During the second SARS-COV-2 pandemic wave in Slovakia (August 2020 - June 2021) reported data did not capture real health care demand and the capacity of hospitals 1 Introduction and mechanical ventilations for COVID-19 patients was exceeded. Real time quantitative polymerase chain reaction 1.1 (RT-qPCR) and lateral flow antigen (LFAg) test incidences were strongly biased due to a variation of the total volume of tests administered and sample selection. Also, confirmation of COVID-19 related deaths was often significantly delayed. Available data thus failed to characterize the true extent of the pandemics. To fill this gap we perform a retrospective analysis of the time series of epidemic indicators and estimate dynamics of the true pandemic state in Slovakia during the pandemic wave. We estimate that on average approximately 20.0% more hospital beds and 19.2% more mechanical ventilators were needed in hospitals than reported bed occupancy in Slovakia during the period November 2020 March 2021. Our estimates rely on a linear relationship between total adjusted incidence in a form of weighted linear combination of RT-qPCR and LFAg incidences and hospitalizations data lagged by 8 days. The linear relationship systematically emerges before and after the epidemic peak and the real epidemic state is estimated by a projection of the observed data on the corresponding linear manifold. The methodology is applicable to epidemic data worldwide.
Reliable data are critical for monitoring of epidemic dynamics and for decision making on public health policies. Despite a vast amount of data on SARS-CoV-2 pandemics there is a large degree of uncertainty in all types of epidemic data including infection incidence, number of hospitalized patients with COVID-19 and number of COVID-19 related deaths.
The sources of uncertainty in the data are diverse:
observed incidence measured by testing programs is limited
by sample size, sample bias, and test parameters,
hospitalization data are subject to limited bed, equipment,
and personnel capacities, particularly during epidemic
peaks, and data on COVID-19 related deaths are limited by
methodological issues including sample bias and staff
shortage during the epidemic peaks [
The uncertainty in data has consequences. Public health policies depend on observed epidemiological data and under- or over- reporting may lead to wrong decisions. It also creates a significant hurdle in epidemic modelling as limitations in observed data impede model calibration. This in turn makes the decision process on health policies and other epidemic mitigation measures even more difficult and partially blind. 1.2
We combine multiple publicly available data sources to identify a robust linear relationship in data that emerges outside of the periods of severe epidemics. During these periods we assume that the true pandemic state is also governed by the same linear relationship, however, the limitations in the observed data violate it and the data points do not lie on the identified linear pandemic manifold. We estimate the true pandemic state by a projection of the observed data onto the linear manifold. The particular form of the projection (orthogonal projection in normalized data sets) reflects an equal distribution of
beds for COVID patients in the Slovak Republic by approximately 20.0% during the peak of the pandemic wave in December 2020 – March 2021. The demand exceeded the capacity of the hospital beds with mechanical lung ventilation for COVID patients in the Slovak Republic by approximately 19.2% during the pandemic wave in November 2020 – February 2021.
The average clinical sensitivity of the lateral flow antigen tests compared to RT-qPCR tests was approximately 37% during the pandemic wave in October 2020 - June 2021 in the Slovak Republic. uncertainty between various sources of the data. The estimate of true pandemic state allows us to measure the extent of a lack of capacity of hospitals and mechanical ventilations during the peak of the pandemic wave.
We apply the methodology developed in this work to the
epidemic data from the Slovak Republic during its second
SARS-CoV-2 pandemic wave (August 2020 - June 2021).
For a period of more than a month during this wave
Slovakia ranked within the top 3 countries with the largest
number of reported COVID-19 related deaths per capita in
the world [
While most countries report their RT-qPCR incidence as a
diagnostic characteristic of their epidemic situation, during
the studied period Slovakia used two types of tests for
monitoring. Individuals could choose between an RT-qPCR
and an LFAg test. While the scope of RT-qPCR test
program was limited, the LFAg testing was conducted on a
massive scale with mass antigen testing in
OctoberNovember 2020 [
Hospitalizations are lagged by 8 days behind the incidence , MLV are lagged for additional 14 days (see Section 3.2 for details) and scaled to fit hospitalizations (see Section 3.3 for details) in March-June 2021.
