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Excess Mortality in Germany 2020-2022

Christof Kuhbandner and Matthias Reitzner

Abstract

The present study estimates the burden of COVID-19 on mortality. The state-of-the-art method

of actuarial science is used to estimate the expected number of all-cause deaths in 2020 to 2022 if there

had been no pandemic. Then, the number of observed all-cause deaths is compared with this expected

number of all-cause deaths, yielding the excess mortality in Germany for the pandemic years 2020 to

2022.

The expected number of deaths is computed using the period life tables provided by the Federal

Statistical Oﬃce of Germany and the longevity factors of the generation life table provided by the

German Association of Actuaries. In addition, the expected number of deaths is computed for each

month separately and compared to the observed number, yielding the monthly development of excess

mortality. Finally, the increase in stillbirths in the years 2020 to 2022 is examined.

In 2020, the observed number of deaths was close to the expected number with respect to the

empirical standard deviation, approximately 4,000 excess deaths occurred. By contrast, in 2021, the

observed number of deaths was two empirical standard deviations above the expected number, and

even more than four times the empirical standard deviation in 2022. The cumulated number of excess

deaths in 2021, and 2022 is about 100,000 deaths. The high excess mortality in 2021, and 2022 was

almost entirely due to an increase in deaths in the age groups between 15 and 79 and started to

accumulate only from April 2021 onwards. A similar mortality pattern was observed for stillbirths

with an increase of about 9.4% in the second quarter and 19.6% in the fourth quarter of the year 2021.

Something must have happened in April 2021 that led to a sudden and sustained increase in

mortality in the age groups below 80 years, although no such eﬀects on mortality had been observed

during the COVID-19 pandemic so far.

1 Introduction

In the last two years, the burden of the COVID-19 pandemic on mortality has been intensively discussed.

Basically, since COVID-19 is an infectious disease that is caused by a new virus, it is expected that many

people have died because of the new virus who otherwise would not have died. In fact, this expectation

represents one of the central justiﬁcations for the taking of countermeasures against the spread of the

virus. Due to this reason, several previous studies have tried to estimate the extent of the mortality

burden that has been brought about by the COVID-19 pandemic.

At ﬁrst glance, it seems obvious to simply estimate the burden of the COVID-19 pandemic on mortality

based on the number of oﬃcially reported COVID-19-related deaths. However, this has been proven to

be diﬃcult due to several reasons.

1

1.1 Reported COVID-19-Deaths: The Problem

A ﬁrst diﬃculty is the problem that it is unclear whether a reported COVID-death died because of a

SARS-CoV-2-infection or only with a SARS-CoV-2-infection. For instance, according to a published

analysis of the German COVID-19 autopsy registry from March 2020 to the beginning of October 2021

[1], only 86% of the autopsied deaths with a COVID-19 diagnosis died from COVID-19. In particular,

a closer look at the diagnostics used in this study suggests that this may be an overestimation. For

instance, 87 of the 1,095 autopsied persons with the autopsy result of an “unspeciﬁc cause of deaths”

were excluded although such persons seem not to have died from COVID-19. In addition, 10% of the

deaths treated as “died from COVID-19” died actually due to bacterial or fungal super-infections or due

to therapy-associated reasons and are thus not directly caused by COVID-19. These examples highlight

the general problem that the answer to the question whether COVID-19 was the actual cause of death

depends on the used deﬁnition of ‘causality’.

A second diﬃculty is that even if a person died from COVID-19, this does not rule out the possibility

that the person would have died as well even if there had been no COVID-19 pandemic. Many of the

people that have died from COVID-19 were highly frail [2], and these people might have died from

other causes of deaths if they had not died from COVID-19. For instance, it has been shown that

rhinovirus infections have a high mortality risk for vulnerable elderly people as well [3]. Thus, even if

there had been no SARS-CoV-2-infection waves, these individuals might instead have died in one of the

rhinovirus-infection waves. Accordingly, even if there is a large number of deaths that were caused by a

SARS-CoV-2-infection, this would not necessarily mean that all these deaths are additional deaths that

would not have occurred if there had been no COVID-19 pandemic.

1.2 All-Cause Mortality: Estimating the Burden of the COVID-19 Pandemic

An obvious way to solve such problems when estimating the burden of the COVID-19 pandemic on

mortality is to compare the number of observed all-cause deaths independently of the underlying causes

of deaths with the number of all-cause deaths that would have been expected if there had been no

pandemic. If there is a new virus that causes additional deaths beyond what is usually expected, the

number of observed all-cause deaths should be larger than the number of usually expected deaths, and

the higher the number of observed deaths is above the number of usually expected deaths, the higher is

the burden of a pandemic on mortality. In particular, beyond the advantage that the above-mentioned

problems with the number of the reported COVID-19-related deaths are avoided, another advantage

is that additional indirect negative impacts of a pandemic on mortality are covered as well, such as a

possible pandemic-induced strain of the health care system.

Due to these reasons, it is not surprising that several attempts have been made to estimate the

increase in all-cause mortality during the COVID-19-pandemic [4, 5, 7, 8, 9, 10, 11]. Since the death of

a person is a clear diagnostic fact, and since highly reliable data on mortality are available for several

countries, at ﬁrst glance, one may expect that the question of whether more people have died during the

COVID-19-pandemic than is usually expected can be clearly answered.

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However, the existing attempts show very large diﬀerences in the estimated increase in all-cause

mortality during the COVID-19-pandemic. This can be illustrated for Germany where highly reliable

data on the number of all-cause deaths even at the level of individual days are available. The estimated

increase in all-cause mortality during the pandemic years 2020 and 2021 varies from 203,000 additional

deaths [5] to only 29,716 additional deaths [7, 8], and for the pandemic year 2020, it has even been

estimated that less all-cause deaths have been observed than usually expected [9].

How can this large variability in the estimated increase in all-cause mortality be explained? The

number of observed all-cause deaths is a clearly deﬁned number (although it seems that even in Germany

it is diﬃcult to determine this number precisely, see Section 3). But the estimation of the usually expected

deaths is relatively complex and entails several choices of mathematical models and parameters, and which

can lead to large diﬀerences in the estimated values (for a detailed discussion, see Sections 1.4 to 1.7 and

Sections 3 and 4 below).

1.3 The Present Study

Against this background, the present article has the objective to provide a best-practice method, the state-

of-the-art method of actuarial science, to estimate the expected all-cause mortality using the example

of all-cause deaths in Germany in the years 2020 to 2022. The underlying standard model in actuarial

mathematics was already used by Euler and Gauß, modern developments take into account mortality

trends and longevity factors. Using this method, the increase in all-cause mortality in Germany for the

pandemic years 2020 to 2022 is estimated.

In addition, an overview and an evaluation of the model and parameter choices that must be made is

provided. This demonstrates that the amount of increase in all-cause mortality varies depending on the

chosen model and parameters.

As described above, there are several studies that have attempted to estimate the increase in mortality

in Germany in 2020, 2021, and 2022 based on diﬀerent methods [5, 7, 8, 9, 11]. However, there are several

unanswered questions:

(1) Only one study [5], which examined only the year 2020, took into account the historical trend in

mortality rates. We use the mathematical model provided by the German Association of Actuaries.

In particular, this includes longevity factors, which are well established in actuarial science.

(2) Although in most of the studies age-standardized estimations were made, age-dependent diﬀerences

in mortality increase were not examined in detail. We use most recent life tables provided by the

Federal Statistical Oﬃce of Germany to calculate age-dependent expectations.

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(3) In none of the previous studies, it was examined how much the mortality estimates depend on

the underlying data and vary with diﬀerent approaches. Here we state the data uncertainty and

calculate the model and parameter sensitivity by comparing the results achieved using diﬀerent life

tables and longevity factors.

(4) In all of the previous studies except a recent study [6] concerning Austria, only the estimated increase

in all-cause deaths was reported, without examining whether the estimated increase exceeds the

usual variation in mortality found across previous years. We estimate the yearly empirical standard

deviation which could be used to obtain conﬁdence intervals.

(5) The increase in mortality over the course of the year has so far only been investigated for 2020

in two studies [5, 7]. The years 2021, and 2022 have not yet been investigated in this respect.

Furthermore, no study has yet determined the increase in mortality over the course of the year for

diﬀerent age groups.

(6) Comparing the results to possible inﬂuencing factors: In none of the previous studies, possible

factors that might contribute to the observed course of the increase in mortality were explicitly

examined on a monthly base during the pandemic years 2020 to 2022.

(7) Monthly increase in the number of stillbirths in the years 2020 to 2022 in Germany: In all previous

studies, the increase in mortality has only been examined for the age groups 0 and above. Whether

changes in mortality are also found at the level of stillbirths has not been investigated so far.

As will be shown, a proper analysis of the increase in all-cause mortality reveals several previously

unknown dynamics that will require a reassessment of the mortality burden brought about by the COVID-

19 pandemic.

1.4 Estimating the Increase in All-cause Mortality: Population-Size and Historical-

Trend Eﬀects

There are two main eﬀects that have to be taken into account when estimating the increase in all-cause

mortality: eﬀects of changes in the size of the population and eﬀects of historical trends in mortality

rates. To illustrate these eﬀects and the resulting potential pitfalls, Fig. 1 shows for the over 80 years old

population in Germany the number of deaths (Fig. 1A), the population size (Fig. 1B), and the mortality

rate (i.e., percentage of deceased persons; Fig. 1C) for the years 2016 to 2021.

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Fig. 1.: Population-size eﬀects and historical-trend eﬀects on the estimation of the increase in

mortality. For the population over 80 years of age in Germany, (A) shows the number of deaths, (B) the

population size, and (C) the mortality rate (i.e., percentage of deceased persons) for the years 2016 to 2021.

Changes in population size have to be taken into account due to the simple fact that the larger a

population is, the more deaths occur. Ignoring existing changes in population size will lead to erroneous

estimations. For instance, regarding the population over 80 years of age in Germany, the number of

deaths increases from year to year (see Fig. 1A). Concluding from this pattern that mortality increased

in the years 2020 and 2021 compared to previous years would make no sense because this increase is fully

attributable to the increase of population size, as shown in Fig. 1B and 1C.

Historical trends in mortality rates have to be be taken into account due to the fact that mortality

rates are not a stable values but inﬂuenced by environmental and societal changes and improvements in

medical treatments. For instance, as can be seen exemplarily in Fig. 1C, in Germany, there is a historical

trend of a continuous decrease in mortality rate that is observed in most age groups. If such a declining

trend in mortality rates is not taken into account, the number of expected deaths is overestimated and

thus the true mortality excess is underestimated.

The pitfall of ignoring changes in population size is for example found in the estimations provided

by the German Federal Statistical Oﬃce [12] where the increase in mortality is estimated based on

a comparison of the observed number of deaths with the median value of the four previous years. As

illustrated in Fig. 2A, estimating the number of expected deaths based on the median of the four previous

years underestimates the number of expected deaths and thus overestimates the true increase in mortality.

The invalidity of this method can be illustrated by the fact that in case of a continuously increasing

population size, as is the case for the population over 80 years of age in Germany, such a method would

conclude for every year that there was an unexpected increase in mortality compared to previous years.

The pitfall of ignoring longer historic trends is for example found in the estimations provided by the

World Health Organization (WHO) [11] where the increase in mortality is estimated based on a thin-

plate spline extrapolation of the number of expected deaths. As illustrated in Fig. 2B, such an estimation

method is highly sensitive to short-term changes in the observed number of deaths. Accordingly, erratic

estimations of expected deaths predictions can occur. For instance, regarding the WHO estimations for

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Germany, the spline extrapolation predicts – based on the short-term decline in deaths in 2019 compared

to 2018 – that a similar decline would occur in the following years as well, although this completely

contradicts the long-term historical trend.

