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Citation: Přibylová L, EclerováV, Májek 0,
Jarkovský J, Pavlík T, Dušek L (2023) Using realtime
ascertainment rate estimate from infection
and hospitalization dataset for modeling the spread
of infectious disease: COVID-19 case study in the
Czech Republic. PLoS ONE 18(7): e0287959.
https://doi.org/10.1371/journal.pone.0287959
Editor: Seth Blumberg, University of California San
Francisco, UNITED STATES
Received: November 6,2021
Accepted: June 9,2023
Published: July 13,2023
Copyright: © 2023 Přibylová et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All data are included
in R_CODES_DATA.zip file. These data are
collected and available: (1) Mobility data are
available without restrictions at https://github.com/
ActiveConclusion/COVID19_mobility/blob/master/
google_reports/mobility_report_europe.zip(2)
Anonymized SARS-CoV-2 positive records (full
official Czech Republic dataset with hospitalization
data modely_05_datumy.csv) are available at
https://onemocneni-aktualne.mzcr.cz/api/account/
dokumentace on request at https://docs.google.
RESEARCH ARTICLE
Using real-time ascertainment rate estimate
from infection and hospitalization dataset for
modeling the spread of infectious disease:
COVID-19 case study in the Czech Republic
Lenka Přibylová 1
* , Veronika Eclerová 1 2
, Ondřej Májek3 , 4
, Jiří Jarkovský3 , 4
,
Tomáš Pavlík3 , 4
, Ladislav Dušek3 , 4
1 Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, Czech Republic,
2 RECETOX, Faculty of Science, Masaryk University, Brno, Czech Republic, 3 Institute of Biostatistics and
Analyses, Faculty of Medicine, Masaryk University, Brno, Czech Republic, 4 Institute of Health Information
and Statistics of the Czech Republic
* pribylova@math.muni.cz
Abstract
We present a novel approach to estimate the time-varying ascertainment rate in almost
real-time, based on the surveillance of positively tested infectious and hospital admission
data. We also address the age dependence of the estimate. The ascertainment rate estimation
is based on the Bayes theorem. It can be easily calculated and used (i) as part of a
mechanistic model of the disease spread or (ii) to estimate the unreported infections or
changes in their proportion in almost real-time as one of the early-warning signals in case of
undetected outbreak emergence. The paper also contains a case study of the COVID-19
epidemic in the Czech Republic. The case study demonstrates the usage of the ascertainment
rate estimate in retrospective analysis, epidemic monitoring, explanations of differences
between waves, usage in the national Anti-epidemic system, and monitoring of the
effectiveness of non-pharmaceutical interventions on Czech nationwide surveillance datasets.
The Czech data reveal that the probability of hospitalization due to SARS-CoV-2 infection
for the senior population was 12 times higher than for the non-senior population in the
monitored period from the beginning of March 2020 to the end of May 2021. In a mechanistic
model of COVID-19 spread in the Czech Republic, the ascertainment rate enables us to
explain the links between all basic compartments, including new cases, hospitalizations,
and deaths.
Introduction
In mathematical epidemiology, compartmental models of type SIR or SEIR are widely used to
describe and explain outbreaks of epidemics [1-8]. The classical SEIR models monitor compartments
(i) S Susceptible individuals (those who have not yet been infected by the disease
and may become so), (ii) E Exposed individuals (those i n the incubation period), (iii) I Infectious
individuals (those able to spread the disease) and (iv) R Recovered/Removed individuals
PLOS ONE I https://doi.org/10.1371/journal.pone.0287959 July 13, 2023 1/17
PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
com/forms/d/e/
1FAIpQLSfncCCbPngFtfdHV8XMBcU5IGiPMMRGj-
6BYVLw2Nj6PvXOFA/viewform or by sending
request to the Department of Data Analysis of The
Institute of Health Information and Statistics of the
Czech Republic Jiri.Jarkovsky@uzis.cz.
Funding: Authors received financial supports: LP,
Mathematical and Statistical modelling projects
(MUIWA/1615/2020, MUIWA/1342/2021 and
MUNI/A/1132/2022, funder Masaryk University,
www.muni.cz), LP, VE, OM, JJ, LD Online platform
for real-time monitoring, analysis and management
of epidemic situations (MUNI/11 /02202001 /2020,
Masaryk University, www.muni.cz), and LD the
Czech Republic Operational Programme eHealth
and Rare Disease (CZ.03.4.74/0.0/0.0/151025/
0015811, Ministry of Health of the Czech Republic,
www.mzcr.cz). VE also received financial support
from the RECETOX research infrastructure
(Ministry of Education, Youth and Sports of the
Czech Republic: LM2018121, www.msmt.cz/
struktu ralni-fondy/g rantove-projekty), the
CETOCOEN EXCELLENCE Teaming 2 project
supported by Horizon2020 (857560, EU, https://ec.
europa.eu/programmes/horizon2020/en/home),
the Czech Ministry of Education, Youth and Sports
(CZ.02.1.01 /0.0/0.0/1 AJ43/0009632, www.
msmt.cz/strukturalni-fondy/grantove-projekty),
RECETOX Rl project (CZ.02.1.01/0.0/0.0/161013/
0001761, www.msmt.cz/strukturalni-fondy/
grantove-projekty).