Throughout this work we use the following public data
sets [
The RT-qPCR test detects viral genetic material
through the reverse transcription quantitative
polymerase chain reaction. The sample is collected
using two nasopharyngeal and one throat swab.
Various unidentified types of RT-qPCR tests were
used during the second pandemic wave in Slovakia
(August 2020 – June 2021). The tests were available
to the public for free in case the individuals were
indicated by the Regional Public Health Authority or
self-indicated due to a presence of COVID-19
symptoms or a close contact with an infected
individual. RT-qPCR tests were also offered on a
commercial basis to the general public. The
incidence is reported daily by the National Health
Information Center [
The LFAg test detects specific SARS-CoV-2
antigens in nasopharynx through a rapid lateral flow
chromatographic immunoassay. Slovakia used
almost 43 million rapid lateral flow antigen tests
during the studied time period. The majority of the
tests used were STANDARD Q COVID-19 Ag (SD
Biosensor) complemented by Panbio COVID-19 Ag
(Abbott). Biocredit COVID-19 Ag (RapiGen) was
used to a limited extent. The tests were available for
free to the general public on a mass scale. The LFAg
incidence data have limitations as they were
significantly updated a few months back in time
repeatedly during the second pandemic wave in
Slovakia and they contained errors on a daily basis.
None of the three types of the LFAg tests were
validated on a large sample in Slovakia by
RTqPCR tests. The incidence is reported daily by the
Ministry of Investments, Regional Development and
Informatization of the Slovak Republic [
Both types of tests had their advantages and disadvantages: different accessibility of testing for the public, duration of test evaluation, number of sample swabs collected from a tested individual, and reliability of the test result. Individuals were often selecting their diagnostic test type based on their current situation, time constraints, presence of disease symptoms, or contacts with positively tested individuals. While RT-qPCR tests were typically used in hospitals in all suspected cases (from January 2021), in the general public many tested individuals opted for a simpler, quicker and easily accessible LFAg test instead, even if they were symptomatic. Negative test results within the last 7 days (or 14 or 21 days) of any of these two types of tests were required in certain regions or in the whole country for a relief from a mandatory home isolation for the most of the duration of the second pandemic wave in Slovakia. Fig. 2 shows the total volume of the tests administered.
Although in some countries each admitted SARS-COV-2 positive case diagnosed by an LFAg test is confirmed by a RT-qPCR test, this was not the case in Slovakia where in many cases only LFAg was performed instead of a RTqPCR test, particularly in many hospitals during the period November - December 2020.
The daily hospitalization data represent the reported
total number of occupied beds in hospitals in the
Slovak Republic by patients with confirmed positive
COVID-19 tests. The capacity of the beds
designated for the COVID-19 patients was adjusted
when possible and needed throughout the second
wave by a reprofilization (repurposing) of the other
types of hospital beds. The data are reported by the
regional hospitals to the Ministry of Health of the
Slovak Republic and published [
The MLV data represent the reported total number
of occupied beds in hospitals by patients connected
to MLV with positive COVID-19 tests. The capacity
of the beds equipped with MLV designated for the
COVID-19 patients was adjusted when possible and
needed throughout the second wave by a
reprofilization of the other types of hospital beds.
The data are reported by the regional hospitals to the
Ministry of Health of the Slovak Republic and
published [
The numbers of occupied beds and MLV are reported
daily. However, the publicly available data on hospital
admissions and discharges also published by the Ministry
of Health of the Slovak Republic [
In addition to data from the Slovak Republic we also
study the data from Spain as an example of a linear
dependence of lagged hospitalizations behind incidence.