Fig. 2.: The pitfalls of ignoring population-size eﬀects and historical-trend eﬀects. The blue squares

in (A) and (B) show the development of the number of deaths in Germany from 2010 to 2021 (all age groups).

The red squares in (A) show the estimations of the number of expected deaths for the years 2020 and 2021 of the

German Federal Statistical Oﬃce [12] which are based on the median of the four previous years. The red squares

in (B) show the estimations of the number of expected deaths for the years 2020 and 2021 of the World Health

Organization [11] which are based on a thin-plate spline extrapolation that is highly sensitive to short-time changes.

As can be seen, both the ignoring of the increase in population size of the older age groups and the ignoring of

longer historical trends leads to an underestimation of the expected deaths and thus to an overestimation of the

true mortality increase in the years 2020 and 2021.

1.5 Methods That Take Into Account Population-Size and Historical-Trends Eﬀects

A ﬁrst and comparatively simple approach to take into account population-size and historical-trends

eﬀects is the attempt to predict the further course of the number of deaths from observed data in

previous years using regression methods. For instance, in a study by Baum [4], the course of the observed

increase in the number of deaths in Germany from 2001 to 2021 compared to the year 2000 was ﬁtted with

a polynomial function of order two, and the yearly residuals were used to estimate the yearly increase

or decrease in mortality, resulting in an estimated increase in mortality in the years 2020 and 2021 of

about 11,000 additional deaths each. While the advantage of this approach is on the one hand that no

parameter choices have to be made as it is the case with the more complex estimation methods (see

below), on the other hand this is at the same time the weakness of this approach: since every data point

is given the same weight, unique outliers may lead to biased estimations, and developments depending

on more complex circumstances cannot be incorporated in this approach.

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To account for unique outliers, it has been tried to estimate the number of expected deaths by a

time-series model based on the number of observed deaths in previous years, and to exclude past phases

of unique excess mortality, as done in the EuroMOMO project [13]. However, beyond the problem that

the resulting estimates depend on the speciﬁc model and parameter choices made (see below), a common

problem for every approach that bases estimations on the raw number of observed deaths is that the

resulting estimations do not take into account possible changes in the age structure within a population,

which can lead to biased estimates.

To take into account changes in the age structure within a population, so-called age-adjustments has

a long tradition in mortality research [14], which is essential especially when estimating the number of

expected deaths in populations where the proportion of elderly people changes over time. The basic

method is to compute mortality rates for a reference period separately for diﬀerent age groups, and to

extrapolate from the age-dependent mortality rates and the population sizes of the diﬀerent age groups

in the to-be-estimated year the number of expected deaths in each of the age groups.

An example is a recent study by Levitt, Zonta, and Ioannidis [10] where the increase in mortality in

the years 2020 and 2021 was estimated based on the reference period of the three pre-pandemic years

2017-2019 using age strata of 0-14, 15-64, 65-75, 75-85, and 85+ years, resulting in an estimated increase

in mortality of about 16,000 additional deaths in the year 2020, and 38,800 additional deaths in the

year 2021. In two studies by De Nicola et al. [5,6], a more reﬁned method (see below) and a more

ﬁne-grained age adjustment was used, resulting in even lower estimates of increased mortality with about

6,300 additional deaths in 2020 and 23,400 additional deaths in 2021.

A problem in both the study by Levitt et al [10]. and the studies by De Nicola et al. [7, 8] is

that possible historical trends in mortality rates are not taken into account. This was, in addition to

an age-adjustment, done in a study by Kowall et al. [9] where the increase in mortality in the year

2020 was estimated for the countries Germany, Spain, and Sweden. Historical trends in mortality rates

were estimated based on the observed decrease in mortality rates in the pre-pandemic years 2016-1019.

For Germany, it was estimated that the number of observed deaths in 2020 was 0.9% higher than the

number of estimated expected deaths, which is in the range of the estimations in the De Nicola et. study.

Estimations with adjustments for changes in historical trends in mortality rates for the year 2021 have

to date not been reported, at least to our knowledge.

1.6 The Inherent Model Uncertainty of Estimates of Increases in Mortality

As has already become apparent in the previous paragraphs, the estimation of the amount of increase in

all-cause mortality entails several model and parameter choices that have to be made. While a proper

analysis necessarily requires the taking into account of changes in population sizes and historical trends

in mortality rates, there remain a number of degrees of freedom how to exactly do this. For instance, an

open question is which previous years are used as a reference and which model is used for the extrapolation

of the expected deaths based on these years.

What a large eﬀect a small change in the chosen perspective can have on the estimation of the amount

of mortality increase is illustrated in Fig. 3 using the German mortality ﬁgures. When trying to estimate

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the increase in the number of all-cause deaths in the years 2020 and 2021 by a comparison with the

number of expected deaths that is estimated based on the course of deaths in the four pre-pandemic

years 2016-2019, one can for instance take two diﬀerent perspectives: one can consider the year 2018

as an unusual outlier above the typical course of the number of deaths, or one can consider the year

2019 as an unusual outlier below the typical course of the number of deaths. Depending on the chosen

perspective, extrapolating the expected number of deaths with either excluding the year 2018 (“outlier

upwards”) or the year 2019 (“outlier downwards”) leads to totally diﬀerent results, with an estimation of

a strong increase in mortality in the former case and an estimation of even a slight decrease in mortality

in the latter case.

Fig. 3.: Possible large eﬀect of small changes in the chosen perspective. The blue squares in (A) and (B)

show the development of the number of observed all-cause deaths in Germany from 2016 to 2021 (all age groups).

The expected all-cause deaths in the years 2020 and 2021 are estimated based on the observed deaths in the years

2016-2019 using a simple linear regression function, excluding the year 2018 as an “unusual outlier upwards” (A)

or the year 2019 as an “unusual outlier downwards” (B), giving the impression of a strong increase in mortality in

the former case and the impression of even a slight decrease in mortality in the latter case.

Since there is no truth criterion that would determine which of the choices is the best one to be made,

there is no such thing as a “true” increase in mortality. Instead, the amount of increase in mortality must

be understood as a construct with an inherent model uncertainty, that varies depending on the chosen

perspective. This fact has at least three important implications:

First, when reporting estimates of the amount of increase in mortality, it is important to show how

strongly the estimates vary with diﬀerent model and parameter choices that can reasonably be made. In

particular, possible choices and the resulting estimates should be communicated to readers in a way so

that they are enabled to draw their own conclusions depending on their speciﬁc questions they would

like to answer (see next point).

Second, when interpreting estimates of the increase in mortality, one has to be aware of the made

model and parameter choices. In particular, when deciding which approach is chosen, one has to clarify

8

which question is tried to answer, and to choose the approach that best ﬁts the to-be-answered question.

For instance, if one is interested in the question of how far the observed number of deaths is above the

usually occurring deaths, excluding outlier years when estimating the amount of increase in mortality

may be a reasonable decision. However, if one is interested in whether the observed number of deaths is

above the extreme values of previous years, excluding outliers may be a less reasonable decision.

Third, despite the inherent uncertainty of the estimates of increases in mortality, the comparison

of increases in mortality between diﬀerent years may nevertheless reveal clear results. If the observed

diﬀerence between the years does not vary as a function of the chosen parameters and model, it can be

assumed that the observed diﬀerences in estimated increases in mortality reﬂects the true fact that there

was a larger increase in mortality in one of the years.

1.7 The Use of the Term ”Excess Mortality“

In many of the previous studies, the observation that the number of observed all-cause deaths is larger

than the number of expected all-cause deaths is designated by the term “excess mortality”. However,

such a use of terms is questionable. The number of deaths from year to year does not follow a straight

line but varies around a common trend.

Fig. 4.: The inﬂationary use of the term “excess mortality”. The colored squares show the number of

all-cause deaths in Germany from 2010 to 2021. The dashed red line shows the common trend across the years

(linear regression). If one were to designate as “excess mortality year” all years in which more deaths are observed

than expected according to the common trend (red-colored squares), one would have to conclude that an “excess

mortality” is observed in six years, and a “mortality deﬁcit” in the other six years.

Accordingly, as illustrated in Fig. 4, if one were to designate as “excess mortality year” all years

in which more deaths are observed than expected according to the common trend, one would have to

9

conclude that an “excess mortality” is observed in about 50% of all years, and a “mortality deﬁcit” in

the other 50% of all years.

Since about half of the years show mortality levels above the common trend, one could use the term

“excess mortality” only for years that show an outstanding increase in mortality above a certain threshold.

One straightforward possibility to establish such a threshold would be to compute the mean variation

(empirical standard deviation) around the common trend across the years, and to designate as “excess

mortality years” only those in which the number of observed deaths exceeds twice the mean variation.

Another possibility would be to search for previous years with peak deviations from the common trend,

and then to compare the deviation observed in the year one is interested in with the peak deviations in

previous years. Such a comparison was for instance made in a recent study by Staub et al. [15] where the

historical dimension of the COVID-19 pandemic was examined for the countries Switzerland, Sweden,

and Spain over a time span of more than 100 years, revealing that the peaks of monthly excess mortality

in 2020 were greater than most peaks since 1918.

Nevertheless, also in this contribution we decided to use the terms “excess mortality” and “mortality

deﬁcit” for a mortality which is just above, respectively below, the estimated value, as in most other

contributions. An attempt to deﬁne an outstanding “excess mortality year“ via mean variations will be

made in Section 3 and Section 4.

2 Yearly expected mortality

2.1 Methods

It is the standard method in actuarial science to use life tables and population tables, and to obtain by a

(suitable modiﬁed) multiplication the expected number of deaths. Historical population tables are used

to estimate the longevity trend which is in addition taken into account.

Thus, the starting point for our investigations are the period life tables and population demographics

available from the Federal Statistical Oﬃce of Germany. As usual in actuarial science, we denote by

•lx,t the number of xyear old male at January 1st in year t;

•ly,t the number of yyear old female at January 1st in year t;

•dx,t the number of deaths of xyear old males in year t;

•dy,t the number of deaths of yyear old females in year t;

•qx,t (an estimate for) the mortality probability for an xyear old male in year t;

•qy,t (an estimate for) the mortality probability for a yyear old female in year t.

Note that dx,t also contains deceased that have been (x−1) years old at January 1st in year tand died

as xyear old. To compensate this problem, the 2017/2019 life table of the Federal Statistical Oﬃce of

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Germany [16] (like most German life tables) uses the method of Farr to estimate qx,t (and analogously

qy,t).

ˆqx,2019 =P2019

t=2017 dx,t

1

2P2019

t=2017(lx,t +lx,t+1) + 1

2P2019

t=2017 dx,t

(1)

The period life table 2017/19 of the Federal Statistical Oﬃce of Germany thus takes into account only

the mortality probabilities in these three years.

A much more complicated task is to compute generation life tables. Generation life tables observe

the mortality development over a long period, roughly 100 years, smoothen the existing data, and in

particular estimate the long term behaviour of the mortality probabilities. These probabilities have been

decreasing within the last 100 years, and the common ansatz is to set

qx,t =qx,t0e−F(x;t,t0), qy,t =qy,t0e−F(y;t,t0).

Here, the German Association of Actuaries (DAV) uses a smoothed life table qx,t0in the base year t0,

and models the trend underlying future mortality, the longevity trend function F(x;t, t0), via regression

separately for the male and female population. In the year 2004, it turned out that the decrease of the

mortality probabilities in the previous years has been steeper than expected, therefore the DAV life table

DAV 2004 R [17] distinguishes between a higher short-term trend and a lower long-term trend. These

trends are of high importance and used for life annuities, whereas for life insurances the trend (at least

the short term trend) is mostly ignored.