Competing interests: The authors have declared
that no competing interests exist.
(those who cannot become infectious anymore; they are either recovered or deceased). Application
of SEIR-type models to the COVID-19 pandemic is problematic due to the existence of
a non-observable variable of infectious that spread the virus asymptomatically. This problem
was studied even before SARS-CoV-2 emergence, and an additional cohort was introduced to
model respiratory infection outbreaks, such as the first SARS outbreak in 2002-2003 [9,10].
These SEIAR models are modifications of the standard SEIR models. Papers [9, 10] provide
valuable information on their derivation, R0 computations, and other related topics. In Supplement,
you will find a basic comparison of the SEIR and SEIAR models and the derivation of _R0
for the SEIAR model. The main issue addressed in the manuscript is the estimation of the
unreported A compartment.
In the real-world data, positive subjects are not necessarily symptomatic and, on the contrary,
not all symptomatic subjects are detected. Without knowing what part of the epidemic is
observed, we cannot rely on case reports. This detected part of the infectious individuals is
known as the ascertainment rate (AR) and it has to be estimated. Although the compartmental
approach is widely used for modeling COVID-19 epidemics in a broad modeling community,
most of the time the observational layer estimation is missing [11,12], or if an undetected
infectious compartment is part of such a model, the A R is calibrated as an unknown parameter
along with other parameters, as demonstrated in several other studies [13-15]. Information
about both the observed and unobserved proportions of the epidemic is crucial for estimating
predicted admissions to hospitals and setting up effective and timely interventions [16]. Of
course, rough estimations can be performed and used as a fixed value for a long period [17],
but changes in the testing and tracking strategy may significantly affect the volume of the
observed epidemic and lead to biased estimates of disease spread.
Excess deaths certainly provide the most accurate information about AR, but their usage
is possible only retrospectively. Similarly, the case fatality rate (CFR) is unsuitable for modeling
an ongoing epidemic in real-time due to the long delay from infection to death [13,18].
Another option to estimate A R is to use virological and serological participatory surveillance,
but again it cannot be done in real time and must also be done retrospectively.
According to some estimates [19], only 31 % of people with symptoms similar to C O VID-19
sought medical attention in a monitored period; such results were confirmed by serological
studies [20].
Here, we offer an approach to A R estimation that can be used both retrospectively and in
real time. We demonstrate the applicability of our approach of real-time A R estimate on data
from hospital admissions due to SARS-CoV-2 infection in the years 2020-21 and show our
results on a specific model. In SEIR models, the number of susceptible individuals determines
the dynamics of the epidemic, so the peak is primarily driven by the rate of connectivity and
mixing. SEIR-type models can still be used if there are reasonably good estimates of the factors
influencing the transmissibility rate (related to mixing, environmental variables, etc.), but
obtaining this information is difficult when a new virus emerges or when new control interventions
are introduced. Therefore, we used modified ZSEIAR model in our case study of
COVID-19 in the Czech Republic that uses the dependence of transmissibility purely on
mobility data, and all other dependencies such as environmental or social variables are moved
to be optimized by feeding Z to S, which replaces social network connectivity or temperature
dependence or other unknown variables to estimate the driving force of real-world epidemics.
The model is described in the Supplement in detail and is enclosed as an R script.
Another challenging issue is that even studies of the early pandemic period monitoring the
almost immune-naive population showed that mild cases of COVID-19 are significantly
understated because the severity of symptoms of COVID-19 is age-dependent [21,22]. We
will therefore address the age-dependency of our A R estimate in the paper as well.
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
Research methods
Ascertainment rate independent estimation principle
The basic principle for the estimation of the moving A R estimate is based on the Bayes rule for
conditional probabilities and on the assumption that the average probability of hospitalization
of an infected person P(H) at time t is given or estimated. We also derive an estimation procedure
depending on the age structure.
