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Abstract and Figures

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.
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Excess mortality in Germany 2020-2022
Christof Kuhbandner1, Matthias Reitzner2,*,
1Department of Human Sciences, University of Regensburg, 93040 Regensburg,
Germany
2Institute for Mathematics, Osnabr¨uck University, 49069 Osnabr¨uck, Germany
* matthias.reitzner@uni-osnabrueck.de
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. By contrast, in 2021, the observed number
of deaths was two empirical standard deviations above the expected number. The high
excess mortality in 2021 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 11
percent in the second 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 1
In the last two years, the burden of the COVID-19 pandemic on mortality has been 2
intensively discussed. Basically, since COVID-19 is an infectious disease that is caused 3
by a new virus, it is expected that many people have died because of the new virus who
4
otherwise would not have died. In fact, this expectation represents one of the central 5
justifications for the taking of countermeasures against the spread of the virus. Due to
6
this reason, several previous studies have tried to estimate the extent of the mortality 7
burden that has been brought about by the COVID-19 pandemic. 8
At first glance, it seems obvious to simply estimate the burden of the COVID-19 9
pandemic on mortality based on the number of officially reported COVID-19-related 10
deaths. However, this has been proven to be difficult due to several reasons. 11
August 10, 2022 1/42
1.1 Reported COVID-19-Deaths: The Problem 12
A first difficulty is the problem that it is unclear whether a reported COVID-death died
13
because of a SARS-CoV-2-infection or only with a SARS-CoV-2-infection. For instance,
14
according to a published analysis of the German COVID-19 autopsy registry from 15
March 2020 to the beginning of October 2021 [1], only 86% of the autopsied deaths with
16
a COVID-19 diagnosis died from COVID-19. In particular, a closer look at the 17
diagnostics used in this study suggests that this may be an overestimation. For instance,
18
87 of the 1,095 autopsied persons with the autopsy result of an “unspecific cause of 19
deaths” were excluded although such persons seem not to have died from COVID-19. In
20
addition, 10 percent of the deaths treated as “died from COVID-19” died actually due 21
to bacterial or fungal super-infections or due to therapy-associated reasons and are thus
22
not directly caused by COVID-19. These examples highlight the general problem that 23
the answer to the question whether COVID-19 was the actual cause of death depends 24
on the used definition of ‘causality’. 25
A second difficulty is that even if a person died from COVID-19, this does not rule 26
out the possibility that the person would have died as well even if there had been no 27
COVID-19 pandemic. Many of the people that have died from COVID-19 were highly 28
frail [2], and these people might have died from other causes of deaths if they had not 29
died from COVID-19. For instance, it has been shown that rhinovirus infections have a
30
high mortality risk for vulnerable elderly people as well [3]. Thus, even if there had 31
been no SARS-CoV-2-infection waves, these individuals might instead have died in one
32
of the rhinovirus-infection waves. Accordingly, even if there is a large number of deaths
33
that were caused by a SARS-CoV-2-infection, this would not necessarily mean that all 34
these deaths are additional deaths that would not have occurred if there had been no 35
COVID-19 pandemic. 36
1.2 Estimating the Burden of the COVID-19 Pandemic Based 37
on All-Cause Mortality 38
An obvious way to solve such problems when estimating the burden of the COVID-19 39
pandemic on mortality is to compare the number of observed all-cause deaths 40
independently of the underlying causes of deaths with the number of all-cause deaths 41
that would have been expected if there had been no pandemic. If there is a new virus 42
that causes additional deaths beyond what is usually expected, the number of observed
43
all-cause deaths should be larger than the number of usually expected deaths, and the 44
higher the number of observed deaths is above the number of usually expected deaths, 45
the higher is the burden of a pandemic on mortality. In particular, beyond the 46
advantage that the above-mentioned problems with the number of the reported 47
COVID-19-related deaths are avoided, another advantage is that additional indirect 48
negative impacts of a pandemic on mortality are covered as well, such as a possible 49
pandemic-induced strain of the health care system. 50
Due to these reasons, it is not surprising that several attempts have been made to 51