One should keep in mind that the life tables DAV2004R and the longevity factors DAV2004R are

tailor made for pensions funds. Since we are interested in predictions concerning the whole German

population, we use the life table for the general population of the Federal Statistical Oﬃce of Germany,

not the life table DAV2004R, and adapt the longevity factors of the DAV2004R to ﬁt for the whole

population. In addition, it seems that the longevity trend was ﬂattening in the last years. Therefore, we

have decided to use half the long-term trend function given by the DAV2004R,

F(x;t, t0) = 1

2(t−2019)Fl,x, F (y;t, t0) = 1

2(t−2019)Fl,y

where the numbers Fl,x and Fl,y are contained in the DAV 2004 R table. We also decided to use the

probabilities ˆqx,2019 and ˆqy,2019 of the life table 2017/2019 by the Federal Statistical Oﬃce of Germany

as the base life table in a ﬁrst step, and thus take t0= 2019. A second possible choice would be to take

t0= 2018, the ‘mean year’ of the table, which only results in minor changes, but we follow the actuarial

standard to set t0as the year when the table was completed.

For a discussion concerning our model parameters, i.e. the inﬂuence of the longevity trend and our

choice using half of it, and the choice of the (non-smoothed) life table 2017/19, we refer to Section 3.

Also, it is well known that mortality probabilities for males and females diﬀer substantially, therefore

these two cases are computed separately.

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Putting things together, we deﬁne the mortality probability of an xyear old male in year tby

qx,t = ˆqx,2019e−1

2(t−2019)Fl,x ,

and for a yyear old female in year tby

qy,t = ˆqy,2019 e−1

2(t−2019)Fl,y .

Now, for each individual, the probability to die at age xis given by qx,t, and hence, in a ﬁrst attempt,

a population of lx,t individuals produces binomial distributed random numbers Dx,t and Dy,t of deaths

for males, respectively females, with expected values

EDx,t =lx,tqx,t ,EDy,t =ly,tqy,t.

As is well known (and already discussed above in connection with Farr’s method), this formula ignores

those individuals which have been of age (x−1) at the beginning of year t, and died as xyear olds. To

compensate for this missing piece, we follow the procedure proposed by De Nicola et al. [7]. Roughly half

of the x−1 year old population at the beginning of the year which is of size lx−1,t dies after its birthday

as xyear old. For them we use the smoothed mortality probability

qx−1,t +qx,t

2.

The other half of the xyear old deceased belongs to the population of xyear old at the beginning of the

year which is of size lx,t. For them we use the smoothed mortality probability

qx,t +qx+1,t

2.

For more details see [7]. Hence, for x= 0,...,101 the random number Dx,t of deaths of age xin year t

is binomial distributed and satisﬁes

EDx,t =1

2lx−1,t

qx−1,t +qx,t

2+lx,t

qx,t +qx+1,t

2(2)

where lx−1,t and lx,t are taken from the population table of the Federal Statistical Oﬃce of Germany [18].

For x= 0 we set l−1,t =l0,t+1 if available, l−1,t =l0,t else, and q−1,t =q0,t. The same considerations lead

to EDy,t.

The 2017/2019 life table by the Federal Statistical Oﬃce of Germany contains the mortality prob-

abilities qx,t and qy,t, and the underlying population table the population size lx,t and ly,t for the age

x= 0,...,100. In principle it would be more precise to use life tables and population tables up to age

113 but these data are not available. The excess mortality is obtained by comparing the expected values

EDx,t +EDy,t to the observed data dx,t +dy,t for t= 2020,2021 and 2022 .

Some remarks are important in order to contextualize the method.

12

•Modelling the longevity factors is a challenging task. For example, the Actuarial Association

of Austria uses factors involving arctant

100 −20.01which has serious advantages. The need for

longevity factors depends heavily on the country, it seems for example that in Japan and in England

the mortality trend has already vanished and the mortality probabilities are more or less constant.

•The mortality probability heavily depends on gender and diﬀers for the male and female population.

However, the resulting excess mortality is nearly the same for the male and female population.

Hence, in the following, we calculate the expected number of deaths separately and show only the

total number of deaths. On the other hand, huge diﬀerences occur for the excess mortality in

diﬀerent age groups, and therefore we present our results for each age group separately.

•The mortality probability not only depends on age and gender, but also signiﬁcantly on social

status, profession, health condition, region, etc. As is common, the German life tables give average

mortality probabilities. Also, it is unclear - at least to the authors - whether the SARS-CoV-2-

infection rate and mortality depends on these factors, too. For a deeper investigation of COVID-19

mortality increase, this should be taken into account, but at the moment, appropriate data are not

available.

•One has to take into account that the year 2020 is a leap year. Therefore we have “added” an

additional day by multiplying the result of the computations described above by 366

365 .

2.2 Results

Following the computations described in the previous section, we obtain the expected number of deaths

in 2020, 2021, and 2022. The expectations EDx,t and EDy,t for each age x, y = 0,1,...,99 and t= 2020,

2021, and 2022 are given in the supplement, Section 8.1. The Federal Statistical Oﬃce of Germany

provides the number of deaths only in certain age groups [19], and for the year 2022 the number of

deaths reported by the Federal Statistical Oﬃce of Germany is still preliminary. The following table

gives the number of deaths in the age groups

¯a∈ {[0,14],[15,29],[30,39],[40,49],[50,59],[60,69],[70,79],[80,89],[90,∞)}.

We set

D¯a,t =X

x∈¯a

Dx,t +X

y∈¯a

Dy,t and d¯a,t =X

x∈¯a

dx,t +X

y∈¯a

dy,t.

To compare the expected ED¯a,t and the observed values d¯a,t, we use the relative diﬀerence

d¯a,t −ED¯a,t

ED¯a,t

.

13

Table 1: Expected deaths and yearly excess mortality for diﬀerent age groups.

t= 2020 t= 2021 t= 2022

age expected abs. rel. expected abs. rel. expected abs. rel.

range observed diﬀ. diﬀ. observed diﬀ. diﬀ. observed diﬀ. diﬀ.

0-14 3,531 3,513 3,517

3,306 -225 -6.38% 3,368 -145 -4.14% 3,527 10 0.28%

15-29 3,944 3,817 3,755

3,844 -100 -2.53% 3,934 117 3.07% 4,115 360 9.59%

30-39 6,626 6,585 6,546

6,668 42 0.64% 6,812 227 3.44% 7,130 584 8.92%

40-49 15,345 14,877 14,601

15,507 162 1.06% 16,095 1,218 8.19% 15,653 1,052 7.20%

50-59 58,641 57,705 56,471

57,331 -1,310 -2.23% 59,350 1,645 2.85% 56,554 83 0.15%

60-69 117,432 118,456 119,983

118,460 1,028 0.88% 126,781 8,325 7.03% 128,370 8,387 6.99%

70-79 198,389 190,335 186,303

201,957 3,568 1.80% 204,839 14,504 7.62% 205,435 19,132 10.27%

80-89 378,459 392,535 404,994

378,406 -53 -0.01% 398,041 5,506 1.40% 421,201 16,207 4.00%

90-∞199,191 201,884 202,375

200,093 902 0.45% 204,467 2,583 1.28% 219,191 16,816 8.31%

total 981,557 989,707 998,545

985,572 4,015 0.41% 1,023,687 33,980 3.43% 1,061,176 62,631 6.27%

The deviation in 2020, 2021, and 2022 must be compared to the deviation inherent in the parameter

choice of our model, and the empirical standard deviation which has occurred in the years before. This

will be done in Section 3 and Section 4 for the total number and for all age groups separately. It will

turn out, that in year 2020 the observed numbers of deaths are extremely close to the expected numbers

with respect to the empirical standard deviation. Yet, in 2021 the diﬀerence between the observed total

number of deaths and the expected number is more than twice the empirical standard deviation, and

in 2022 it is even beyond four times the standard deviation. This mortality excess is mainly caused

by the following age groups: in 2021 we observe an excess mortality of more than twice the empirical

standard deviation in the age groups [60,69],[70,79], and an excess mortality of ﬁve times the empirical

standard deviation in the age group [40,49]; in 2022 we observe an excess mortality of more than twice

the empirical standard deviation in the age groups [15,29],[30,39],[60,69],[70,79],[80,89], and an excess

mortality of four times the empirical standard deviation in the age groups [40,49],[90,∞). The other

age groups are below twice the standard deviation.

14

Fig. 5 illustrates that the deviation of the observed mortality from the expected mortality is not

uniform over the diﬀerent age groups, and, in particular, that the structure changes from 2020 to 2021,

and 2022. A closer look reveals that the excess mortality observed in 2021 is almost entirely due to an

above-average increase in deaths in the age groups between 15 and 79. The highest values are reached in

the age group 40-49, where an increase in the number of deaths is observed that is 9% higher than the

expected values. In 2022 the excess mortality is above 6% for nearly all age groups in the range [15,∞).

An exception for all three years is the age group [50,59] where, in contrast to the surrounding age groups,

a substantially lower excess mortality is observed. This is also visible if the 2017/2019 life table by the

Federal Statistical Oﬃce of Germany is replaced by a life table from another year. We are not aware

of an explanation for this fact, an interesting avenue for future research may be to explore what factors

make this age group so resilient.

Fig. 5: Yearly excess mortality 1.The red bars show the excess mortality in 2020 (left panel), 2021 (middle

panel), and 2022 (right panel) in diﬀerent age groups, the grey bars the total excess mortality.

It should be pointed out that in the last twenty years the maximal excess mortality in a year was

about 25,000 deaths, and the authors are not aware of an excess mortality of more than 60,000 deaths –

or in two consecutive years about 100,000 deaths – in the last decades.

1For infants something unexplained happens. In the beginning of 2020 there were 774,870 people of age 0, during the year

2,373 children of age 0 died, yet at the end of 2020 there were 783,593 (!) people of age 1. This is maybe due to migration

eﬀects, but we do not have suﬃcient precise data to model this eﬀect. And for our investigations concerning COVID-19

excess mortality, the infant mortality can be ignored.

15

3 Data uncertainty and model uncertainty

The most basic data set for estimating excess mortality is the number of all-cause deaths in each year.

The Federal Statistical Oﬃce of Germany publishes each week the number of reported deaths. After the

end of the year, the Federal Statistical Oﬃce of Germany undertakes a “plausibility check” and then

publishes about September next year the corrected ﬁnal number of deaths. E.g., for 2019, this resulted

in a change of at least 20,000 data sets yielding a cumulative change of nearly 3,000 deaths, and for

2021 we observe a cumulative change of more than 2,000 deaths. Hence, even in a country like Germany,

already the number of observed deaths seems to have an intrinsic uncertainty of 2,000 to 3,000 deaths.

One should keep in mind that the life tables of the Federal Statistical Oﬃce of Germany are calculated

using this reported number of deaths, and hence also the life tables contain this data uncertainty.

Computing the expected number of deaths using a life table, several parameters for modelling mor-

tality probabilities essentially inﬂuence the results. One could replace the 2017/2019 life table of the

Federal Statistical Oﬃce of Germany by the life tables 2016/18 or 2015/17. And one could use diﬀerent

longevity factors, or ignore them totally. The question, whether a serious excess mortality occurs for

2020, 2021, and 2022, heavily depends on this underlying data sets. In the next table we present the

total expected number of deaths over all age groups

EDt=

101

X

x=0

EDx,t +

101

X

y=0

EDy,t

using diﬀerent life tables and taking into account either none, or half, or the full longevity trend.

