The ongoing pandemic of the novel coronavirus disease COVID-19 provides enormous global challenges for public health, society and economy. As of November 03, 2020, there have been more than 47 million confirmed cases of SARS-CoV-2 infections and more than 1.2 million associated deaths worldwide (ECDC). Germany has also been affected with at least 560,379 confirmed cases and 10,661 associated deaths (RKI, a).
These reported numbers for Germany can be used to estimate a "naive" measure of mortality, yielding a case fatality rate (CFR) of 1.9% (as of November 03, 2020). The CFR is defined by the number of COVID-19 associated deaths divided by the number of confirmed cases. This "naive" measure should be interpreted with caution during an active phase of the pandemic, as potential deaths tend to occur several weeks after the confirmation of SARS-CoV-2 infections. More importantly, as the CFR is generally based on the number of confirmed cases, it largely depends on implemented testing policies, which differ locally and are also changing within a given region throughout the course of the pandemic. For example in Germany, the number of conducted SARS-CoV-2 tests increased from 586,620 in calendar week 31 (end of July) to 1,121,214 tests in calendar week 35 (end of August) (RKI, b), reflecting a doubling in weekly conducted tests, partly due to increased testing in the context of the summer holiday season.
As a more realistic measure of mortality, the infection fatality rate (IFR) is defined by the number of COVID-19 associated deaths divided by the total number of infections. In contrast to the CFR, the IFR is not based on the number of confirmed cases and should therefore not be biased by potential drifts in testing policies. However, as the total number of infections with SARS-CoV-2 is generally unknown, the IFR can only be estimated based on available surveillance and seroprevalence data. Many seroprevalence studies have been conducted worldwide with the aim of estimating the true numbers of infections and resulting IFRs. Also in Germany, several local seroprevalence studies are being conducted; a completed study from the early phase of the pandemic in the high-prevalence region of Gangelt, Heinsberg, reports an estimated population-averaged IFR of 0.41% (95% confidence interval [0.33%; 0.52%]), based on 8 observed deaths until 20th of April, two weeks after the end of the study acquisition period (Streeck et al., 2020) . Overviews of completed studies and reported results from a wide range of countries can for example be found in Meyerowitz-Katz and Merone (2020) and Ioannidis (2020) . The meta-analysis of Meyerowitz-Katz and Merone (2020) yields an estimated population-averaged IFR of 0.68% [0.53%; 0.82%], while in the meta-analysis of Ioannidis (2020) estimated population-averaged IFRs range from 0.02% to 0.86% with a median IFR of 0.26%. Although the two meta-analyses differ heavily regarding their results and conclusions, both observe a high heterogeneity in population-averaged IFR estimates among different studies and emphasize the importance of obtaining reliable age-stratified estimates.
More recent meta-analyses (O'Driscoll et al., 2020; Levin et al., 2020) investigate age-specific mortality by estimating infection fatality rates for different age groups. As the risk of death from COVID-19 is estimated to increase exponentially with age (Levin et al., 2020) , it is crucial that the age distribution of infections is taken into account when interpreting estimated "overall" (populationaveraged) IFRs from different seroprevalence studies in different regions. In fact, a large proportion of the variability in estimated IFRs between studies in different regions may be explained simply by differences in demographics, particularly the age structure of populations. In addition, other factors such as the prevalence of certain comorbidities, the access to intensive medical care and systematic differences in social networks might also contribute to the variability of COVID-19 associated mortality in different regions.
However, even when considering only a particular region, the observed mortality of SARS-CoV-2 may vary over time, as the distribution of infections over different age groups may be changing throughout the pandemic, e.g. due to different and changing risk behaviours. In a recent study regarding the analysis of "the foreshadow of a second wave" in Germany, Linden et al. (2020) estimate the effective IFR as a time-dependent measure, by considering the infection fatality rate given the evolving age distribution of confirmed cases. Based on age-specific IFR estimates derived from the recent meta-analysis of Levin et al. (2020) , the authors estimate the effective IFR in Germany to be 0.86% in July, 0.48% in August, 0.70% in early September and 0.96% in late September. The effective IFR in Linden et al. (2020) is estimated based on the assumption that the distribution of confirmed cases reflects the distribution of true infections. This may not necessarily be the case, as e.g. infections with SARS-CoV-2 may be more likely detected in older age groups since these tend to experience a more serious disease course. Furthermore, as the focus of Linden et al. (2020) has been on the analysis of the potential beginning of a second wave of infections during late summer 2020, effective IFRs have not been reported for the earlier phase of the pandemic in Germany.