estimate the increase in all-cause mortality during the COVID-19-pandemic [410]. 52
Since the death of a person is a clear diagnostic fact, and since highly reliable data on 53
mortality are available for several countries, at first glance, one may expect that the 54
question of whether more people have died during the COVID-19-pandemic than is 55
usually expected can be clearly answered. 56
However, the existing attempts show very large differences in the estimated increase
57
in all-cause mortality during the COVID-19-pandemic. This can be illustrated for 58
Germany where highly reliable data on the number of all-cause deaths even at the level
59
of individual days are available. The estimated increase in all-cause mortality during 60
the pandemic years 2020 and 2021 varies from 203,000 additional deaths [5] to only 61
August 10, 2022 2/42
29,716 additional deaths [6, 7], and for the pandemic year 2020, it has even been 62
estimated that less all-cause deaths have been observed than usually expected [8]. 63
How can this large variability in the estimated increase in all-cause mortality be 64
explained? While the number of observed all-cause deaths is a fixed and clearly defined
65
number, the estimation of the usually expected deaths is relatively complex and entails
66
several choices of mathematical models and parameters and which can lead to large 67
differences in the estimated values 68
Against this background, the present article has three objectives: 69
1. To provide an overview and an evaluation of the choices that must be made. 70
2. To demonstrate that the amount of increase in all-cause mortality must be 71
understood as an inherent fuzzy construct that varies depending on the chosen 72
perspective. 73
3. To provide a best-practice method how to estimate and interpret the increase in 74
all-cause mortality using the example of observed all-cause deaths in Germany in
75
the years 2020 to 2022. As will be shown, a proper analysis of the increase in 76
all-cause mortality reveals several previously unknown dynamics that will require a
77
reassessment of the mortality burden brought about by the COVID-19 pandemic.
78
1.3 Estimating the Increase in All-cause Mortality: 79
Population-Size and Historical-Trend Effects 80
There are two main effects that have to be taken into account when estimating the 81
increase in all-cause mortality: effects of changes in the size of the population and 82
effects of historical trends in mortality rates. To illustrate these effects and the resulting
83
potential pitfalls, Fig. 1 shows for the over 80 years old population in Germany the 84
number of deaths (Fig. 1A), the population size (Fig. 1B), and the mortality rate (i.e., 85
percentage of deceased persons; Fig. 1C) for the years 2016 to 2021. 86
87
Fig. 1.: Population-size effects and historical-trend effects on the estimation of 88
the increase in mortality. For the population over 80 years of age in Germany, (A) shows 89
the number of deaths, (B) the population size, and (C) the mortality rate (i.e., percentage of 90
deceased persons) for the years 2016 to 2021. 91
Changes in population size have to be taken into account due to the simple fact that
92
the larger a population is, the more deaths occur. Ignoring existing changes in 93
population size will lead to erroneous estimations. For instance, regarding the 94
population over 80 years of age in Germany, the number of deaths increases from year 95
to year (see Fig. 1A). Concluding from this pattern that mortality increased in the years
96
2020 and 2021 compared to previous years would make no sense because this increase is
97
fully attributable to the increase of population size, as shown in Fig. 1B and 1C. 98
August 10, 2022 3/42
Historical trends in mortality rates have to be be taken into account due to the fact
99
that mortality rates are not a stable values but influenced by environmental and societal
100
changes and improvements in medical treatments. For instance, as can be seen 101
exemplarily in Fig. 1C, in Germany, there is a historical trend of a continuous decrease
102
in mortality rate that is observed in most age groups. If such a declining trend in 103
mortality rates is not taken into account, the number of expected deaths are 104
overestimated and thus the true mortality decrease is underestimated. 105
The pitfall of ignoring changes in population size is for example found in the 106
estimations provided by the German Federal Statistical Office [11] where the increase in
107
mortality is estimated based on a comparison of the observed number of deaths with the
108
median value of the four previous years. As illustrated in Fig. 2A, estimating the 109
number of expected deaths based on the median of the four previous years 110
underestimates the number of expected deaths and thus overestimates the true increase
111
in mortality. The invalidity of this method can be illustrated by the fact that in case of