Table 2: Expected deaths for diﬀerent life tables.

longevity

trend life table ED2020 ED2021 ED2022

2015/17 1,010,478 1,025,768 1,041,319

none 2016/18 999,592 1,014,802 1,030,423

2017/19 988,288 1,003,270 1,018,827

2015/17 989,964 998,213 1,006,620

half 2016/18 986,021 994,294 1,002,869

2017/19 981,557 989,707 998,545

2015/17 969,896 971,451 973,159

full 2016/18 972,649 974,230 976,105

2017/19 974,875 976,341 978,263

observed 985,572 1,023,687 1,061,176

It turns out that the life tables have a signiﬁcant eﬀect on the question whether an excess mortality

exists. For example, the use of the life table 2015/17 of the Federal Statistical Oﬃce of Germany without

16

the longevity trend yields for the ﬁrst two Corona-years 2020 and 2021 even a mortality deﬁcit. And

when keeping half the longevity trend, in 2021 the excess mortality of 31,723 deaths for the life table

2017/19 should be compared to the smaller excess mortality of 23,217 deaths when using the life table

2015/17, the total diﬀerence being 8,506 deaths. In other words, the life tables of the Federal Statistical

Oﬃce of Germany have a serious ﬂuctuation over the years which should be taken into account as the

model uncertainty. Yet all parameter choices lead to a serious mortality excess for the year 2022.

For a more convenient view we present the excess mortality using the relative diﬀerence, see Table 3

and Fig. 6.

Table 3: Excess mortality for diﬀerent life tables.

longevity

trend life table 2020 2021 2022

2015/17 -2.46% -0.20% 1.91%

none 2016/18 -1.40% 0.88% 2.98%

2017/19 -0.27% 2.04% 4.16%

2015/17 -0.44% 2.55% 5.42%

half 2016/18 -0.05% 2.96% 5.81%

2017/19 0.41% 3.43% 6.27%

2015/17 1.62% 5.38% 9.04%

full 2016/18 1.33% 5.08% 8.72%

2017/19 1.10% 4.85% 8.48%

In the light of these results, we have decided to choose a model which avoids the extremes and includes

half of the longevity factor in Section 2.1. In this case, the range between the three models – which is an

indicator for the model uncertainty – is in both years approximately 8,500 deaths per year.

Yet in all these results obtained by life tables of recent years of the Federal Statistical Oﬃce of

Germany, and in most other models [4, 5, 7, 8, 9], the main point coincides with our results: for 2020 the

number of deaths is close to the expected value, whereas for 2021 there is a noticeable excess mortality,

and for 2022 there is a huge mortality excess which has not been observed during the last decades.

17

Fig. 6: The model sensitivity. The bars show the mortality deﬁcit, respectively the excess mortality in 2020

(left panel), 2021 (middle panel), and 2022 (right panel) for diﬀerent life tables and longevity trends.

A more detailed analysis of all the age groups introduced in Section 2.2 shows that, independently

of the model used, the excess mortality in 2021, and 2022 is far above the values for 2020. These more

detailed results are given in the supplement, Section 8.2.

4 The empirical standard deviation

As remarked in Section 2.2, to contextualize the deviation in 2020 – 2022, it must be compared to

the model uncertainty, and to the empirical standard deviation occurred in the years before. Since the

precise value of the empirical standard deviation – like the expectation – heavily depends on the underlying

mathematical model, and since we are only interested in a rough approximation of the empirical standard

deviation, we use an extremely simple model: we approximate the expected number of deaths using a

linear regression model and calculate the empirical standard deviation in this model.

18

Fig. 7: The empirical standard deviation. The red squares show the number of all-cause deaths in Germany

from 2010 to 2019. The blue line shows the regression line.

The regression leads to

dt=

100

X

x=0

dx,t +

100

X

x=0

dy,t ≈L(t) = −21,936,713.9 + 11,336.2·t

which shows that each year we expect an increase of approximately 11,300 deaths in Germany. Observe

that we have taken into account that the years 2012 and 2016 have been leap years and the number of

deaths has been normalized to 365 days per year.

Table 4: Linear regression of the observed deaths.

year lin. reg. observed

t L(t)dt

2010 849,062 858,768

2011 860,398 852,328

2012 871,735 867,206

2013 883,071 893,825

2014 894,407 868,356

2015 905,743 925,200

2016 917,079 908,410

2017 928,416 932,263

2018 939,752 954,874

2019 951,088 939,520

19

Calculating in this simple model the empirical standard deviation gives

ˆσ= ˆσ(dt) = 14,162.(3)

We do not claim that this is a precise estimate of the standard deviation σ(Dt), yet we are convinced that

this at least reﬂects the order of magnitude. To check whether this order of magnitude is plausible, we

also computed the empirical standard deviation for the years 2000-2009, using again the linear regression

model. For these years, the empirical standard deviation is approximately 12,600 which is the same order

as (3).

At ﬁrst sight this empirical standard deviation is somehow surprising and seems to be in contrast

to the model used for modelling Dx,t described in Section 2.1. As is common, we assumed that the

number of deaths follows a binomial distribution. This is the most natural assumption. It would imply

that the variance VDx,t =lx,t(1 −qx,t)qx,t is approximately the number of deaths lx,tqx,t, since for the

large majority of xthe mortality probabilities are close to zero. This assumption and the independence

property of the binomial model would lead to a total variance of approximately one million, and a standard

deviation of approximately 1,000 in Germany. Thus, in actuarial science, a further randomization of qx,t

is introduced which keeps the expectation unchanged – and thus our results in Sections 2.1–3 are still

valid – but increases the variance to the observed 14,000.

We compare the excess mortality of approximately 4,000 deaths in 2020, 34,000 deaths in 2021, and

63,000 in 2022 to the empirical standard deviation ˆσ. In 2020, this leads to

d2020 −ED2020 ≈0.28ˆσ,

hence the number of deaths in 2020 is very close to the expected number. For 2021, we have

d2021 −ED2021 ≈2.40ˆσ

and for 2022

d2022 −ED2022 ≈4.42ˆσ.

In many applications, an observed deviation beyond twice the standard deviation is called ‘signiﬁcant’

because for normal distributed random variables the 5% conﬁdence interval leads to this bound. For a

normal distributed random variable, a bound of 2.4 times the standard deviation (occurring in 2021) leads

approximately to a 1.6% conﬁdence interval, and a bound of 4.4 times the standard deviation (occurring

in 2022) leads approximately to a 0.006% conﬁdence interval. We want to point out that the probability

distribution of the number of deaths as a random variable is unknown and thus we avoid the use of

the words “conﬁdence interval” for excess mortality. Comparing the excess deaths against the empirical

standard deviation just enables the reader to compare the deviation in 2020, 2021, and 2022 to historical

ﬂuctuations.

In addition one should also have in mind the data uncertainty of 2,000 to 3,000 deaths, and the model

uncertainty of approximately 4,250 deaths.

20

The same method can be applied to certain age groups ¯a, using a linear regression model only for the

development of the number of deaths in this age group, and estimating the observed empirical variance

ˆσ(d¯a,t). This leads to the following results, for the details see Section 8.3 in the supplement.

Table 5: Empirical standard deviations for age groups ¯a.

age range ˆσ(da,t)

0-14 158

15-29 148

30-39 245

40-49 237

50-59 868

60-69 3,646

70-79 6,101

80-89 7,770

90-∞4,005

total 14,162

Comparing these to the values in Table 1 shows that the excess mortality in 2021 is more than twice

the empirical standard deviation in the age groups [40,49],[60,69],[70,79], and in 2022 more than twice

the empirical standard deviation in all age groups except [0,14] and [50,59], whereas in 2020 for all age

groups the excess deaths are close to the expected value compared to the empirical standard deviation.

5 Monthly expected mortality

5.1 Methods

In the following two sections, we present a more detailed analysis of the number of deaths during the

years 2020 to 2022. It is well known that the mortality probabilities are not constant but diﬀer from

month to month with peaks at the beginning and the end of the year, and also sometimes in summer

when the weather is too hot (and depending on many other circumstances).

Unfortunately, the data basis for such investigations provided by the Federal Statistical Oﬃce of

Germany is somehow weak. Therefore, again several approximation steps have to be applied. We denote

by dx,t,m, respectively dy,t,m , the number of deaths of xyear old male and yyear old female in year tin

month m. The Federal Statistical Oﬃce of Germany oﬀers tables for d¯x,t,m and d¯y,t,m in the age groups

¯x, ¯y∈ {[0,14],[15,29],[30,34],[35,39],...,[90,94],[95,∞)}which we use for the years t= 2010,...,2022,

see [19].

Denote by fmthe estimated proportion of deaths in month m,m= 1,...,12. I.e., we distribute d¯x,t

21

onto the monthly number of deaths d¯x,t,m via

f¯x,m =1

10

2019

X

t=2010

d¯x,t,m

d¯x,t

,

12

X

m=1

f¯x,m = 1,

where we modify the formula slightly to take into account that 2012 and 2016 were leap years. We list the

obtained estimates in the supplement, Section 8.4. These mortality factors have been highly concentrated

around their mean f¯x,m during the last years, the empirical standard deviation being below 1.5% for all

age groups, and mainly around 0.5%. In the supplement, Section 8.4, we list the empirical standard

deviations for all months and age groups.

Then, we distribute the expected number of deaths for year t= 2020,2021,2022 according to the

factors f¯x,m and f¯y,m ,

ED¯x,t,m =f¯x,mED¯x,t ,ED¯y,t,m =f¯y,mED¯y,t ,

yielding the expected number of deaths in month m. For ¯aa suitable interval in [0,∞), consistent with

the age groups deﬁned by the Federal Statistical Oﬃce of Germany, we set

ED¯a,2021,m =X

¯x⊂¯a

ED¯x,t,m +X

¯y⊂¯a

ED¯y,t,m.

Again, for 2020 we take into account that this is a leap year with one additional day in February. These

expected values should be compared to the observed data d¯a,t,m for m= 1,...,12. The remarks made

at the end of Section 2.1 apply similarly to the computations made in this section.

The monthly mortality probabilities implicitly estimated with this method clearly reﬂect the excess

deaths caused by the usual infections in winter (and possible high temperature weeks in summer), and

thus lower mortality probabilities in spring and fall. This is what is expected, and the COVID-19 excess

deaths must be compared to these expected mortality waves in winter.

Also note, that we do not assume that the population or the age structure is constant during a year

when distributing the expected number of deaths to months. We just assume that the mean population

change in the last years is comparable to the situation in 2020-2022, and thus the changes from January

to December to January in the last ten years mimic the changes in 2020-2022.

Note that the reported number of deaths in 2022 are from the most current data set of the Federal

Statistical Oﬃce of Germany: this is still preliminary, and in particular in the second half of 2022, there

will still be monor changes within the next weeks and months.

5.2 Results

Following the computations described in the previous section, we calculate the expected number of deaths

ED¯a,2021,m for all months m= 1,...,12 in the years t= 2020,2021,2022.

22

To compare the expected and the observed values, we again use the relative diﬀerence

d¯a,2021,m −ED¯a,2021,m

ED¯a,2021,m

.

