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. On the contrary, the majority of hospitalized patients were in the elderly population. The anonymized dataset 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. and our recent analyses of Czech data. Using this estimate and the known age structure of the reported infectious individuals, we obtain a rough estimate of the time-dependent probability of hospitalization that varies according to the age of the infected  where  and  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 , which is employed later on. Average PCR(H), estimations (3) and P(H65+) = 12P(H65-) imply formula (4) in the form 
The method based on HFR 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%, which is assumed to be close to the IFR. Our estimate  together with an average of 22% HFR (from MZ CR data) 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 non-indicated 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  since the consequence of this PCR(H) level was the estimation of AR 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) is highly correlated with the epidemic wave and gave approximately 0.5 excess unreported deaths to each reported patient who died with COVID-19 (using an anonymized dataset created from ISID data that can be downloaded from the MZ CR portal), 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 AR estimate at that time and is also in agreement with a study published by Pullano et al..
An 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 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  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.
We have been using a compartmental model including AR for epidemic monitoring during the years 2020-21 in the Czech Republic (see S1 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 AR 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 AR estimate average baseline . 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 during Christmas time. They postponed hospitalizations more than usual. And 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 AR 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 AR 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 AR 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 , we calculated p(35) 0.13 from the data of hospitalized subjects by Bayes formula (1) at the beginning of April 2020.
The usefulness of using moving AR is shown in Fig 3. On 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 OKD mines led to mass PCR testing in the whole of OKD mines employees (OKD mines carry out hard coal mining in Karvina part of the Ostrava-Karvina 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 AR (different AR in the two waves visibly led to different CFR or hospitalization rate in the two waves); therefore, the fluctuation due to the change in testing during the OKD outbreak is not reproducible at all.
The AR estimate can be used as an early-warning indicator. As we saw in the previous subsection, retrospective analysis provides a good explanation of the OKD 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 AR 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 AR estimate was a signal of contact tracing and testing system overload. There are other declines:during the epidemic waves (see Oct-Nov). 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 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.
Another possible application of AR 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).
Time series for fits to cases, estimated exposed, hospitalized, and deaths are depicted in Fig 2.