Japanese Foundation for Cancer Research (JFCR), Tokyo, Japan.
9Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS),RIKEN, Saitama, Japan. 10 Science Groove Inc., Fukuoka, Japan.
Received: 25 January 2022 Accepted: 22 July 2022

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