We will use the following notation:
(PI) P(Det\H)—the probability that a person admitted to a hospital for the infectious disease
was previously detected at time t; to estimate this probability, we use the 7-day or 14-day
moving proportion of patients reported before admission to hospital from all patients hospitalized
for the infectious disease (including those not detected before admission to hospital)
with respect to the date of the positive test report
(P2) P(H\Det)—the probability that if an individual was detected at time t, he or she would be
hospitalized; to estimate this probability, we use the 7-day or 14-day moving proportion of
all reported hospitalized patients detected before admission to the hospital, from all that
time already detected subjects with respect to the date of the positive test report, that is,
except those detected in the hospital afterward who are part of the undetected compartment
at time t
(P3) P(H)—the probability that an infected individual is / was / will be hospitalized for the
infectious disease (irrespective of whether it was detected or not) at time t; should be
derived for each community/country separately since it is highly dependent on the structure
of the age of the population, a possible estimation method is described below
We can calculate the estimation ofp(t) = P(Det) as the moving average relative to the date t
of the positive test report according to the Bayes formula
. . P(Det\H)P(H)
pit) = P(Det) = - / / \ '. 1
y y
' y
' P(H\Det) y
'
Therefore, to summarize the principle of estimation, we estimate the invisible part of the
epidemic using knowledge about late-detected individuals (i.e., undetected infectious subjects)
who end up in hospitals with a severe symptoms of the infectious disease and who are confirmed
there afterward. At least a 7-day window has to be used for the moving average due to
the week oscillations. A 14-day window is possible in case there is a low number of hospitalizations
since a low number of hospitalizations results in greater variability of all estimates. O n
the other hand, 14 days moving average flattens the curve also in case of sudden changes,
which can hide early information about the outbreak.
A n unknown value in the Bayes formula (1) is the specific probability of hospitalization P
(H), which needs to be estimated, for example, from surveillance studies [23-25]. Other methods
can also be used without an additional surveillance data set: (1) infection-fatality rate (IFR)
and hospitalization-fatality rate (HFR) estimates, since the average probability of hospitalization
due to the infection is HFR/IFR, (2) nationwide non-indicated antigen test screening data
with comparison to hospitalized and observed infectious cases or retrospectively also (3) excess
deaths data. A l l these methods are presented in the Results section for the data on the C O V I D -
19 epidemic in the Czech Republic. These indirect pieces of evidence support our estimate of
the average probability of hospitalization due to COVID-19 infection for the Czech population
in the monitored period. The approach can be adapted by analogy for use in another commu-
nity/country.
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
Because the probability of hospitalization due to an infectious disease is usually age-dependent,
we propose a method based on a rough estimate of the relationship between total and
age-specific probabilities of hospitalization and the age structure of the monitored population
divided into n age categories, that is
n
P(H) = ^>,.P(H,.), (2)
1=1
where w, > 0, satisfying Xw=i w
i =
1>a r e
proportions of the age-structured monitored population,
and P(Hi) are the age-specific probabilities of hospitalization due to infection for given
i = 1,..., n. In case of COVID-19, the minimum necessary division of the weighted average is
into two groups: non-seniors (65-, i.e. individuals under 65) and seniors (65+, i.e. individuals
over 65).
Necessary data for real-time AR estimation
To estimate A R in real time, it is necessary to continuously collect both hospital data and laboratory
data. The minimum personal record must include age or age group, date of positivity,
and if hospitalized, then date of admission to the hospital due to the infectious disease.
In the Czech Republic, data on reported SARS-CoV-2-positive individuals and COVID-19
patients and their hospital stays are collected and processed by the Information System of
Infectious Diseases (ISID) almost in real-time [26]. ISID includes a complete health care information
record for a person; we used a dataset (shared at [27] that includes variables describing
the infection case: district and regional number, sex, age group (0-19,20-64, over 65), date of
the first symptom, date of sampling collection, date of a positive result, date of report, date of
isolation, date of admission to a hospital, end of hospitalization, date of recovery, date of
death.
A n unspoken but important assumption is that all hospital patients are tested for the infectious
disease. In case of a positive test, the result should be entered into the data collection system
even if the patient had not tested positive before admission. However, this is common
practice in most developed countries. In the case of COVID-19, persons requiring hospital
health care with severe respiratory problems are almost all tested for SARS-CoV-2 immediately
after admission.
Ascertainment rate usage in modeling epidemics
SEIR-type models are commonly used for epidemic monitoring, primarily to predict the number
of severe cases that require hospital care. It is common practice to assume a fixed proportion
of observed infectious or hospitalized individuals during the course of an epidemic, unless
new drugs are discovered, the population structure changes significantly, or other major epidemiological
factors come into play. The compartment of hospitalized subjects H is usually calibrated
through observed epidemics as its fixed part [17,28-30]. Using accurate real-time data
to estimate ARp(t) can improve epidemic and hospitalization surveillance. Fig 1 shows a
scheme of the classical SEIR/SEIAR model with the observed (bright colored circles with solid
border) and unobserved (pale colored circles without border). Here, P(H) is the probability of
hospitalization due to an infectious disease, as discussed in the previous subsection.
Results
Our main result is a novel method to estimate A R and its changes that can be used in real time.