Thus, our study aims to investigate the time-dependent evolution of effective IFR over the course of the COVID-19 pandemic in Germany, by combining age-specific IFR estimates from multiple studies with publicly available German surveillance data. We compare the estimated evolution of effective IFR based on the age distribution of confirmed cases with estimates derived from the age distribution of estimated infections, obtained through estimated age-and time-dependent dark figures. Results are presented based on age-specific IFR estimates from four different studies, illustrating the remaining uncertainty regarding age-specific mortality. As the distributions of confirmed cases and of estimated infections change substantially during the course of the pandemic, the effective IFR is estimated to vary largely over time. The estimated evolution of effective IFR is compared with the observed evolution of CFR and the underlying assumptions for the presented analysis are discussed.
For this study we make use of publicly available surveillance data: the German disease control and prevention agency, the Robert Koch Institute (RKI, c), provides data of individual confirmed cases with date of onset of disease (or date of confirmation of SARS-CoV-2 infection, if disease onset was not available) and information on deaths associated with COVID-19. Data on age of confirmed cases and deaths are available for the following age groups A = {0-4, 5-14, 15-34, 35-59, 60-79, 80+}.
Here we consider cumulative data for each calendar week, so that potential weekday-specific fluctuations in confirmed cases and deaths are eliminated. Let C a,t denote the number of confirmed cases for age group a ∈ A in calendar week t and let C t = a∈A C a,t denote the total number of confirmed cases in week t. Similarly, let I a,t and I t denote the number of true infections and let D a,t and D t denote the number of deaths, for age group a and week t. Note that deaths typically occur weeks after the onset of symptoms from COVID-19 with an estimated mean number of 16 days from onset of symptoms to death (Khalili et al., 2020) , while infections with SARS-CoV-2 occur several days prior to manifestation of disease with an estimated median incubation period of 5 days . However, here the time point t in C a,t , I a,t and D a,t always refers to the same week where the infection has been manifested or confirmed for the first time. Based Figure 1 : Absolute numbers of weekly confirmed cases C a,t (left plot) and deaths D a,t (right plot) for the different age groups in Germany, as of November 03, 2020 (RKI, c). In both graphics, the time points t refer to the respective weeks of disease onset (or of confirmation of cases if disease onset was not available).
on the RKI data, Figure 1 depicts the evolution of the absolute numbers of weekly confirmed cases C a,t and deaths D a,t for the different age groups a ∈ A in Germany.
The observed CFR in week t is defined by the number of deaths D t (resulting from infections in week t) divided by the number of confirmed cases C t in week t, i.e. CFR t = Dt Ct . On the other hand, the effective IFR in week t is defined based on the total number of infections, i.e. IFR eff,t = Dt It . As the number of infections I t is unknown, we estimate the weekly effective IFR by taking into account the time-dependent distribution of infections in the different age groups a ∈ A as well as age-specific IFR estimates from four different studies. In particular, we consider one modelling study (Verity et al., 2020) estimating age-specific IFR for China based on individual-case data from the early phase of the pandemic (until 25th of February), one seroprevalence study (Perez-Saez et al., 2020) in Geneva, Switzerland (until 1st of June) specifically designed for age-stratified estimation of IFR, as well as two recent meta-analyses (Levin et al., 2020; O'Driscoll et al., 2020) combining the evidence from multiple seroprevalence studies worldwide. As the age groups in the four sources of estimated IFR slightly differ from the age groups A considered in the German RKI data, age-specific IFR estimates are adjusted to match with the age groups A via weighted averaging of estimates, taking into account the age structure of the German population (based on data from Statistisches Bundesamt).
Let IFR (i) a denote the resulting estimated IFR for age group a from study i, with the index i referring to one of the four literature sources (i ∈ {O'Driscoll, Verity, Perez-Saez, Levin}). The true IFR eff,t can be estimated by a weighted average of age-specific IFR estimates, i.e.
whereω a,t denotes an estimator for the fraction of infections I a,t /I t in age group a in week t. Note that estimator (1) for the effective IFR eff,t is based on the crucial assumption that age-dependent infection fatality rates IFR a do not change over the course of the pandemic and that the estimates of IFR a from the four international studies are applicable to Germany.
In the following, we consider three different estimators for the fraction of infectionsω a,t , which are derived under different assumptions regarding the distribution of infections.
(a) Under the theoretical assumption that the risk of infection is independent of age and time (compare also Linden et al., 2020) , the effective IFR -denoted by IFR (i) DE -is constant over time and estimated by equation (1) 
where the population numbers P a in age groups a ∈ A and the total population number P = a∈A P a of Germany are regarded as constant over time.