112
a continuously increasing population size, as is the case for the population over 80 years
113
of age in Germany, such a method would conclude for every year that there was an 114
unexpected increase in mortality compared to previous years. 115
The pitfall of ignoring longer historic trends is for example found in the estimations
116
provided by the World Health Organization (WHO) [10] where the increase in mortality
117
is estimated based on a thin-plate spline extrapolation of the number of expected 118
deaths. As illustrated in Fig. 2B, such an estimation method is highly sensitive to 119
short-term changes in the observed number of deaths. Accordingly, erratic estimations 120
of expected deaths predictions can occur. For instance, regarding the WHO estimations
121
for Germany, the spline extrapolation predicts based on the short-term decline in 122
deaths in 2019 compared to 2018 that a similar decline would occur in the following 123
years as well, although this completely contradicts the long-term historical trend. 124
125
Fig. 2.: The pitfalls of ignoring population-size effects and historical-trend effects. 126
The blue squares in (A) and (B) show the development of the number of deaths in Germany 127
from 2010 to 2021 (all age groups). The red squares in (A) show the estimations of the number
128
of expected deaths for the years 2020 and 2021 of the German Federal Statistical Office [11] 129
which are based on the median of the four previous years. The red squares in (B) show the 130
estimations of the number of expected deaths for the years 2020 and 2021 of the World Health
131
Organization [10] which are based on a thin-plate spline extrapolation that is highly sensitive 132
to short-time changes. As can be seen, both the ignoring of the increase in population size of 133
the older age groups and the ignoring of longer historical trends leads to an underestimation of
134
the expected deaths and thus to an overestimation of the true mortality increase in the years 135
2020 and 2021. 136
August 10, 2022 4/42
1.4 Methods That Take Into Account Population-Size and 137
Historical-Trends Effects 138
A first and comparatively simple approach to take into account population-size and 139
historical-trends effects is the attempt to predict the further course of the number of 140
deaths from observed data in previous years using regression methods. For instance, in
141
a study by Baum [4], the course of the observed increase in the number of deaths in 142
Germany from 2001 to 2021 compared to the year 2000 was fitted with a polynomial 143
function of order two, and the yearly residuals were used to estimate the yearly increase
144
or decrease in mortality, resulting in an estimated increase in mortality in the years 145
2020 and 2021 of about 11,000 additional deaths each. While the advantage of this 146
approach is on the one hand that no parameter choices have to be made as it is the case
147
with the more complex estimation methods (see below), on the other hand this is at the
148
same time the weakness of this approach: since every data point is given the same 149
weight, unique outliers may lead to biased estimations, and developments depending on
150
more complex circumstances cannot be incorporated in this approach. 151
To account for unique outliers, it has been tried to estimate the number of expected
152
deaths by a time-series model based on the number of observed deaths in previous years,
153
and to exclude past phases of unique excess mortality, as done in the EuroMOMO 154
project [12]. However, beyond the problem that the resulting estimates depend on the 155
specific model and parameter choices made (see below), a common problem for every 156
approach that bases estimations on the raw number of observed deaths is that the 157
resulting estimations do not take into account possible changes in the age structure 158
within a population, which can lead to biased estimates. 159
To take into account changes in the age structure within a population, so-called 160
age-adjustments has a long tradition in mortality research [13], which is essential 161
especially when estimating the number of expected deaths in populations where the 162
proportion of elderly people changes over time. The basic method is to compute 163
mortality rates for a reference period separately for different age groups, and to 164
extrapolate from the age-dependent mortality rates and the population sizes of the 165
different age groups in the to-be-estimated year the number of expected deaths in each
166
of the age groups. 167
An example is a recent study by Levitt, Zonta, and Ioannidis [9] where the increase
168
in mortality in the years 2020 and 2021 was estimated based on the reference period of
169
the three pre-pandemic years 2017-2019 using age strata of 0-14, 15-64, 65-75, 75-85, 170