To show this method's successful use, we present a case study of the COVID-19 epidemic
PLOS ONE I https://doi.org/10.1371/journal.pone.0287959 July 13, 2023 4 / 1 7
PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
Observed epidemics
© P(H) f i V ^
Unobserved epidemics
Fig 1. SEIAR model scheme. The compartment i-f of hospitalized patients due to infectious disease is recruited from the exposed compartment £ with
a given probability P(H). We can estimate the size of the exposed compartment using the estimation of the ascertainment rate p(t) at time t from the
reported epidemic (i.e., the observed cases I), while the compartment A remains unobserved.
https://doi.org/10.1371/journal.pone.0287959.g001
based on data from monitored period from March 2020 to May 2021 in the Czech Republic,
where we used this real-time estimation in a model that is described in detail in SI Appendix.
The model code in R is also included.
Probability of hospitalization due to SARS-CoV-2 infection—Derivation of
the age-structure dependence
In our case study on the Czech population, the probability of hospitalization due to SARS—
CoV-2 infection described by formula (2) can be simplified using the minimum division into
non-senior and senior population parts as
since about 1/5 of the population is over 65 years old in the Czech Republic [31]. On the contrary,
the majority of hospitalized patients were in the elderly population. The anonymized
dataset [27] from ISID used for real-time COVID-19 monitoring contains information about
age. There are three age groups: those aged 0 to 19 years (0-19y) and 20 to 26 years (20-64y)
will be referred to as non-senior or 65- population, whereas those aged 65 years or older will
be referred to as senior or 65+ population. The long-term proportion of the hospitalized population
over 65 was 3/4 resulting in a risk ratio of 12 for the senior versus the non-senior population.
Similarly, a rough estimate of the age-dependent hospitalization risk ratio can be made
for other data sets.
This rough estimate is also consistent with the log-linear relationship between the infection
fatality rate (IFR) and age, as published in a study by Levin et al. [32] and our recent analyses
of Czech data [33, 34]. Using this estimate and the known age structure of the reported infectious
individuals, we obtain a rough estimate of the time-dependent probability of
PCR(H) fP(H6 5 _)+iP(H6 5 + ), (3)
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
hospitalization that varies according to the age of the infected
P(H) = Pt5-P(H6S_) + p + + P ( H 8 B + ) = ( l + l l p + + ) p ( H 6 5 _ ) , (4)
where pjj"B+ andpg5 _ are 7-day moving averages of the senior and non-senior (children
included) population ratio in the reported cases. We obtained the dependence of the probability
of hospitalization on the ratio of the infected senior population.
Note that it is necessary to use some average level of probability of hospitalization due to
COVID-19 infection in the Czech Republic PCR(H). This level can be estimated using various
methods based on indirect evidence, but high precision is not important for short- and
medium-term predictions of hospital admissions. However, a good estimate of this average
level of hospitalization probability is necessary for long-term forecasts and herd immunity estimates.
In the following, we present the options that can be used to estimate the average level of
hospitalization probability. For the Czech Republic, we used PCR (H) = ^, which is employed
later on. Average PCR(H), estimations (3) and P(H65+) = 12P(H6 5 _) imply formula (4) in the
form
P ( H ) = I i 5 ( l + l l p + + ) .
The method based on H F R and IFR ratio must rely on data from countries where contact
tracing was performed more thoroughly. Countries such as South Korea, Malaysia, Thailand,
and Singapore showed an observed case-fatality ratio 0.5% [35-38], which is assumed to be
close to the IFR [18,22, 32, 39,40]. Our estimate PCR{H) = ^ together with an average of 22%
H F R (from M Z C R data [27]) during autumn 2020 gives IFR of 0.44%.
The estimate of the average probability of hospitalization due to the infection can be supported
by surveillance studies, or we can use wide non-indicated or other representative testing
data as full-scale testing of employees, schoolchildren, and students. The nationwide nonindicated
antigen test screening in the Czech Republic showed an average test positivity of 5%
in December 2020. This is another strong supporting argument for estimating the probability
of hospitalization as P c f i ( H ) = ^ since the consequence of this PC J ?(H) level was the estimation
of A R around 0.25, and half a million infected active cases fully corresponds to the observed
quarter of 120,000 active cases at that time.
Retrospectively, excess mortality can be used. In the Czech Republic, excess mortality during
autumn and winter 2020 (from [41]) is highly correlated with the epidemic wave and gave
approximately 0.5 excess unreported deaths to each reported patient who died with C O V I D -
19 (using an anonymized dataset created from ISID data that can be downloaded from the M Z
CR portal [27]), see Fig 2.
Since about half of the hospitalized were detected only after admission to a hospital (which
implies that half of them were not reported during the infectious period), meaning that seriously
ill people were detected late, we can deduce that the observed part of the infectious compartment
could be around 33% at the beginning of the year 2021 which corresponds to the A R
estimate at that time and is also in agreement with a study published by Pullano et al. [19].
A n improvement can be made in the age structure dependence of the probability of hospitalization.