We use two data sources: hospitalizations [
Note that to eliminate natural weekly oscillations in all sources of data we systematically use moving 7-day averages or 7-day totals. Each 7-day average and total is identified with the day in the middle to eliminate the time shift introduced by the averaging and summation. 3
3.1
The RT-qPCR and LFAg incidences are typically added up
to describe the total incidence. This is also the practice of
the COVID automaton policy in Slovak Republic enforced
by the Ministry of Health that monitors the epidemic
situation weekly in 79 individual counties and nationwide
[
Total Adjusted Incidence =
RT-qPCR Incidence
Incidence + c *LFAg
Here we set the coefficient of RT-qPCR incidence to be
equal to one without loss of generality. The weight
coefficient c of the LFAg incidence can be interpreted as a
ratio of relative diagnostic performance of the tests, i.e. a
multiplicator characterizing how many samples tested by
LFAg would be positive on average per one LFAg positive
test if the samples were tested by RT-qPCR tests. There are
numerous studies of sensitivity of various types of LFAg
tests compared to (various types) of RT-qPCR tests (see the
comprehensive summary in the SI of [
We select the value of the coefficient c based on the best
linear fit between the total adjusted incidence and the
lagged hospitalization data outside of the epidemic peaks
(see the next section for details). We consider the values of
c in the interval [
Existing studies and datasets identify the proportion of
cases that required a hospitalization from reported positive
COVID-19 cases [
A linear relationship is apparent in some countries between the observed incidence and the lagged hospitalizations, see an example of Spain in Fig. 3. of the red line is 0.66 (95% CI: 0.61-0.70). The intercept with the vertical axis is at 6759 (95% CI: 6013-7505).
For Slovakia such a good approximation by a linear relationship cannot be identified for any reasonable lags (014 days) if incidence is measured solely by the RT-qPCR tests or solely by the LFAg tests. Thus we search for a linear relationship of the total adjusted incidence that includes both RT-qPCR and LFAg tests and the reported hospitalization data lagged by D days. To keep the number of parameters of the model as small as possible, we do not introduce a parameter for the mutual lag between RT-qPCR LFAg incidence. Its inclusion influences our results only very marginally (not shown).
Hospitalizations (t+D) = a+b*Total Adjusted Incidence (t)
The linear pandemic manifold serves as a basis of an estimate of the true pandemic state. Outside of the epidemic peaks and the periods of mass testing, the data lie on the manifold. We calculate the parameters of the linear manifold by a linear regression of the total adjusted incidence and lagged hospitalizations (both with 7-day moving averages) on the data outside of the epidemic peak - 60 days at the onset of the wave (Oct 10 – Dec 9, 2020) and 60 days at the tail of the wave (Apr 16 - Jun 15, 2021). The relative diagnostic sensitivity parameter c and the lag D were optimized simultaneously by minimization of the residuals of the linear regression along with parameters a and b, see Fig.4. The cut-off dates (Dec 9 and Apr 16) were selected to obtain a robust data fit that does not significantly change when the interval is shortened or extended by a few days (eventually, these dates can be selected simultaneously as a part of the optimization process). The optimization was performed in MATLAB©. Fig. 4. Sum of squares of residuals of a linear regression of total adjusted incidence and the lagged 7-day moving average of hospitalizations over the time periods Oct 10 – Dec 9, 2020 and Apr 16 - Jun 15, 2021. The optimized parameters are the coefficient of relative test sensitivity c and the lag of hospitalization data D. The dark color corresponds to low values of the error.
The optimal parameters c = 2.68 and D = 8 days correspond to approximately 37% clinical sensitivity of LFAg tests on a large predominantly asymptomatic sample compared to the RT-qPCR tests and 8 days lag of reported hospitalization behind the reported incidence. Note that the value of c in the interval (2, 3) does not significantly alter the total error. The value of the slope parameter b = 0.48 (95% CI: 0.46-0.50) of the pandemic manifold has a practical implication: the bed occupancy in hospitals is approximately a half of the 7-day total adjusted incidence 8-days ago. The vertical intercept is at a = 68.35 (95% CI: 25.85-110.86). If we take into account the large volume of the total tests (RT-qPCR+LFAg) administered, the calculated rate b is in agreement with the estimate for Spain (Fig. 3).