and 85+ years, resulting in an estimated increase in mortality of about 16,000 additional
171
deaths in the year 2020, and 38,800 additional deaths in the year 2021. In two studies 172
by De Nicola et al. [5,6], a more refined method (see below) and a more fine-grained age
173
adjustment was used, resulting in even lower estimates of increased mortality with 174
about 6,300 additional deaths in 2020 and 23,400 additional deaths in 2021. 175
A problem in both the study by Levitt et al [9]. and the studies by De Nicola et 176
al. [6, 7] is that possible historical trends in mortality rates are not taken into account. 177
This was, in addition to an age-adjustment, done in a study by Kowall et al. [8] where 178
the increase in mortality in the year 2020 was estimated for the countries Germany, 179
Spain, and Sweden. Historical trends in mortality rates were estimated based on the 180
observed decrease in mortality rates in the pre-pandemic years 2016-1019. For Germany,
181
it was estimated that the number of observed deaths in 2020 was 0.9 percent higher 182
than the number of estimated expected deaths, which is in the range of the estimations
183
in the De Nicola et. study. Estimations with adjustments for changes in historical 184
trends in mortality rates for the year 2021 have to date not been reported, at least to 185
our knowledge. 186
August 10, 2022 5/42
1.5 The Inherent Fuzziness of Estimates of Increases in 187
Mortality 188
As has already become apparent in the previous paragraphs, the estimation of the 189
amount of increase in all-cause mortality entails several model and parameter choices 190
that have to be made. While a proper analysis necessarily requires the taking into 191
account of changes in population sizes and historical trends in mortality rates, there 192
remain a number of degrees of freedom how to exactly do this. For instance, an open 193
question is which previous years are used as a reference and which model is used for the
194
extrapolation of the expected deaths based on these years. 195
What a large effect a small change in the chosen perspective can have on the 196
estimation of the amount of mortality increase is illustrated in Fig. 3 using the German
197
mortality figures. When trying to estimate the increase in the number of all-cause 198
deaths in the years 2020 and 2021 by a comparison with the number of expected deaths
199
that is estimated based on the course of deaths in the four pre-pandemic years 200
2016-2019, one can for instance take two different perspectives: one can consider the 201
year 2018 as an unusual outlier above the typical course of the number of deaths, or one
202
can consider the year 2019 as an unusual outlier below the typical course of the number
203
of deaths. Depending on the chosen perspective, extrapolating the expected number of
204
deaths with either excluding the year 2018 (“outlier upwards”) or the year 2019 205
(“outlier downwards”) leads to totally different results, with an estimation of a strong 206
increase in mortality in the former case and an estimation of even a slight decrease in 207
mortality in the latter case. 208
209
Fig. 3.: Possible large effect of small changes in the chosen perspective. The blue 210
squares in (A) and (B) show the development of the number of observed all-cause deaths in 211
Germany from 2016 to 2021 (all age groups). The expected all-cause deaths in the years 2020
212
and 2021 are estimated based on the observed deaths in the years 2016-2019 using a simple 213
linear regression function, excluding the year 2018 as an “unusual outlier upwards” (A) or the
214
year 2019 as an “unusual outlier downwards” (B), giving the impression of a strong increase in
215
mortality in the former case and the impression of even a slight decrease in mortality in the 216
latter case. 217
Since there is no truth criterion that would determine which of the choices is the 218
best one to be made, there is no such thing as a “true” increase in mortality. Instead, 219
the amount of increase in mortality must be understood as an inherent fuzzy construct
220
that varies depending on the chosen perspective. This fact has at least three important
221
implications: 222
August 10, 2022 6/42
First, when reporting estimates of the amount of increase in mortality, it is 223
important to show how strongly the estimates vary with different model and parameter
224
choices that can reasonably be made. In particular, possible choices and the resulting 225
estimates should be communicated to readers in a way so that they are enabled to draw
226
their own conclusions depending on their specific questions they would like to answer 227
(see next point). 228
Second, when interpreting estimates of the increase in mortality, one has to be aware
229
of the made model and parameter choices. In particular, when deciding which approach
230
is chosen, one has to clarify which question is tried to answer, and to choose the 231