(b) In practice, the (non-uniform) age distribution of infections is likely to be changing over the course of the pandemic. Under the assumption that the distribution of confirmed cases approximately reflects the distribution of true infections in the different age groups, i.e.
one can estimate the fraction of unknown infections in age group a in week t by the corresponding fraction of confirmed casesω
Assumption (3) and the resulting estimator (4) correspond also to the effective IFR defined in Linden et al. (2020) . Note that assumption (3) implies that dark figures of undetected infections are approximately independent of age.
(c) Without specific assumptions as in (a) and (b), the number of infections I a,t can be alternatively estimated by considering age-and time-dependent dark figures viâ
wheref (i) a,t denotes the estimated factor for the dark figure in age group a in week t based on study i. Thus, an alternative estimator for the fraction of all infections in age group a in week t is given byω
As the number of true infections I a,t should be larger or equal than the number of confirmed cases C a,t , the corresponding factor f a,t = I a,t / C a,t for the dark figure should be lower bounded by 1. Thus, in the following we use the estimatorf
Due to relatively small numbers of observed deaths in younger age groups, we combine the age groups a ∈ {0-4, 5-14, 15-34, 35-59} yielding joint estimatesf (i) 0−59,t of time-dependent dark figures for ages 0 to 59. To further stabilize the estimation procedure, we estimate monthly (instead of weekly) dark figures based on age-specific CFRs observed for each month.
In this study we hence compare three different estimators for the effective IFR, each depending on different assumptions. Estimator (a) leads to a time-constant effective IFR (i) DE , while estimators (b) and (c) actually take into account the evolving age distribution of confirmed cases and estimated infections, respectively. All three estimators depend on the four available age-specific estimates IFR (i) a from the literature (Table 1 ).
As there is still an ongoing discussion and remaining uncertainty with respect to one of the most important figures in the context of the current SARS-CoV-2 pandemic, our analysis is based on four different IFR estimates that take the age of infected individuals into account. Table 1 provides a summary of age-group specific IFR estimates based on the four considered studies (Levin et al., 2020; Perez-Saez et al., 2020; Verity et al., 2020; O'Driscoll et al., 2020) . The general tendency that the risk of death increases substantially (approximately exponentially) with increasing age is apparent from all four studies and is also supported by the number of observed deaths D a,t per age group in Germany (see Figure 1 ). However, estimates for specific age groups show larger discrepancies between the different studies; as for example in age group 80+, the IFR estimate from Perez-Saez et al. (2020) The estimated population-averaged infection fatality rates IFR (i) DE , based on different age-specific IFR estimates, can be interpreted as reference mortality figures for the German population in order to compare them to other countries. They have a rather theoretical meaning as they do not reflect the actual distribution of infections over different age groups in the population. Figure 2 depicts the evolution of the distribution of weekly confirmed cases over the age groups (central plot) in comparison to the age distribution of the population in Germany (left plot). It can be observed that the age distribution of confirmed cases shifted considerably towards older age groups during the first wave in Germany in March and April 2020. In the summer time period with a relatively low incidence of COVID-19 in Germany, confirmed cases were predominantly observed in younger age groups. However, since September 2020, percentages of confirmed cases among the elderly have been rising again.
The described trend in the distribution of confirmed cases over time is directly reflected in the corresponding evolution of estimated effective IFR (based on method (b) in Section 2). The left side in Figure 3 shows that the resulting estimated effective IFR sharply increases from values between 0.5% and 1% in March to values between 1.5% and 3.5% in the middle of April. After this peak, the estimated effective IFR has been declining to values between 0.2% and 0.5% in the end of August, corresponding to a relatively young age distribution of confirmed cases. This observation may be partly explained by an increased mobility of younger age groups during the summer holiday period. Since September 2020, as the distribution of confirmed cases has been shifting more towards older age groups, effective IFR estimates are rising again with an increasing gradient observed for the end of October. This indicates that with larger SARS-CoV-2 incidences (see Figure 1 ) it may become increasingly difficult to effectively protect vulnerable risk groups and to prevent the spread of the virus from younger to older age groups (see also Linden et al., 2020) .