During the peak of the epidemic wave, we estimate the true epidemic state by a perpendicular projection of the observed data to the linear pandemic manifold. Note that perpendicular projection is not invariant to scaling of the axes. Therefore we first normalize the observed data averages (Total Adjusted Incidence and Hospitalizations) to the same mean over the studied period (Total Adjusted Incidence was scaled in this calculation by a factor F = 0.36, not shown in the figure). The perpendicular projection thus distributes uncertainty in both data series equally. Fig. 5 shows the resulting linear manifold and also illustrates the projection in the rescaled variables. Fig. 6 shows a comparison of the observed data time series and the inferred estimate of the true pandemic state. The result of the projection method is not a simple linear interpolation of the underlying data: in some phases, the true pandemic state is closer to the hospitalizations, and in others to the total adjusted incidence.
Fig. 5. Total adjusted 7-day incidence vs. the 7-day moving average of hospitalizations lagged by 8 days. The linear pandemic manifold (shown in red) was calculated as a linear regression line for the subset of the data (indicated by blue and yellow, respectively, for the first and last 60 data points). A perpendicular projection is displayed for illustration (red dashed lines) at two data points. The projection was calculated for the incidence rescaled to fit the means of the two data sets.
Fig. 6. The inferred estimate of the true pandemic state compared to observed data time series. Hospital admissions are lagged and rescaled to match
hospitalizations in March-June 2021.
The obtained estimates of the dynamics of the true pandemic state allow us to additionally estimate the lack of capacity in the hospitals for both the total bed capacity for COVID-19 patients and for mechanical lung ventilations. Fig. 7 shows a comparison of the estimated real demand for hospital beds by the projection method. During the period December 2020 – March 2021 we estimate that about 20.0% of the demand exceeded the hospital's capacity. These patients would be hospitalized if they fell ill outside of the period of epidemic peak.
We see that the capacity for hospital beds was saturated at the end of November. Even the increase of the bed capacity through reprofilization of beds in the next few months could not meet the steadily increasing demand for hospitalizations. The demand exceeded the capacity the most around the end of December. After a short improvement in the first half of January a worsening trend in the second half of January followed. After a two week stagnation the excess of demand started to shrink significantly and it disappeared completely in the middle of March.
A partial check of our estimate can be performed using the hospital admission data. As discussed above, the reported hospital admission data do not agree with the reported hospital bed occupancy by patients with COVID19 due to reporting issues with the health system. However, here we use them as an independent data set. Fig. 6 shows that our estimated true pandemic state agrees well with the hospital admissions from mid-October to mid-November (a much better fit than hospitalization and incidence data that were used in the projection method to obtain the estimate). Later the hospital admissions start to deviate from the true epidemic state similarly to hospitalization data due to the lack of hospital capacity. However, the hospital admissions data have a local peak starting after Jan 1, 2021 that is not present in the hospitalization data but it is reflected in the estimate of the true pandemic state. Finally, the hospital admissions have a peak in February followed by a systematic long-term decrease in agreement with the true pandemic state estimate.
A similar analysis provides an estimate of an excess of demand for the mechanical lung ventilations over the hospital capacity. Here we first need to transform the MLV data by a proper rescaling and a time lag to agree with the scale of the total adjusted incidence data. To remove any potential bias, we fit the MLV data to the hospitalization data outside of the epidemic peaks. The reciprocal value of the resulting scaling factor 1/k = 0.14 captures the proportion of hospital beds with COVID-19 patients occupied by patients on MLV. It corresponds to an average ratio of MLV and hospitalizations over a long time period. A lag D2 = 14 days represents the average lag of the MLV occupancy data behind the hospital bed occupancy. Fig. 8 shows the comparison of the estimated real demand for MLV by the projection method to the linear pandemic manifold with the reported MLV occupancy. During the period November 2020 – February 2021 we estimate that about 19.2% of the demand exceeded the mechanical ventilation bed capacity of the health system.
The dynamics of the excess demand for MLV agrees with the excess demand for the total hospital bed occupancy with a few differences: (i) our estimate captures one additional wave of excess demand for MLV in October - November 2020, (ii) hospitalizations excess demand is lagging approximately 2 weeks behind the MLV excess demand.