approach that best fits the to-be-answered question. For instance, if one is interested in
232
the question of how far the observed number of deaths is above the usually occurring 233
deaths, excluding outlier years when estimating the amount of increase in mortality may
234
be a reasonable decision. However, if one is interested in whether the observed number
235
of deaths is above the extreme values of previous years, excluding outliers may be a less
236
reasonable decision. 237
Third, despite the inherent fuzziness of the estimates of increases in mortality, the 238
comparison of increases in mortality between two years may nevertheless reveal clear 239
results. If the observed difference between the two years does not vary as a function of
240
the chosen parameters and model, it can be assumed that the observed differences in 241
estimated increases in mortality reflects the true fact that there was a larger increase in
242
mortality in one of the years. 243
1.6 The Use of the Term ”Excess Mortality“ 244
In many of the previous studies, the observation that the number of observed all-cause
245
deaths is larger than the number of expected all-cause deaths is designated by the term
246
“excess mortality”. However, such a use of terms is questionable. The number of deaths
247
from year to year does not follow a straight line but varies around a common trend. 248
Accordingly, as illustrated in Fig. 4, if one were to designate as “excess mortality year” 249
all years in which more deaths are observed than expected according to the common 250
trend, one would have to conclude that an “excess mortality” is observed in about 50 251
percent of all years, and a “mortality deficit” in the other 50 percent of all years. 252
253
Fig. 4.: The inflationary use of the term “excess mortality”. The colored squares 254
show the number of all-cause deaths in Germany from 2010 to 2021. The dashed red line shows
255
the common trend across the years (linear regression). If one were to designate as “excess 256
mortality year” all years in which more deaths are observed than expected according to the 257
August 10, 2022 7/42
common trend (red-colored squares), one would have to conclude that an “excess mortality” is
258
observed in six years, and a “mortality deficit” in the other six years. 259
Since about half of the years show mortality levels above the common trend, one 260
could use the term “excess mortality” only for years that show an outstanding increase
261
in mortality above a certain threshold. One straightforward possibility to establish such
262
a threshold would be to compute the mean variation (empirical standard deviation) 263
around the common trend across the years, and to designate as “excess mortality years”
264
only those in which the number of observed deaths exceeds twice the mean variation. 265
Another possibility would be to search for previous years with peak deviations from
266
the common trend, and then to compare the deviation observed in the year one is 267
interested in with the peak deviations in previous years. Such a comparison was for 268
instance made in a recent study by Staub et al. [14] where the historical dimension of 269
the COVID-19 pandemic was examined for the countries Switzerland, Sweden, and 270
Spain over a time span of more than 100 years, revealing that the peaks of monthly 271
excess mortality in 2020 were greater than most peaks since 1918. 272
Nevertheless, also in this contribution we decided to use the terms “excess mortality”
273
and “mortality deficit” for a mortality which is just above or below the estimated value,
274
as in most other contributions. An attempt to define an outstanding “excess mortality 275
year“ via mean variations will be made in Section 3 and Section 4. 276
1.7 The Present Study 277
The aim of the present study is to provide the state-of-the-art method of actuarial 278
science to estimate the expected number of all-cause deaths and thus to estimate the 279
increase in all-cause mortality for the pandemic years 2020 to 2022. In particular we 280
evaluate the all-cause deaths in Germany. The following four questions are investigated:
281
(1) Yearly Increase in mortality in the years 2020 to 2022 in Germany: As described 282
above, there are several studies that have attempted to estimate the increase in 283
mortality in Germany in 2020 and 2021 based on different methods [5–8, 10]. 284
However, there are several unanswered questions: 285
First, only one study [5], which examined only the year 2020, took into account 286
the historical trend in mortality rates. We use the best available mathematical 287
model provided by the German Association of Actuaries, where well established 288
longevity factors are used for estimating the trend. 289
Second, although in most of the studies age-standardized estimations were made, 290