We improved it to depend on three age groups from May 2021. Until then, the
probability of hospitalization in the Czech Republic was distinguished only between two age
groups (under/over 65), but this started to be non-sufficient since COVID-19 spread across
younger people and mandatory testing has been introduced in schools. The children were
hardly tested before due to the age specificity of the symptoms. Retrospective data analysis
shows that people under 20 were hospitalized with almost zero probability and three age
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
2500
2000
deaths 2011
deaths 2012
deaths 2013
deaths 2014
deaths 2015
deaths 2016
deaths 2017
deaths 2018
deaths 2019
deaths 2020
deaths 2021
+ reported 2020
+ reported 2021
-19 deaths 2020
-19 deaths 2021
-500
10 20 30 40 50
Week number
Fig 2. Excess deaths in the Czech Republic. Excess deaths in weeks of the years 2020 (red thick dashed line) and 2021 (blue
thick dashed line) according to average week deaths in years 2011-2019 (thin solid lines show excess deaths in weeks of the years
2011 to 2019) [41], weekly reported SARS-CoV-2 positive subjects that died (red and blue thick solid lines), and weekly reported
deaths (red and blue thin solid lines) [27]. Our model works with data related to the date of the report, including fatal reports
(thick line), and excess deaths (dashed line) and COVID-19 deaths (thin line) are related to the day of death.
https://doi.org/10.1371/journal.pone.0287959.g002
compartments (under 20,20-65, and over 65) appear to be enough for the basic improvement
of the P(H) estimate in the form
where p+_ is a 7-day moving average of the under 20 population ratio in the reported cases.
The need for this modification is also visible in Fig 3.
Ascertainment rate estimate usage—Epidemic monitoring
We have been using a compartmental model including A R for epidemic monitoring during
the years 2020-21 in the Czech Republic (see SI Appendix). The initial outbreak is set to the
time when the first person in the Czech Republic tested positive for SARS-CoV-2 at the beginning
of March 2020. The period described here includes the dominance of wild-type and alpha
variants of SARS-CoV-2 until May 2021. Later, we used more complex models of SEIAR and
SEIARS types, which included data on vaccinations and reinfections. Fig 3 shows the estimated
exposed using real new cases divided by A R estimate, real new cases, real daily admissions to
hospitals and deaths (black circles) and the optimized fit to hospitalization data (solid blue
line) with the A R estimate average baseline PCR{H) = ±. You can see here that the model
explains all the compartments in the whole period except for a short period of Christmas. This
period was burdened with several unknowns. The main one was the arrival of the new alpha
variant; in the Czech Republic at that time, there was no surveillance either by variant multiplex
PCR method or sequencing, so we could not change the model parameters according to
the variants in proper time and proportion. The second unknown was people's behavior
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
Exposed to SARS-CoV-2 SARS-CoV-2+ new cases
2020-03-01 2021 2020-03-01 2021
New daily COVID-19 hospitalized Daily COVID-19 deaths
2020-03-01 2021 2020-03-01 2021
Fig 3. Epidemic monitoring. Exposed (new cases divided by AR estimate), new cases, hospital admissions and daily deaths (black), and model fit
optimized to hospital admissions (blue).
https://doi.org/10.1371/journal.pone.0287959.g003
during Christmas time. They postponed hospitalizations more than usual. A n d the third was a
significant change in testing strategy before Christmas—the government introduced free antigen
tests for everyone. This should be the minor problem since the estimate is robust against
testing strategy changes outside hospitals. Despite these unknowns, the model fits very well.
We used the A R estimate and the transmissibility rate as strictly dependent on the number
of contacts or mobility. Other dependencies were included in the estimate of affected susceptible
compartment S, which was supplied from an additional fitted compartment. This simulated
the first outbreak very well from April 4, 2020, until the summer; see Fig 4. During the
first 35 days, the laboratories' capacity and the testing and tracing system capacity increased
substantially. For this period, we could not even estimate the A R due to the small number of
hospitalized. We let it constantly grow due to the lack of data at the beginning of the outbreak,
since it corresponds to an increase in test capacity in this period, and started to compute the
moving average of A R from time t = 35 (t = 0 marks the first detection day—March 1,2020).
Assuming that the probability that the infected person will be hospitalized is P C K ( H ) = ^ we
calculatedp(35)=0.13 from the data of hospitalized subjects by Bayes formula (1) at the beginning
of April 2020.
The usefulness of using moving A R is shown in Fig 3. O n a logarithmic scale, it shows the
first two waves (the waves differed in scale), including the period of July 2020. During this
period, a local outbreak in O K D mines led to mass PCR testing in the whole of O K D mines
employees (OKD mines carry out hard coal mining in Karviná part of the Ostrava-Karviná
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
SARS-C0V-2+ new cases (constant AR) SARS-C0V-2+ new cases (time-dependent AR)
0
CD
1 8
CD
E
0
Si
1 s
Ascertainment rate (AR) New daily COVID-19 hospitalized
S 0
o
Q.