Table 6: Expected deaths and monthly excess mortality over all age groups.

t= 2020 t= 2021 t= 2022

expected expected expected

observed rel.diﬀ. observed rel.diﬀ. observed rel.diﬀ.

m=1 89,441 90,492 91,328

84,980 -4.99% 106,803 18.02% 89,440 -2.07%

m=2 88,627 86,593 87,400

80,030 -9.70% 82,191 -5.08% 82,796 -5.27%

m=3 92,263 93,345 94,203

87,396 -5.28% 81,901 -12.26% 93,719 -0.51%

m=4 81,088 82,022 82,762

83,830 3.38% 81,877 -0.18% 86,179 4.13%

m=5 79,013 79,895 80,592

75,835 -4.02% 80,876 1.23% 81,767 1.46%

m=6 74,508 75,331 75,979

72,159 -3.15% 76,836 2.00% 79,412 4.52%

m=7 78,389 79,268 79,960

73,795 -5.86% 76,704 -3.24% 85,878 7.40%

m=8 76,809 77,661 78,334

78,742 2.52% 76,402 -1.62% 86,359 10.25%

m=9 73,745 74,564 75,208

74,243 0.68% 77,931 4.52% 80,664 7.26%

m=10 80,294 81,209 81,926

79,781 -0.64% 85,080 4.77% 93,881 14.59%

m=11 80,143 81,061 81,779

85,989 7.30% 93,915 15.86% 87,966 7.57%

m=12 87,237 88,266 89,075

108,792 24.71% 103,171 16.89% 113,115 26.99%

In the following sections, we investigate in detail the monthly excess mortality in the age ranges

¯a= [0,14], [15,29], [30,49], [50,59], [60,79], and [80,∞). The next ﬁgure shows the results for these age

groups.

23

Fig. 8: Development of the monthly excess mortality. For six age groups the black lines show the monthly

excess mortality from January 2020 to December 2022. The red-shaded areas show the the time periods where

a mortality increase was observed; the green-shaded areas show the time periods where a mortality deﬁcit was

observed.

5.2.1 Children [0,14]

In the the age group [0,14], the number of deaths is small and dominated by the relatively large infant

mortality in the ﬁrst year of life. The expected number of deaths in a month is approximately 300, and

hence in the binomial model – which as we know from the investigations in Section 4 heavily under-

estimates the standard deviation – we would already expect oscillations at least of the order

2σ(D[0,14],t,m)≥2qD[0,14],t,m ≈35.

Yet such deviations already lead to an excess mortality of more than 10%. The graph in Fig. 8 shows

in fact such abrupt oscillations, hence we think that any conclusion relying on these numbers has to be

24

taken with great care. The maybe only notable results are, ﬁrst, the well accepted fact that children are

extremely robust with respect to SARS-CoV-2-infections, and the curve seems to be independent of the

usual SARS-CoV-2-infection waves. Second, the presumably diﬀerent social behavior during the Corona

crises seems to lead to a mortality deﬁcit in the younger age groups which is visible here. An exception

are the months May and November 2021, and June and November 2022, with a visible positive mortality

excess, and December 2022 with a serious mortality excess. In the supplement, Section 8.5, we state the

table with precise results.

5.2.2 Young Adults [15,29]

The age group [15,29] is the ﬁrst age group where we list the results in detail.

Table 7: Expected deaths and monthly excess mortality [15,29].

t= 2020 t= 2021 t= 2022

expected expected expected

observed rel.diﬀ. observed rel.diﬀ. observed rel.diﬀ.

m=1 336 327 321

329 -2.22% 290 -11.19% 345 7.39%

m=2 322 301 296

330 2.63% 278 -7.73% 299 0.87%

m=3 333 323 318

320 -3.88% 296 -8.39% 364 14.51%

m=4 325 315 310

288 -11.37% 327 3.69% 306 -1.38%

m=5 335 325 320

311 -7.18% 324 -0.36% 351 9.72%

m=6 329 320 315

329 -0.12% 385 20.43% 316 0.46%

m=7 353 342 337

335 -5.08% 357 4.23% 370 9.80%

m=8 339 329 324

357 5.35% 313 -4.83% 350 8.17%

m=9 325 315 310

305 -6.08% 344 9.15% 368 18.68%

m=10 322 313 308

320 -0.65% 358 14.53% 344 11.85%

m=11 313 304 299

309 -1.22% 321 5.74% 337 12.83%

m=12 312 303 298

311 -0.33% 341 12.61% 365 22.52%

25

As for the age group 0-14 the number of expected and observed deaths is small. Hence, again the

observed excess mortality has to be interpreted with great care. The numbers until March 2021 are

mostly negative and reﬂect the minimal number of deaths by the two COVID-19 waves in this age

range. Somehow unexpectedly, in June 2021, a signiﬁcant excess mortality is observed, followed by a

decrease. However, other than at the beginning of the year, excess mortality remains above zero – with

the exceptions of August 2021 and April 2022 – and has visible peaks in October and December 2021,

again in March and May 2022 and increases drastically in December 2022.

5.2.3 Adults [30,49]

The age group [30,49] is the largest group, and we expect approximately 1,800 deaths per month.

Table 8: Expected deaths and monthly excess mortality [30,49].

t= 2020 t= 2021 t= 2022

expected expected expected

observed rel.diﬀ. observed rel.diﬀ. observed rel.diﬀ.

m=1 1,949 1,908 1,880

1,964 0.78% 1,952 2.29% 1,954 3.96%

m=2 1,850 1,750 1,724

1,782 -3.69% 1,702 -2.75% 1,788 3.69%

m=3 1,957 1,917 1,889

1,924 -1.68% 1,965 2.49% 1,934 2.37%

m=4 1,816 1,779 1,753

1,929 6.25% 1,893 6.41% 1,926 9.88%

m=5 1,841 1,804 1,778

1,845 0.24% 1,998 10.77% 1,894 6.55%

m=6 1,789 1,753 1,726

1,788 -0.06% 1,837 4.82% 1,859 7.68%

m=7 1,839 1,802 1,776

1,834 -0.29% 1,842 2.22% 1,992 12.19%

m=8 1,814 1,778 1,752

1,821 0.38% 1,838 3.39% 1,875 7.04%

m=9 1,736 1,701 1,677

1,762 1.48% 1,865 9.61% 1,802 7.47%

m=10 1,804 1,768 1,742

1,766 -2.12% 1,927 9.02% 1,864 7.03%

m=11 1,744 1,709 1,684

1,756 0.67% 1,944 13.75% 1,844 9.50%

m=12 1,831 1,794 1,768

2,004 9.46% 2,144 19.52% 2,051 16.04%

26

As in the age group [15,29], the numbers in year 2020 are mostly unremarkable, and reﬂect the

minimal number of deaths by the ﬁrst COVID-19 wave in April 2020, and a visible excess mortality in

December 2020 in this age range. Then, the excess mortality ﬂuctuates around zero until March 2021.

From an actuarial perspective, we would expect this to continue until winter.

Somehow unexpectedly, in April and mainly in May 2021, a signiﬁcant increase in excess mortality is

observed, occurring one month before the similar excess mortality in the age group [15,29]. The excess

mortality in May is followed by a decrease up to August. However, other than at the beginning of the

year, excess mortality remains above zero so that the increase in excess mortality in April and May is not

compensated for. In September, there is again a signiﬁcant excess mortality which increases in November

and reaches 20% in December 2021. In 2022 the excess mortality stays always positive, ﬂuctuating around

8%, and reaches again a serious excess mortality of 16% in December.

5.2.4 The exceptional age group [50,59]

The age group [50,59] seems to be exceptional resilient against the factors that drive the excess mortality

in the other age groups. There are neither huge peaks of mortality excess nor serious mortality deﬁcits,

the mortality excess ﬂuctuates around zero.

The numbers in year 2020 are close to zero, ignoring the ﬁrst COVID-19 wave in April 2020, and

show a some mild excess mortality in winter 2020 in this age range. There is a visible peak in April 2021

and December 2021. In 2022, the excess mortality is always close to zero, only in December there is some

positive excess mortality.

This leads to the surprising result, visualized in Fig. 5, that in all pandemic years 2020 to 2022 this

age goup has – in contrast to all neighboring age groups – no serious excess mortality. In the supplement,

Section 8.5, we state the table with precise results.

5.2.5 The retirement age [60,79]

This group consists of the ages [60,79], a ‘mixed’ group where parts of this population are still in health,

and parts are already vulnerable, and for these a SARS-CoV-2-infection can be dangerous. This is visible

in the results for 2020.

27

Table 10: Expected deaths and monthly excess mortality [60,79].

t= 2020 t= 2021 t= 2022

expected expected expected

observed rel.diﬀ. observed rel.diﬀ. observed rel.diﬀ.

m=1 28,409 27,857 27,627

27,905 -1.77% 32,372 16.21% 28,787 4.20%

m=2 27,910 26,423 26,205

26,369 -5.52% 26,505 0.31% 25,643 -2.15%

m=3 29,147 28,569 28,326

28,708 -1.51% 27,195 -4.81% 28,883 1.97%

m=4 26,058 25,555 25,347

27,314 4.82% 27,839 8.94% 26,916 6.19%

m=5 25,811 25,320 25,119

25,201 -2.36% 27,507 8.64% 26,443 5.27%

m=6 24,481 24,022 23,836

23,960 -2.13% 25,274 5.21% 25,620 7.49%

m=7 25,625 25,148 24,954

24,683 -3.68% 25,440 1.16% 27,620 10.68%

m=8 25,105 24,631 24,437

25,595 1.95% 24,939 1.25% 27,153 11.11%

m=9 24,060 23,603 23,415

24,568 2.11% 25,164 6.62% 25,690 9.71%

m=10 25,923 25,422 25,216

26,101 0.69% 27,119 6.67% 29,139 15.56%

m=11 25,669 25,164 24,954

27,211 6.01% 29,519 17.31% 27,621 10.69%

m=12 27,623 27,076 26,848

32,802 18.75% 32,747 20.95% 34,290 27.72%

The results show a decent peak for April 2020, followed by a signiﬁcant peak around December 2020.

The peak of December 2020 continues in January 2021, but then turns into a mortality deﬁcit. In April

2021 we observe a serious excess mortality for two months. In September and October 2021 we see a

decent, and in November and December 2021 again a signiﬁcant excess mortality. The year 2022 starts

with an unremarkable mortality deﬁcit which again in April turns into an excess mortality which remains

for the rest of the year on a high level, and is even above 27% in December.

28

5.2.6 Old ages [80,∞)

The last group consists of the ages ≥80 (beyond the expected life time in Germany which is approximately

at the age of 80), where large parts of the vulnerable population belong to, and a SARS-CoV-2-infection is

particularly dangerous. We list the total expected monthly number of deaths ED¯a,t,m for the population

above the age of the average life expectancy, ¯a= [80,∞), the observed number of deaths and the relative

diﬀerence.

Table 11: Expected deaths and monthly excess mortality [80,∞).

t= 2020 t= 2021 t= 2022

expected expected expected

observed rel.diﬀ. observed rel.diﬀ. observed rel.diﬀ.

m=1 53,222 54,945 56,154

49,408 -7.17% 66,478 20.99% 53,070 -5.49%

m=2 53,232 53,052 54,210

46,559 -12.54% 48,891 -7.84% 50,303 -7.21%

m=3 55,264 57,043 58,287

51,095 -7.54% 47,315 -17.05% 57,447 -1.44%

m=4 47,779 49,324 50,404

49,267 3.12% 46,397 -5.93% 52,079 3.32%

m=5 45,848 47,332 48,365

43,459 -5.21% 45,728 -3.39% 48,251 -0.23%

m=6 42,944 44,334 45,299

41,260 -3.92% 44,312 -0.05% 46,808 3.33%

m=7 45,469 46,936 47,955

42,018 -7.59% 44,096 -6.05% 50,826 5.99%

m=8 44,518 45,954 46,952

46,067 3.48% 44,384 -3.42% 52,037 10.83%

m=9 42,744 44,125 45,083

42,870 0.30% 45,689 3.54% 48,112 6.72%

m=10 47,087 48,612 49,669

46,579 -1.08% 50,414 3.71% 57,455 15.68%

m=11 47,343 48,873 49,932

51,720 9.25% 56,735 16.09% 53,308 6.76%

m=12 52,200 53,887 55,059

68,197 30.65% 62,069 15.18% 70,696 28.40%

29

The results for the age group [80,∞) show a decent peak for April 2020, and a huge peak around

December 2020. The peak of December 2020 continues in January 2021, but then turns into a mortality

deﬁcit until April 2021, where the downwards trend stops. In September and October, we see a decent,

and in November and December 2021 again a serious excess mortality. The year 2022 starts with a

mortality deﬁcit which again in June turns into an excess mortality which remains on a high level for the

rest of the year, and reaches a extremum in December with more than 28% excess mortality.