As the age distribution of confirmed cases may not generally reflect the age distribution of true infections, in a further analysis we account for age-and time-dependent dark figures (see method (c) in Section 2). The right hand side of Figure 2 depicts the evolution of estimated true infections based on IFR estimates from Levin et al. (2020) . It can be seen that the evolution of estimated infections is similar in shape as the observed evolution of confirmed cases. However, in particular following the high phase of the first wave of infections in April (compare Figure 1) , the estimated distribution of infections is shifted towards younger age groups in comparison to the distribution of confirmed cases. This shift results from dark figures of infections which are estimated to be considerably larger in younger age groups in comparison to the age group 80+ during this particular time. A plausible explanation for this observation might be that in times of limited testing capacities, preferential testing of individuals in age group 80+ has been more pronounced, as these patients are more likely to show (severe) symptoms from COVID-19 requiring medical intervention. Note that similar effects on estimated infections during the first wave are also observed when using agespecific IFR estimates from Verity et al. (2020) and O'Driscoll et al. (2020) , whereas the number of infections in age group 80+ are estimated to be comparatively larger based on Perez-Saez et al. (2020), as the age-specific IFR estimate for the oldest group is smaller in this study (see Table 1 , detailed results on estimated infections not shown). Since summer 2020, there seems to be a close alignment of estimated infections with confirmed cases, as age-dependent factors for dark figures are estimated to be close to 1 (and would have partly been even below 1). This may indicate that a large proportion of infections has been detected with recent testing policies. Another potential (partial) reason for this observation may be that age-specific IFR estimates from Levin et al. (2020) overestimate recent infection fatality rates, for example due to improvements in treating COVID-19 patients (Horwitz et al., 2020) or due to other factors.
The right hand side of Figure 3 depicts the resulting estimated evolution of effective IFR when accounting for age-and time-dependent dark figures. It can be seen that the adjustment for dark q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q eff,t for Germany based on four different age-specific IFR estimates (i). The left plots refer to the estimates with method (b) based on the age distribution of confirmed cases, while the plots on the right refer to the estimates with method (c) based on the age distribution of infections via estimated ageand time-dependent dark figures. The lower plots additionally display the evolution of observed CFR in Germany. Due to the expected time lag between infections and deaths, observed CFRs are only shown until the end of September; similarly, effective IFR estimates with method (c) are only accounted for estimated dark figures (based on observed CFRs) until the end of September.
figures of infections has a particular effect during the first wave of the pandemic in Germany, where estimated effective IFRs tend to be smaller in comparison to the unadjusted estimates based on confirmed cases (compare to left hand side of Figure 3 ). However, even when adjusting for age-dependent dark figures, there still remains a pronounced increase in estimated effective IFRs during the first wave of infections, indicating that this increase in mortality cannot exclusively be explained by preferential testing, but that there has been an actual and considerable change in the age distribution of infected individuals. Since summer 2020, the age distribution of estimated infections more closely aligns with the age distribution of confirmed cases and thus the estimates of effective IFR adjusted for dark figures are very similar to the unadjusted estimates. Figure 3 also shows the evolution of estimated effective IFR in comparison with the evolution of observed CFR in Germany. It can be seen that trends in observed CFR closely resemble trends in effective IFR estimated based on the age distribution of confirmed cases (as well as true infections). This implies that the evolving age distribution of infections is a major determinant (and predictor) for the resulting mortality associated with COVID-19. Despite this, it can be observed that the gap between the CFR and the IFR has been declining during the course of the pandemic in Germany; in fact, observed CFRs in August and September are even lower than some estimates of effective IFR. There may be multiple possible reasons for the decline in CFR, which are independent of the age distribution of infections: The first and probably largest contribution to the observed decline in CFR is the steady and considerable increase in conducted SARS-CoV-2 testing in Germany (RKI, d). Another plausible reason for a further decline in observed CFR may be due to improvements in treatment of COVID-19 (Horwitz et al., 2020) or other factors leading to a decrease in age-specific IFR.
Limitations of this study include that the analysis is based on the assumptions that age-specific IFRs are constant over time and that IFR estimates from the four considered studies are applicable to Germany. Furthermore, in this work we have focused on estimating the effective IFR based on the evolving age distribution of infections; however, in practice many other factors may also contribute to the variability in mortality of COVID-19, such as the distribution of different comorbidities as well as the sex distribution of infected individuals. Another limitation of this study is that, due to relatively small numbers of observed deaths in young age groups, monthly dark figures are estimated jointly for the wide age group of 0 to 59 years, even though true dark figures may further differentiate in practice (see e.g. Hippich et al., 2020 for a recent study of dark figures for children in Germany).
In conclusion, based on the example of Germany we have illustrated that the effective IFR is estimated to vary largely during the course of the pandemic, as the age distribution of infections is changing over time. In fact, it can be observed that a large fraction of the time-dependent variability in CFR can be explained by the evolution of the age distribution of infections. The additionally observed decline in the gap between the CFR and effective IFR requires further investigation in order to disentangle the contributions of increased testing and of other factors that may induce changes in mortality. In particular, obtaining reliable and timely age-specific IFR estimates for Germany is an important issue for further research.