O !M
May Jul Sep Oct May Jul Sep Oct
Fig 4. Comparison of using constant AR and moving AR. The model (blue) fitted to the data (black) of new hospitalizations (bottom right)
reproduces the actual new cases (top black) only when using moving AR, whereas using constant AR does not account for either the number of cases in
the first wave or the fluctuation due to the change in testing during the OKD outbreak (Jul).
https://doi.org/10.1371/journal.pone.0287959.g004
district in Moravian-Silesian region). Since the prevalence of COVID-19 was low in other
regions of the Czech Republic at that time, this outbreak enormously changed AR. It is obvious
that the model fitted to hospitalizations corresponds to real data of new cases compartments
only in the case of using moving AR, while it does not in the case of using the average constant
A R (different A R in the two waves visibly led to different C F R or hospitalization rate in the
two waves); therefore, the fluctuation due to the change in testing during the O K D outbreak is
not reproducible at all.
Ascertainment rate estimate usage—Early warning
The A R estimate can be used as an early-warning indicator. As we saw in the previous subsection,
retrospective analysis provides a good explanation of the O K D wave: new cases did not
significantly affect the dynamics of hospitalized patients and deaths (see Fig 4). Similarly, if the
increase in the number of new cases is offset by a decline in AR, the outbreak may remain
hidden.
Actually, the A R estimate shows the likely decrease in the effectiveness of tracing by the
Regional Public Health Authorities (RPHAs), which occurred in the second half of September
(Fig 5, grey period). A significant decrease in the A R estimate was a signal of contact tracing
and testing system overload. There are other declines—during the epidemic waves (see OctNov).
But these are related to the peak period of the epidemic. The detection rate was limited
due to a more or less linear increase in capacities of RPHAs and call centers, but an
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
2020-10-01
CD
o
LO
o
o
CO
o
CM
o
o
o
2020-09-15
Sep Nov Jan
Fig 5. Insufficiency of tracing in the second half of September 2020. The peak of the second wave is marked on 2020-11-04. The dates in the chart
denote: 2020-09-01—the beginning of the school year; 2020-09-15 to 2021-10-01—tracing overload; 2021-11-04—peak of the second wave; and 2021-
01-01—New Year.
https://doi.org/10.1371/journal.pone.0287959.g005
approximately exponential increase of cases. The September decline before the wave was an
early-warning signal.
Between mid-September and mid-December 2020, we observed a high correlation of the
proportion of newly hospitalized non-detected in the community 1 - P(Det\H) and the subsequent
number of COVID-19 patients treated at intensive care units (Fig 6, correlation coefficient
0.92). This confirmed the suitability of this parameter as a good indicator of the future
burden of hospital care. Based on the fact that the positivity of the tests changed after the introduction
of antigen screening tests (and therefore ceased to be a proxy variable for AR), we
instead introduced this indicator to the epidemic Risk Index in the Czech Republic.
Ascertainment rate estimate usage—Effectiveness of non-pharmaceutical
interventions (NPIs)
Another possible application of A R is the opportunity to monitor the effectiveness of NPIs retrospectively.
By estimating the actual number of infections, it is possible to visualize the data,
including the 'invisible' part (with a proper delay given by the incubation period and the mean
time to the report), and discuss directly how the established NPIs have possibly worked in the
Czech Republic. Fig 7 shows several specific dates: 2020-09-01 (beginning of the school year),
2020-10-22 (partial lockdown), 2020-11-14 (announcement of the first measure release, partial
school reopening 2020-11-18), 2020-12-03 (shops reopening), 2020-12-27 (a partial lockdown),
2021-01-11 (partial school reopening), and 2021-02-23 (mandatory and widespread
introduction of respirators followed with partial lockdown).
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
60% 10000
Fig 6. Correlation with ICUs. Proportion of undetected in the community (PU) and number of COVID-19 patients
treated in intensive care units after 30 days (ICU30d).
https://doi.Org/10.1371 /jou rnal.pone.0287959.g006
Time series for fits to cases, estimated exposed, hospitalized, and deaths are depicted in
Fig 2.
Discussion and conclusion
In the paper, we presented a method for moving A R estimates that can be continuously monitored
during the epidemic in real-time. It can also be used retrospectively. The method is
based on the Bayes formula and compiling information from independent datasets of positively
tested infectious individuals and hospital admissions with a principal diagnosis of the
monitored infectious disease.
2020-10-22 ! 2021-02-23 Time in dates
2020-12-27
Fig 7. Exposed population and NPIs. Exposed estimated using new cases divided by AR estimate compared to the introduction of
NPIs: 2020-09-01 (beginning of the school year), 2020-10-22 (a partial lockdown), 2020-11-14 (announcement of the first measure
releases after 2020-11-18), 2020-12-03 (shops reopening), 2020-12-27 (a partial lockdown), 2021-01-11 (partial school reopening),
2021-02-23 (mandatory respirators).
https://doi.org/10.1371/journal.pone.0287959.g007
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
The main advantage of this method is that the estimate is rigorous and independent, so
there is no need for calibration in a compartmental model with undetected compartments.