Although the trend in the age groups [60,79] and [80,∞) looks parallel, it is interesting to point out

the diﬀerences. As can be seen in in Fig. 8, the curve for the age group [80,∞) is below and somehow

parallel to the curve for the age group [60,79]. The main diﬀerence is the deviation of the age group

[60,79] in April and May 2021 where a jump in the mortality behaviour for the this age group is visible.

The age group [80,∞) seems to be more resistant to mortality causes at a larger scale than other age

groups. At certain moments, some people die some months before or after the ‘expected‘ time of death,

but the curve for the excess mortality mostly oscillates around the 0% axes. A visible mortality deﬁcit

until October 2020 with exceptions in April and August is followed by a huge mortality excess peak at

the turn of the year 2020/2021. This in turn is more or less compensated by the mortality deﬁcit in

January to July 2021, the peak around November and December 2021 is nearly compensated in February

to March 2022. Something surprisingly, since August 2022 the mortality excess stays continuously on a

very high level and even increases again to nearly 30% in December 2022.

It is interesting to make this observations visible by calculating the cumulative excess mortality since

January 2020 in absolute numbers. Maybe due to a comparably mild ﬂue saison in 2019/2020, we start

with a negative value. In July 2020, up to 20,000 people more than expected are still alive, which is

compensated in December 2020 to February 2021, where the curve is 10,000 above the expectation, and

then the curve ﬂuctuates to −10,000, to +10,000, and until July 2022 where it is approximately 7,000.

This shows that a mortality deﬁcit or a excess mortality in the age group [80,∞) usually just shifts the

time of death by some months. This changes in the last months of 2022 where where we see a cumulated

excess mortality of 42,000 deaths at the end of the year.

This is in contrast to the situation for the age group [60,79]. The cumulative excess mortality is

steadily increasing up to 55,000 deaths at the end of year 2022. The diﬀerence between both age groups

is demonstrated in Fig. 9.

30

Fig. 9: The cumulative excess mortality. The green areas show the regions of a cumulative mortality deﬁcit,

the red areas of a cumulative excess mortality from January 2020 to December 2022. The age group [80,∞) is

oscillating, the age group [60,79] nearly monotone increasing.

5.3 Stillbirths in the years 2019 to 2022 in Germany

In all previous studies on excess mortality during the COVID-19 pandemic, only the increase in mortality

for the age groups 0 and above have been examined. In the following, it is examined whether similar

increases in mortality than that found for the age groups 0 and above are also found at the level of

stillbirths.

One problem with analyzing excess mortality at the level of stillbirths in Germany is that the deﬁnition

of a ‘stillbirth’ has been changed at the end of 2018. Up to this point, a stillborn child was considered

a stillbirth if a birth weight of at least 500 grams was reached. Since the end of 2018, a stillborn child

is considered a stillbirth if at least 500 grams or the 24th week of pregnancy was reached, which led to

a diagnostically related increase in stillbirths. This means that the ﬁgures on stillbirths are only validly

comparable from 2019 onwards. Thus, estimating excess mortality at the level of stillbirths based on a

modeling of long-term trends in mortality is problematic. Furthermore, the empirical standard deviation

occurred in the years before cannot be determined. Due to these reasons, we only descriptively report

the course of stillbirths from 2019 on.

When analyzing the number of stillbirths, it is important to note that they must be placed in relation

to the number of total births, because an increase or decrease in the number of total births is automatically

31

accompanied by an increase or decrease in stillbirths. Fig. 10 shows in the ﬁrst panel the number of live

births per quarter [20] and in the second panel the number of stillbirths per quarter [21] since 2019. As

can be seen from the shift in the seasonal peaks of the stillbirths compared to the seasonal peaks of the

live births, stillbirths precede live births from the same pregnancy cohort by about one trimester. Thus,

to correctly control for the eﬀect of a general increase or decrease in the number of total births, the

number of total births must be calculated as the sum of the number of stillbirths in a quarter and the

number of live births in the following quarter.

Fig. 10 shows in the third panel the number of stillbirths per 1,000 total births, and in the fourth

panel the quarterly increase in the number of stillbirths per 1,000 total births in the years 2021, and 2022

compared to the mean across the years 2019 and 2020. Note that the number of stillbirths per 1,000 total

births cannot be determined for the third quarter 2022 because the number of live births in the fourth

quarter 2022 has not yet been published by the Federal Statistical Oﬃce of Germany. Also note that the

quarterly pattern observed in the year 2022 cannot be validly interpreted because only preliminary data

is available based on the reporting month, with the data being assigned to the month of death only with

the publishing of the ﬁnal data by the Federal Statistical Oﬃce of Germany.

Fig. 10: Stillbirths in the years 2019 to 2022 in Germany. The ﬁrst panel shows the number of live births

per quarter from 2019 to 2022, the second panel the number of stillbirths per quarter from 2019 to 2022, the third

panel the number of stillbirths per 1,000 total births (sum of the number of stillbirths in a quarter and the number

of live births in the following quarter) per quarter from 2019 to 2022, and fourth panel the quarterly increase in

32

the number of stillbirths per 1,000 total births in the years 2021, and 2022 compared to the mean across the years

2019 and 2020.

Until the end of 2021, the number of live births shows a stable course with a regularly repeating

seasonal pattern. In the ﬁrst quarter 2022, a sudden and sustained drop in the number of births is

observed. Regarding the number of stillbirths, a stable course is observed until the end of the ﬁrst

quarter of 2021. In the second quarter of 2021, a sudden increase in stillbirths is observed, despite the

stable course of live births until the end of 2021. Compared to the quarterly number of stillbirths per

1,000 total births in the years 2019 and 2020, the number of stillbirths increased by 9.4% in the second

quarter of 2021 and by 19.6% in the fourth quarter of 2021. This is similar to the increases in mortality

observed for the age groups 0 and above: whereas in the year 2020 no change in stillbirths is found

compared to the previous year, in the year 2021, a sudden increase in stillbirths is observed in the second

quarter of 2021 which reaches a high level in fourth quarter of 2021.

6 Discussion

In the previous sections, we estimated the expected number of all-cause deaths and the increase in all-

cause mortality for the pandemic years 2020 to 2022 in Germany. The results revealed several previously

unknown mortality dynamics that require a reassessment of the mortality burden brought about by the

COVID-19 pandemic.

The analysis of the yearly excess mortality showed a marked diﬀerence between the pandemic years

2020, 2021, and 2022. Whereas in the year 2020 the observed number of deaths was extremely close to

the expected number with respect to the empirical standard deviation, in 2021, the observed number

of deaths was far above the expected number (more than twice the empirical standard deviation), and

further increases in 2022 (above four times the standard deviation). An age-dependent analysis showed

that the strong excess mortality observed in 2021, and 2022 was mainly due to an above-average increase

in deaths in the age groups between 15 and 79. A detailed analysis of the monthly excess mortality showed

that the high excess mortality observed in the age groups between 15 and 79 started to accumulate from

April 2021 onwards. A similar pattern was observed for the number of stillbirths which was similar to

the previous years until March 2021, after which also a sudden and sustained increase was observed.

Taken together, these ﬁndings raise the question what happened in spring 2021 that led to a sudden

and sustained increase in mortality, although no such eﬀects on mortality had been observed during the

COVID pandemic so far. In the following sections, possible explanatory factors are explored.

6.1 Possible factors inﬂuencing mortality

As already mentioned, apart from the population structure, the number of deaths in a year depends

on several diﬀerent factors, the most important being maybe the severity of the ﬂue, and the number

of extremely hot weeks. The ﬂuctuations between diﬀerent years, and thus the approximation of the

empirical standard deviation ˆσ(Dt) in Section 4, includes all these factors. It is unclear, rather subjective,

33

and most probably impossible to precisely deﬁne ‘extreme events’, to calculate the inﬂuence of such

extreme events, and to adjust mortality to ‘entirely normal’ years. Thus our calculations gives the

expected number of deaths taking into account all these extreme and non-extreme eﬀects which are

contained in the diﬀerent life tables. We tried to quantify the sensitivity of our approach in Section 3

and Section 4 against the background of extreme events in the last years.

For the pandemic years 2020 to 2022, it is clear that the number of deaths has been inﬂuenced directly

and indirectly by COVID-19. First, clearly, there has been a serious number of COVID-19 deaths, either

as the only reason for death or in combination with several other causes, which also might have caused

death independently of COVID-19, see e.g. the discussion in Section 5.2.6 and in the forthcoming

Section 6.2. Second, the vaccination campaign which started in 2021 should be visible in a reduced

excess mortality, or even better as a mortality deﬁcit. An attempt to compare our results to the number

of vaccinations is the content of Section 6.3.

Third, the indirect eﬀects on mortality due to the COVID-19 measures are extremely harder to

quantify. Several aspects may contribute to an excess mortality or a mortality deﬁcit. In Germany, strict

control measures since 2020 limited personal freedom, schools were partially closed, there were severe

lockdowns. This substantially inﬂuenced the risk of road accidents and other outdoors casualties. On the

other hand, many clinical services have been delayed or avoided in 2020, 2021, and 2022. All these and

many more factors inﬂuenced mortality in diﬀerent directions and on diﬀerent time scales, but most of

them are hard to measure, many eﬀects are highly correlated, and it seems to be impossible to quantify

the overall impact of the control measures on the number of deaths.

6.2 COVID-19 deaths and mortality

In this section we compare the excess mortality since March 2020 to the reported number of COVID-19

deaths by the German Robert Koch Institute. The Robert Koch Institute provides the weekly number

of COVID-19 deaths [22] for the age groups [0,9], [10,19], . .. (which diﬀer from the age groups used by

the Federal Statistical Oﬃce of Germany); in addition these numbers are incomplete because all numbers

below 4 are not stated due to data security reasons.

Even when the reporting system in Germany seems to be imprecise and partially insuﬃcient, there

should be a serious correlation between the reported number of deaths and the excess mortality. To

make the diﬀerence between the excess deaths and the COVID-19 deaths visible, we show the monthly

development of the number of reported COVID-19 deaths and the excess mortality in Fig. 11, and on

the same scale the diﬀerence between both in Fig. 12.

34

Fig. 11: COVID-19 deaths versus excess mortality. The blue squares show the number of reported COVID-

19 deaths, the red squares the mortality deﬁcit, respectively the excess mortality from January 2020 to December

2022.

Fig. 12: Diﬀerence of COVID-19 deaths and excess mortality. The line shows the diﬀerence between the

number of excess deaths and the number of COVID-19 deaths from January 2020 to December 2022.

Until July 2020, the number of excess deaths is below the number of reported COVID-19 deaths, and

except for April 2020, a mortality deﬁcit is observed despite the reporting of COVID-19 deaths. From

August 2020 to December 2020, the numbers of excess deaths and reported COVID-19 deaths largely

coincide. However, after that, the number of COVID-19 deaths stays on a high level while all-cause

mortality decreases, and in February and March 2021, a noticeable all-cause mortality deﬁcit is observed

despite a high number of reported COVID-19 deaths of up to 10,000. Starting in September 2021, a

marked increase in excess mortality is observed that is not accompanied by a comparable increase in

reported COVID-19 deaths. From January 2022 onwards, both curves decouple, showing partly oppo-

site patterns of increases versus decreases. From June 2022 onwards, the number of excess deaths is

35

increasingly larger than the number of reported COVID-19 deaths, reaching very large diﬀerences in De-

cember 2022 where over 24,000 excess deaths are observed but only slightly more than 3,500 COVID-19

deaths reported. It thus is obvious that the number of reported COVID-19 deaths is ﬂuctating somehow

independently of the excess mortality, and contains a large number of ‘expected’ deaths.