We have estimated the dynamics of the true pandemic state during the second pandemic wave in Slovak Republic. The true pandemic state is a characterization of the true demand for the hospital beds and the corresponding expected epidemic incidence. It does not characterize the true number of infected individuals in the population as the available data do not offer any direct characterization of this quantity. Nevertheless, the hospitalizations are typically of the main interest during peaks of epidemic outbursts and thus the estimated true pandemic state provides a useful characterization of the epidemic situation.
We also provide estimates of the lack of capacity of hospital beds and mechanical lung ventilations. These estimates may serve for an evaluation of the hospital capacity during the future waves, in a design of programs of patient reallocation and in a retrospective evaluation of the real pandemic costs.
One of the interesting features brought by the analysis is a comparison of the observed RT-qPCR and particularly LFAg incidence with the estimated true pandemic state. From mid November 2020 to March 2021 the total adjusted incidence is above (and often significantly above) the true epidemic state. We suspect that it indicates that the information provided by the observed incidence at that time was overestimating the true pandemic state, particularly, during the period mid-December 2020 to mid-January 2021. The positive case detection was thus higher during the period close to the peak of the pandemic wave. There are multiple reasons that may cause this effect: a larger proportion of tested individuals in an early stage of infection causing higher detection rate by the LFAg tests, larger public awareness of the pandemic situation and higher willingness of potentially infected individuals to get tested, a larger proportion of cases within hospitals with better testing surveillance, a larger overall testing capacity relative to the need. Additional factors can also be involved. Note that if the detection rate were not overestimating the true pandemic state, the estimate of the excess demand for hospital beds and mechanical lung ventilations would be even higher than presented here.
Also note that we do not evaluate any effects of testing efforts to mitigate pandemics, just the information value of the observed incidence for the estimation of the true pandemic state.
We have also derived a coefficient of relative test
sensitivity c between RT-qPCR and LFAg antigen tests.
The estimated value that corresponds to a sensitivity ratio
approximately 37% is well below the values in validation
studies (typically 50-70% see [
The true pandemic state estimates can be compared
with reported deaths and excess deaths (excess mortality)
[
Additional factors can improve the match of the estimated true pandemic state to the fitted data. A natural choice is an addition of lagged incidences exponentially discounted in time to total adjusted incidence to reflect the distribution of the admission of the hospitalized patients at a given time. The total quadratic residuals error of the linear pandemic manifold from the fitted data points can be decreased by adding two additional lagged incidences (by 4 and by 8 days) by approximately 20%. Although these factors change the dynamics of the estimated true pandemic state, the estimate for scope and time of the excess need for hospitalizations and MLV change only very marginally. Therefore we do not show these improved results here.
Our methodology has some limitations. The genomic data from Slovakia reveal that during the period September 2020 - June 2021 the dominant variant of the virus was changing. These different variants of SARS-COV-2 may eventually have different epidemiological parameters relative sensitivity of detection by LFAg test compared to RT-qPCR tests and ratio of hospitalized patients to the total number of cases. Thus the constant parameters derived within our analysis are just a crude approximation of eventual time dependencies. Also, our method relies on available data with significant limitations and possible inaccuracies.
Although the analysis is limited to the second pandemic wave in the Slovak Republic and involves particular data limitations that may not be applicable elsewhere, the overall methodology of using a linear pandemic manifold for an estimation of a true pandemic state during the pandemic peaks is universal with potential application in other geographical locations.
The authors thank the reviewers for their useful comments and suggestions that helped us to improve the manuscript. We also thank the initiative of Slovak scientists and specialists Veda pomáha – COVID-19 for support and fruitful discussions during the preparation of the manuscript. This work has been supported by the Slovak Research and Development Agency under the Contract Nos. APVV-18-0308 (RK), PP-COVID-20-0017 (RK, KB) and by the Scientific Grant Agency of the Slovak Republic under the Grants Nos. 1/0755/19 (RK) and 1/0521/20 (KB).