From 2021 onwards, many modeling teams used mechanistic compartmental models as a
starting point and developed their own models to monitor the epidemic [42]. Those models
are collected on platforms such as the European Covid-19 Forecast Hub [43,44], coordinated
by the European Centre for Disease Prevention and Control (ECDC). European Covid-19
Forecast Hub is coordinated by E C D C , and the models for the ongoing COVID-19 pandemic
are collected and monitored at this platform; model performances are evaluated, and ensemble
models are made [43,45]. In some models from the European Covid-19 Forecast Hub, the
authors empathize with the role of the observable part of the epidemic (such as the FIAS_FZJ-EpilGer
team [46], or the DSMPG-bayes team [47]) and use their own methodology to
model it. The compartmental approach is used in a broader community than just at E C D C
Forecast Hub, but usually either A R estimation is missing [11,12] or it is calibrated as an
unknown parameter [13-15]. To the best of our knowledge, there is no other modeling group
that estimates the time-varying observable part of the epidemic independently from hospital
data in real time. We are unaware of any other published method of estimating time-varying
real-time A R (except using proxy variables that include the proportion of positive tests, unfortunately,
sensitive to changes in testing strategies). We submitted predictions with outstanding
evaluations [48] since March 2021 at the E C D C Forecast Hub [43], our model continuously
showed high-quality predictions for the Czech Republic. The relative measure of our model's
two-week forecast performance called relative weighted interval score (relative WIS [44]) evaluated
for the first 30 weeks was the best of all modeling teams (0.37) and even better than the
Eurocovidhub-ensemble score (0.42). We believe that if a similar real-time hospital data collection
system exists in other countries, the method may significantly refine and simplify many
existing models despite the type of infectious disease.
The A R estimate can also monitor the overall increase of infected in the population and real
prevalence, so it can monitor possible approaching herd immunity (in case of a low rate of
reinfections) and better predict possible scenarios for later epidemic dynamics. For the Czech
Republic, this provides indirect evidence that it was very far from approaching herd immunity
in summer 2020. When we calibrated and optimized our model with the average level
PCR{H) = ^, we obtained a total number of at most 80,000 overall infected individuals (0.8%
of the population) in May 2020 (after the first wave). This estimate is entirely consistent with
the SARS-CoV-2-CZ-Preval prevalence study [49] that estimated the range of prevalence values
for SARS-CoV-2 antigen positives between 0% and 0.22% in regions where active cases at
that time corresponded to 40 positively tested per 100,000 inhabitants and between 0% and
0.4% in regions where active cases at that time corresponded to 140 positively tested per
100,000 inhabitants.
If it is possible to use the method to calculate the A R estimate at the regional level, it can be
used to compare the situation in selected regions. A problem for regional use may be the high
variability in the case of low numbers of hospital admissions, which can be reduced by extending
the moving average window from 7 to 14 days. Supplement elaborates on these limitations
and possess an illustrative example with a restriction and comparison of the A R estimate calculation
from the whole Czech Republic and the Moravian-Silesian region, enclosed in R code
AR_comparison.R. Regional reduction of the data at periods of low disease prevalence implies
high variability and inaccuracy of the estimate, whereas at times of outbreak, stratification and
calculation for the affected region gives a better estimate for the area. Moreover, assuming that
there is no significant difference in the probability of hospitalization in individual regions, we
can measure the relative A R by the ratio of regional A R with respect to the reference region.
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
This measure was used as one of the regional tracing effectiveness indicators in the Czech
Republic during the epidemic in 2020-2021 in regional weekly reports for key stakeholders.
There are also limitations of the described approach. Some principal limits are related to
data collection. First of all, hospitalized subjects' data must be collected continuously in almost
real-time. Data collection may be inaccurate if, for example, hospitals are overwhelmed. In the
Czech Republic, there has never been a period in which emergency care was not provided during
the COVID-19 epidemic, but during some periods when case numbers peaked, there were
delays of several days in reporting. The second limit is the lack of data due to low hospitalizations
(for example, during European summer). Extending the moving window for calculations
flattens the A R estimate. There is also more significant variability at times of lower hospitalization
numbers. However, the technique seems robust to different testing regimens because it
relies on unmissable severe cases (and if oxygen support data collection in real-time is available,
it can also be used). Various testing regimes were held in the Czech Republic (also screening
with antigen tests) during the monitored period, and the method was effective over time.
There are limitations related to the used method based on the estimated fixed average probability
of hospitalization due to the disease. Changes in the virulence or other characteristics of
viral variants must be considered. The estimate PCR (H) = ^ is valid for wild-type coronavirus
lineages. In 2021, a new variant alpha B. 1.1.7 spread in the Czech Republic and dominated.