It is also elucidating to compare the cumulative number of COVID-19 deaths to the cumulative

number of excess deaths in Fig. 13. The cumulative number of reported COVID-19 deaths is increasingly

higher than the cumulative number of excess deaths.

Fig. 13: Cumulative COVID-19 deaths versus cumulative excess mortality. The blue squares show the

cumulative number of reported COVID-19 deaths, the blue squares the cumulative mortality deﬁcit, respectively

the excess mortality from March 2020 to December 2022.

Because the Federal Statistical Oﬃce of Germany uses diﬀerent age groups, direct comparisons are

made diﬃcult, e.g. for the age group [15,29] used in the previous section. Therefore we divide the number

of COVID-19 deaths in the age group [10,19] into two equal parts to obtain the number of COVID-19

deaths in the age groups [0,14] and [15,29], estimate the number of deaths for those weeks with less than

4 deaths, and divide each week where two months overlap between these two months.

We list the number of excess deaths in six age groups and compare these to the COVID-19 deaths,

as a timetable we use the ﬁrst pandemic year 04/2020 – 03/2021 and compare this to the second year

04/2021 – 03/2022. It seems diﬃcult to ﬁnd a convincing pattern which explains the dependence of the

excess deaths on the COVID-19 deaths.

36

Table 12: Expected vs. observed deaths, and excess deaths vs. COVID-19 deaths

04/20–03/21 04/21–03/22

age exp. exp.

range obs. abs.diﬀ. obs. abs.diﬀ.

COVID COVID

0-14 3,519 3,514

3,195 -324 3,425 -89

25 73

15-29 3,904 3,801

3,729 -175 4,078 277

81 120

30-49 21,790 21,380

22,124 334 22,964 1,584

617 1,256

50-59 58,269 57,383

57,385 -884 58,775 1,392

2,094 3,072

60-79 313,204 308,100

323,507 10,303 328,861 20,761

21,351 18,645

80-∞580,971 598,030

594,121 13,150 600,644 2,614

53,882 30,465

0-∞981,656 992,209

1,004,061 22,405 1,018,747 26,538

78,050 53,631

In the age groups [0,14], [15,29] and [50,59], mortality deﬁcits occur but the number of COVID-19

deaths is positive. In the age group [30,49], the number of excess deaths seems to ﬁt the number of

COVID-19 deaths. In the age group between 60 and 79 years, of the approximately 40,000 COVID-19

deaths reported until the end of March 2022, approximately 9,000 did not show up as excess deaths.

The strongest divergence is found in the age group over 80 years, where of the approximately 84,000

COVID-19 deaths reported until the end of March 2022, approximately 69,000 did not show up as excess

deaths. Taken together, of the 132,000 reported COVID-19 deaths in the age groups over 20 years,

more than 83,000 did not show up as excess deaths and are thus contained in the ‘expected’ number

of deaths. Since other factors beyond COVID-19, such as delayed or avoided clinical services, may have

contributed to the number of excess deaths, the number of reported COVID-19 deaths which actually

37

represent expected deaths and not excess deaths is probably even higher.

Taken together, it seems to be misleading to measure the risk of the COVID-19 pandemic only using

the reported deaths. One should rather use the excess mortality curve than the number of reported

COVID-19 deaths, or a combination of both, to carve out the moments of high risk, and to evaluate the

total risk of a pandemic.

Beyond the problem that the number of reported COVID-19 deaths cannot be validly used to assess

the eﬀects of the COVID-19 pandemic on mortality, it seems also unlikely that the high excess mortality

in 2021 in the age groups under 80 years can be explained by COVID-19 deaths because the marked

increases in excess mortality in April to June 2021 - the mortality increases abruptly by 13% from March

to April 2021 in the age group between 15 and 59 - and also in October to December 2021 were not

accompanied by comparable increased in the number of COVID-19 deaths. Furthermore, it seems also

very unlikely that the abrupt increase of the mortality is due to delayed or avoided clinical services which

should lead to much smoother changes, or due to side eﬀects of COVID-19 measures. This is the more

unlikely in the year 2022 where excess mortality increases even further despite a decrease in reported

COVID-19 deaths, and despite the fact that clinical care should slowly return to normal. Thus, it remains

to investigate the factors which could lead to the surprising jumps in excess mortality in April to June

2021, in October and November 2021, and in the year 2022.

6.3 COVID-19 vaccination and mortality

In April 2021, an extensive vaccination campaign started in Germany. Comparing the number of vac-

cinations [23] to the excess mortality should show the sum of two theoretically possible eﬀects of vacci-

nations: the prevention of infections and deaths through immunization should decrease the number of

excess deaths, and, if there were side eﬀects in the form of deaths, the occurrence of such side eﬀects

should increase the number of excess deaths. The following Fig. 14 shows on the left scale the number

of excess deaths, respectively death deﬁcit, and on the right scale the number of vaccinations.

38

Fig. 14: Number of vaccinations versus excess mortality. The red line shows the death deﬁcit, respectively

the excess deaths, the four dashed lines the number of vaccinations from January 2021 to June 2022.

As can be seen in Fig. 14, a visible positive eﬀect of the vaccinations on excess mortality does not

occur. Instead, the opposite is observed. Although at the beginning of September 2021, 82.7% of the

population over 60 years and 65.2% of the population from 18–59 years were fully vaccinated, the number

of excess deaths nevertheless started to increase strongly, reaching a level of almost 15,000 excess deaths

in December 2021, where 86.1% of the population over 60 years and 75.7% of the population from 18–59

years were fully vaccinated. At the beginning of March 2022, 88.6% of the population over 60 years and

83.3% of the population from 18–59 years were fully vaccinated, and 77.4% of the population over 60

years and 60.6% of the population from 18–59 years had even received a third vaccination. Despite this

high level of population-wide vaccinations, excess mortality starts to monotonically increase, reaching a

level of more than 24,000 excess deaths in December 2022.

The observation that excess mortality increased with increased vaccinations not only casts some

doubts on the eﬀectiveness of the vaccinations. From the perspective of pharmacovigilance, such an

observation represents a safety signal because such a temporal relationship between the number of excess

deaths and the number of vaccinations would occur if the vaccinations caused unwanted deaths as a side

eﬀect. This impression is strengthened by a closer inspection of the courses of the numbers of vaccinations

and excess deaths in the second pandemic year April 2021 to March 2022. The number of excess deaths

closely follows the course of the number of vaccinations, showing an increase as soon as the number of

vaccinations increases, and a decrease as soon as the number of vaccinations decreases. A similar safety

signal in terms of a temporal relationship between an increase in vaccinations and deaths is also observed

at the level of stillbirths. Exactly with the beginning of the vaccinations in the age group [18,59], the

number of stillbirths suddenly increased after being stable for at least the two previous years.

Safety signals such as the observation of a temporal relationship between the administration of vaccines

39

and the occurrence of adverse events do not necessarily imply a causal relationship since there may be

potential third variables that inﬂuence both the course of vaccinations and the course of excess deaths.

Thus, a safety signal does not indicate a causal relationship between a side eﬀect and a drug but is only

a hypothesis that calls for further assessment.

However, there is ﬁrst evidence from autopsy studies that vaccinations can at least cause deaths. For

instance, in a study of a research team led by Peter Schirmacher [24], out of 35 bodies found unexpectedly

dead at home with unclear causes of deaths within 20 days following COVID-vaccination, autopsies

revealed causes of death due to pre-existing illnesses in only 10 cases. From the remaining 25 cases, in three

cases it was concluded from the autopsies that vaccination-induced myocarditis was the likely cause of

death, and in two cases it was concluded that this was possibly the case. According to [24], Supplementary

Table 1, vaccination was the cause of deaths in further cases as well. For instance, a 38-year-old man with

no relevant preexisting disease died due to vaccine-induced thrombotic thrombocytopenia; a 23-year- old

woman with no relevant preexisting disease died due to pulmonary embolism, which may also suggest

vaccination as the cause of death.

These ﬁndings indicate two important aspects. First, the ﬁndings show that COVID-vaccinations can

cause deaths as a side eﬀect. Second, the ﬁndings show that deadly side eﬀects of COVID-vaccinations

are not extreme exceptional cases. The authors of the paper correctly conclude that epidemiological

conclusions in terms of incidence or risk estimation cannot validly drawn from their study. However,

the fact that the re-examination of only 35 deaths of only one speciﬁc type (bodies found unexpectedly

dead at home) in only a small region in Germany (catchment area of the Heidelberg University Hospital)

already reveals so many deaths that have likely or probably been caused by a COVID-vaccination at least

suggests that COVID-vaccine-induced deaths are not extremely unlikely.

Given the temporal relationship between the increase in vaccinations and excess mortality from the

beginning of the vaccinations campaign onwards, it seems surprising that a respective safety signal has

not been detected in the pharmacovigilance by the Paul-Ehrlich-Institut (PEI), which is responsible for

the safety monitoring of drugs in Germany. A closer inspection of the methods used by the PEI to

monitor possibly deadly side eﬀects of the COVID-vaccinations reveals that a ﬂawed safety analysis is

used that will not indicate a safety signal even if a vaccine causes extremely large numbers of unexpected

deaths.

The PEI uses a so-called observed-versus-expected analyses where the expected number of all-cause

deaths in the vaccinated group is compared to the number of deaths that is has been reported to the PEI

with a suspected connection to a COVID-vaccination. If the number of reported suspected vaccine-related

deaths is not signiﬁcantly higher than the number of expected all-cause deaths, the PEI concludes that

there is no safety problem. For instance, in the safety report of August 19, 2021 [25], the PEI calculates

that 75,284 all-cause deaths within a period of 30 days after the vaccinations are expected for the group

of people that have been vaccinated with Corminaty. From the fact that the number of 926 reported

suspected vaccine-related deaths does not exceed the threshold of the expected 75,284 all-cause deaths,

the PEI concludes that there is no warning signal of increased post-vaccination mortality for Corminaty.

Such a safety analysis is profoundly absurd. For a safety signal to occur, more suspected vaccine-

related deaths would need to be reported than are caused by all other causes of death (cancer, heart

40

disease, stroke, etc.) combined. Thus, it is not surprising that a safety signal has not been detected in

the pharmacovigilance by the PEI, because the occurrence of safety signals is essentially impossible due

to the used ﬂawed methods.

Since the available mortality data do not allow to determine the expected and observed numbers of

deaths for the vaccinated group only, it is impossible to examine what would have been observed if the

PEI had applied a correct safety analysis. However, to at least demonstrate how a proper observed-

versus-expected analysis should be done, two time periods can be compared: the time of April 2020 to

March 2021 (ﬁrst pandemic year) can be used as a rough estimate of the expected number of excess

deaths without vaccinations. The estimated expected excess deaths can be compared with the observed

number of excess deaths in the time of April 2021 to March 2022 (second pandemic year) where large

parts of the population were vaccinated. The following Table 13 shows the results of such an analysis for

six age groups.

Table 13: Expected deaths and excess mortality 04/20–03/22.

04/20–03/21 04/21–03/22

age exp. exp.

range obs. abs.diﬀ. rel.diﬀ. obs. abs.diﬀ. rel.diﬀ.