According to studies [50, 51], we increased the transmissibility rate to a 1.5 times higher level
during this time in the model; higher average probabilities of hospitalization and death were
also to be taken into account. The data show interesting information that the mutation did not
affect the ratio of death rates in hospitals between categories 65- and 65+, more precisely P
(Death|H and 65-)/P(Death|H and 65+) = 1/4 during the entire epidemic before the vaccination
introduction. Let us note that the ratio then changed to the disadvantage of non-seniors,
which can be considered evidence of the effectiveness of vaccines against hospitalization
(seniors were prioritized in the vaccination schedule). The unaltered ratio of non-senior and
senior death rates implies that the increase in the death rates due to the new variant in both
age categories must also be proportional. Our model fits a rough estimate of a 10% increase in
the hospital death rate and an approximately 25% increase in the probability of hospitalization
during the spring outbreak of a new variant B.l.1.7 in 2021.
Another issue that has to be incorporated is the ongoing vaccination process, which
strongly influences the probability of hospitalization. It is a more complicated problem. This
paper shows the modeling method on data with the model fixed and conserved in time before
summer 2021 (before delta variant B.1.617.2 dominance). In the model whose predictions we
continuously published on the E C D C hub until June 2022, we solved this problem in a simplified
way, namely by reducing the transmissibility rate and the probability of hospitalization
and death proportionally to the percentage of vaccinated people based on effectiveness computed
in [33, 34]. Various later recalibrations and minor model changes were continuously
specified in the description of the M U N I _ D M S - S E I A R model at the E C D C Forecast Hub [43]
as they arose over time due to new variants' emergence or vaccination, etc. The model
described in the Supplement is limited to the period when the population was mostly unvaccinated—not
because the A R estimation method could not be used, but because of the chosen
model. The effects of vaccination and its waning change the probability of hospitalization of
the vaccinated, and the models we used further were more complicated. We wanted to show
the usage of A R in various ways in the most simple but realistic model. The lineages from the
delta variant B.l.617.2 dominance period are not included in this paper for the same reason.
Challenging issues arise with lineages evolving from the omicron variant, where the probability
of hospitalization dropped significantly. The omicron variant is really a game-changer
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
that shows an increase in the number of cases included in hospitalization dataset that are
"with" covid, and not "for" covid when it is the principal diagnosis. Patients hospitalized for
COVID-19 generally need some respiratory support, so if there is enough data, those data can
be used instead of only hospital admissions data. In our data set, we also have the date of the
oxygen support recorded. However, patients usually arrive in a serious condition with a delay,
so it is more convenient to work with hospital admission data. A n open dataset [52] shows that
during the analyzed period 2020-21, the proportion of the most severe cases (ICU need) was
very stable. That also justifies using hospital admissions data in the Czech Republic in the
monitored period. That is not true for the omicron variant period. Moreover, the level of reinfections
became significant during the omicron variant dominance period. In that case, the
SEIARS-type model had to be used instead of the SEIAR-type model.
Another issue that arises from using A R in the SEIR-type model using data is related to
data stratification. In the case of the Czech Republic, stratification is not necessary and our
approach can be applied without stratification due to the relatively small size of the country
and the homogeneity of population density. However, in the case of countries with varying
population densities, stratification may be necessary. Our approach can be easily adapted to
include such stratification, by using region-specific A R estimates, mobility data or other relevant
variables to capture the spatial dynamics of the epidemic.
We believe that a moving estimate of A R is essential for monitoring the ongoing epidemic,
and our approach brings a credible estimate in almost real time. We hope that our results will
be helpful both for the modeling community in other countries and for further research in the
field, as many countries collect data from hospitals [53].
Supporting information
SI Appendix. Supplement.
(PDF)
SI Data. Data and R code.
(ZIP)
Acknowledgments
We would like to express our gratitude to Academic Editor Prof. Seth Blumberg, as well as to
all anonymous reviewers who assisted in improving the manuscript.
Author Contributions
Conceptualization: Lenka Přibylová, Veronika Eclerová, Tomáš Pavlík.
Data curation: Jiří Jarkovský.
Formal analysis: Lenka Přibylová, Veronika Eclerová, Ondřej Májek.
Funding acquisition: Ladislav Dušek.
Investigation: Lenka Přibylová, Veronika Eclerová, Ondřej Májek.
Methodology: Lenka Přibylová, Veronika Eclerová.
Project administration: Ladislav Dušek.
Resources: Ladislav Dušek.
Software: Veronika Eclerová.
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PLOS ONE Real-time ascertainment rate estimate for modeling the spread of infectious disease
Supervision: Lenka Přibylová, Ladislav Dušek.
Validation: Lenka Přibylová, Veronika Eclerová, Ondřej Májek.
Visualization: Lenka Přibylová, Ondřej Májek.
Writing - originál draft: Lenka Přibylová, Veronika Eclerová, Ondřej Májek.
Writing - review & editing: Lenka Přibylová, Veronika Eclerová, Ondřej Májek, Jiří Jarkovský,
Tomáš Pavlík, Ladislav Dušek.
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