0-14 3,519 3,514

3,195 -324 -9.20% 3,425 -89 -2.54%

15-29 3,904 3,801

3,729 -175 -4.48% 4,078 277 7.28%

30-49 21,790 21,380

22,124 334 1.53% 22,964 1,584 7.41%

50-59 58,269 57,383

57,385 -884 -1.52% 58,775 1,392 2.43%

60-79 313,204 308,100

323,507 10,303 3.29% 328,861 20,761 6.74%

80-∞580,971 598,030

594,121 13,150 2.26% 600,644 2,614 0.44%

For alle age groups under eighty years, a signiﬁcant mortality increase is observed in the second pan-

demic year where large parts of the population were vaccinated. According to the empirical standard de-

viations for the diﬀerent age groups given in Section (8.3), ˆσ(d[0,14]) = 158,ˆσ(d[15,29]) = 148,ˆσ(d[30,49]) =

427,ˆσ(d[50,59]) = 868,ˆσ(d[60,79])=5,088 and ˆσ(d[80,∞))=9,924, excess mortalities which are far beyond

2ˆσdid not occur in the in the ﬁrst pandemic year without vaccinations but only in the second pandemic

with vaccinations (age groups [30,49] and [60,79]; these are highlighted in the table). Thus, the amount of

excess mortality observed in the second pandemic year with vaccinations is much higher than the amount

of excess mortality in the ﬁrst pandemic year without vaccinations. This on the one hand contrasts with

41

the expectation that the vaccination should decrease the number of COVID-19 deaths, and on the other

hand indicates a safety signal.

The only exception is the last age group [80,∞), where in the ﬁrst year a larger number of excess

deaths was observed than in the second year. However, when interpreting this ﬁnding, it has to be taken

into account that in this age range, there was a huge mortality deﬁcit from 2019 until October 2020,

which was compensated in November, December 2020, and January 2021. Such an eﬀect could not occur

a second time within one year. Furthermore, even if the mortality decrease from the ﬁrst to the second

pandemic year observed in the age group [80,∞) would be a direct eﬀect of the vaccinations, this would,

at least according to the results in the Table 13, not justify the vaccination of the whole population

independently of age. In total, the decrease of the number of excess deaths by 10,550 in the age groups

above 80 and the increase of the number of deaths by 14,657 in the younger age ranges yield a negative

net eﬀect.

Taken together, although one would expect that vaccinating large parts of the population should have

reduced excess mortality, the contrary is observed. Both excess mortality and the number of stillbirths

increased with increased vaccinations, and excess mortality was in all age groups below 80 years higher

in the second year of the pandemic, where large parts of the population were vaccinated, than in the ﬁrst

year, where almost nobody was vaccinated. These observations are surprising and should lead to several

more detailed investigations from diﬀerent scientiﬁc ﬁelds to rule out that these safety signals occur due

to the existence of unknown side eﬀects of the COVID-vaccines.

7 Conclusion

The present study used the state-of-the-art method of actuarial science to estimate the expected number

of all-cause deaths and the increase in all-cause mortality for the pandemic years 2020 to 2022 in Germany.

In 2020 the observed number of deaths was extremely close to the expected number, but in 2021

the observed number of deaths was far above the expected number in the order of twice the empirical

standard deviation, and in 2022 above the expected number even more than four times the empirical

standard deviation. The analysis of the age-dependent monthly excess mortality showed, that a high

excess mortality observed in the age groups between 15 and 79 starting from April 2021 is responsible

for the excess mortality in 2021, and 2022. An analysis of the number of stillbirths revealed a similar

mortality pattern than observed for the age group between 15 and 79 years.

As a starting point for further investigations explaining this mortality patterns, we compared the

excess mortality to the number of reported COVID-19 deaths and the number of COVID-19 vaccina-

tions. This leads to several open questions, the most important being the covariation between the excess

mortality and the COVID-19 vaccinations.

42

References

[1] von Stillfried S., B¨ulow R. D., R¨ohrig R., Boor P., for the German Registry of COVID-19 Autopsies

(DeRegCOVID): First report from the German COVID-19 autopsy registry. The Lancet Regional

Health - Europe 15, 100330 (2022). doi.org/10.1016/j.lanepe.2022.100330

[2] Saragih I. D., Advani S., Saragih I. S., Suarilah I., Susanto I., Lin C. J.: Frailty as a mortality

predictor in older adults with COVID-19: A systematic review and meta-analysis of cohort studies.

Geriatric Nurs. 42(5), 983–992 (2021). doi.org/10.1016/j.gerinurse.2021.06.003

[3] Hung I. F. N., Zhang A. J., To K. K. W., Chan, J. F. W. and Zhu S. H. S., Zhang R., Chan T.,

Chan K., Yuen, K.: Unexpectedly Higher Morbidity and Mortality of Hospitalized Elderly Patients

Associated with Rhinovirus Compared with Inﬂuenza Virus Respiratory Tract Infection. Int. J. Mol.

Sci. 18, 259 (2017). doi.org/10.3390/ijms18020259

[4] Baum K.: Considerations on excess mortality in Germany in the year 2020 and 2021. Dtsch. Med.

Wochenschr. 147(7), 430–434 (2022).

[5] COVID-19 Excess Mortality Collaborators: Estimating excess mortality due to the COVID-19 pan-

demic: a systematic analysis of COVID-19-related mortality, 2020–21. The Lancet 399, 1513–1536

(2022). www.thelancet.com/action/showPdf?pii=S0140-6736%2821%2902796-3

[6] Bauer, R., Speringer, M., Fr¨uhwirt, P., Seidl, R., Trautinger, F.: Assessing excess mortality in

Vienna and Austria after the ﬁrst year of the COVID-19 pandemic. Vienna Yearbook of Population

Research 20, 1–22, (2022). https://doi.org/10.1553/populationyearbook2022.dat.1

[7] De Nicola, G., Kauermann, G., H¨ohle, G.: On assessing excess mortality in Ger-

many during the COVID-19 pandemic (Zur Berechnung der ¨

Ubersterblichkeit in Deutsch-

land w¨ahrend der COVID-19-Pandemie). AStA Wirtsch Sozialstat Arch 16, 5–20, (2022).

link.springer.com/article/10.1007/s11943-021-00297-w

[8] De Nicola G., Kauermann G.: An update on excess mortality in the second year of the COVID-19

pandemic in Germany (Ein Update zur ¨

Ubersterblichkeit im zweiten Jahr der COVID-19 Pandemie in

Deutschland). AStA Wirtsch Sozialstat Arch 16, 21–24 (2022). doi.org/10.1007/s11943-022-00303-9

[9] Kowall B., Standl F., Oesterling F., Brune B., Brinkmann M., Dudda M., Pﬂaumer P., J¨ockel K.,

Stang A.: Excess mortality due to Covid-19? A comparison of total mortality in 2020 with total

mortality in 2016 to 2019 in Germany, Sweden and Spain. PLoS ONE 16(8): e0255540 (2021).

doi.org/10.1371/journal.pone.0255540

[10] Levitt M., Zonta F., Ioannidis J.P.A.: Comparison of pandemic excess mortality

in 2020–2021 across diﬀerent empirical calculations. Environ Res. 213: 113754 (2022)

doi.org/10.1016/j.envres.2022.113754

43

[11] World Health Organization: Global Excess Deaths Associated with COVID-19, Jan-

uary 2020–December 2021. https://www.who.int/data/stories/global-excess-deaths-associated-with-

covid-19-january-2020-december-2021 (Accessed on May 6, 2022)

[12] Federal Statistical Oﬃce of Germany: Number of deaths and excess mortality (Sterbefallzahlen und

¨

Ubersterblichkeit). www.destatis.de/DE/Themen/Querschnitt/Corona/Gesellschaft/bevoelkerung-

sterbefaelle.html (Accessed on January 19, 2023)

[13] Mølbak K., Mazick A.: European monitoring of excess mortality for public health ac-

tion (EuroMOMO). European Journal of Public Health 23 suppl. 1, ckt126.113 (2013).

doi.org/10.1093/eurpub/ckt126.113

[14] Keiding N., Clayton D.: Standardization and control for confounding in observational studies: a

historical perspective. Stat. Sci. 29, 529–558 (2014). doi.org/10.1214/13-STS453

[15] Staub K., Panczak R., Matthes K.L., et al.: Historically High Excess Mortality During the COVID-

19 Pandemic in Switzerland, Sweden, and Spain. Annals of Internal Medicine 175 (4), 523–532

(2022). www.acpjournals.org/doi/10.7326/M21-3824

[16] Federal Statistical Oﬃce of Germany: Life tables 2015/17, 2016/18, 2017/2019. www-

genesis.destatis.de/genesis//online?operation=table&code=12411-0006 (Accessed on January 19,

2023)

[17] German Association of Actuaries (DAV): Life table DAV 2004R. www.aktuar.de

[18] Federal Statistical Oﬃce of Germany: Population statistics. www-

genesis.destatis.de/genesis//online?operation=table&code=12411-0006 (Accessed on January

19, 2023)

[19] Federal Statistical Oﬃce of Germany: Death statistics. www.destatis.de/DE/Themen/Gesellschaft-

Umwelt/Bevoelkerung/Sterbefaelle-Lebenserwartung/Tabellen/sonderauswertung-sterbefaelle.html

(Accessed on January 19, 2023)

[20] Federal Statistical Oﬃce of Germany: Number of births. www-

genesis.destatis.de/genesis//online?operation=table&code=12612-0002 (Accessed on January

19, 2023)

[21] Federal Statistical Oﬃce of Germany: Number of stillbirths. Available upon request from the FSOG.

(Accessed on January 24, 2023)

[22] Robert Koch Institut: COVID-19 Todesf¨alle nach Sterbedatum.

www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Projekte RKI/ COVID-

19 Todesfaelle.html (retrieved on January 19, 2023)

44

[23] Robert Koch Institut: Digitales Impfquoten-Monitoring COVID-19.

www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Daten/ Impfquotenmonitor-

ing.xlsx? blob=publicationFile (retrieved on January 19, 2023)

[24] Schwab, C., Domke, L.M., Hartmann, L. et al.: Autopsy-based histopathological characterization of

myocarditis after anti-SARS-CoV-2-vaccination. Clin. Res. Cardiol. (2022). doi.org/10.1007/s00392-

022-02129-5

[25] Paul-Ehrlich-Institute: Bericht ¨uber Verdachtsf¨alle von Nebenwirkun-

gen und Impfkomplikationen nach Impfung zum Schutz vor COVID-19.

https://www.pei.de/SharedDocs/Downloads/DE/newsroom/dossiers/sicherheitsberichte/sicherheitsbericht-

27-12-bis-31-07-21.pdf (retrieved on January 29, 2023)

Christof Kuhbandner Matthias Reitzner

Universit¨at Regensburg Universit¨at Osnabr¨uck

Institut f¨ur Experimentelle Psychologie Institut f¨ur Mathematik

93040 Regensburg 49069 Osnabr¨uck

Germany Germany

45

8 Supplementary Material

8.1 Yearly Mortality Excess

In Section 2.2, we have stated the total expected number of deaths EDtin 2020–2022 only for certain

age groups; in the following table, we list the detailed expected number of deaths EDx,t for males and

EDy,t for females and for each age x, y separately.

age EDx,2020 EDy,2020 EDx,2021 EDy,2021 EDx,2022 EDy,2022

0 1051 837 1053 838 1051 837

1 411 330 403 322 408 327

2 70 56 68 54 67 53

3 50 41 49 40 49 39

4 43 37 43 37 42 37

5 38 34 39 34 39 35

6 35 29 35 29 36 30

7 32 24 32 24 33 24

8 29 21 29 21 30 21

9 29 19 29 19 29 19

10 29 22 28 21 28 21

11 29 26 28 26 28 26

12 31 29 31