age-dependent differences in mortality increase were not examined in detail. We 291
use most recent life tables provided by the Federal Statistical Office of Germany 292
to calculate age-dependent expectations, applying the standard model in actuarial
293
mathematics which was already used by Euler and Gauß. 294
Third, in none of the studies, it was examined how much the mortality estimates 295
vary with different approaches. Here we calculate the model and parameter 296
sensitivity by comparing the results achieved using different life tables and 297
longevity factors. 298
And fourth, in all of the previous studies, only the estimated increase in all-cause
299
deaths was reported, without examining whether the estimated increase exceeds 300
the usual variation in mortality found across previous years. We give an estimate
301
for the empirical standard deviation which can be used to obtain confidence 302
intervals. 303
(2)
Monthly increase in mortality in the years 2020 to 2022 in Germany: The increase
304
in mortality over the course of the year has so far only been investigated for 2020
305
August 10, 2022 8/42
in two studies [5,6]. The years 2021 and 2022 have not yet been investigated in 306
this respect. Furthermore, no study has yet determined the increase in mortality 307
over the course of the year for different age groups. 308
(3) Comparing the results to possible influencing factors: In none of the previous 309
studies, possible factors that might contribute to the observed course of the 310
increase in mortality were explicitly examined on a monthly base during the 311
pandemic years 2020 to 2022. 312
(4) Monthly increase in the number of stillbirths in the years 2020 to 2022 in 313
Germany: In all previous studies, the increase in mortality has only been 314
examined for the age groups 0 and above. Whether changes in mortality are also
315
found at the level of stillbirths has not been investigated so far. 316
2 Yearly expected mortality 317
2.1 Methods 318
The starting point for our investigations are the period life tables and population 319
demographics available from the Federal Statistical Office of Germany. As usual in 320
actuarial science, we denote by 321
lx,t the number of xyear old male at January 1st in year t;322
ly,t the number of yyear old female at January 1st in year t;323
dx,t the number of deaths of xyear old males in year t;324
dy,t the number of deaths of yyear old females in year t;325
qx,t (an estimate for) the mortality probability for an xyear old male in year t.326
qy,t (an estimate for) the mortality probability for a yyear old female in year t.327
Note that
dx,t
also contains deceased that have been (
x
1) years old at January 1st in
328
year tand died as xyear old. To compensate this problem, the 2017/2019 life table of 329
the Federal Statistical Office of Germany [15] (like most German life tables) uses the 330
method of Farr to estimate qx,t (and analogously qy,t). 331
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 332
into account only the mortality probabilities in these three years. 333
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,t0eF(x;t,t0), qy,t =qy,t0eF(y;t,t0)..
Here the German Association of Actuaries (DAV) uses a smoothed life table
qx,t0
in the
base year t0, and models the trend underlying future mortality, the longevity trend
function
F
(
x
;
t, t0
), via regression for the male and female population, separately. In the
year 2004 it turned out that the decrease of the mortality probabilities in the previous
August 10, 2022 9/42
years has been steeper than expected, therefore the DAV life table DAV 2004 R [16]
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. 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 DAV 2004 R,
F(x;t, t0) = 1
2(t2019)Fl,x, F (y;t, t0) = 1
2(t2019)Fl,y
where the numbers Fl,x and Fl,y are contained in the DAV 2004 R table. We also 334
decided to use the probabilities ˆqx,2019 and ˆqy ,2019 of the life table 2017/2019 by the 335
Federal Statistical Office of Germany as the base life table in a first step, thus
t0
= 2019.
336
For a discussion concerning our model parameters, i.e. the influence on the longevity 337
trend and our choice using half of it, and the choice of the (non-smoothed) life table 338
2017/19, we refer to Section 3. Also, it is well known that mortality probabilities for 339
males and females differ substantially, therefore these two cases are computed separately.
340
Putting things together, we define the mortality probability of an
x
year old male in
year tby
qx,t = ˆqx,2019e1
2(t2019)Fl,x ,
and for a yyear old female in year tby
qy,t = ˆqy,2019 e1
2(t2019)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,t qy,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 (x1) 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. [6]. Roughly half of the x1 year old
population at the beginning of the year which is of size
lx1,t
dies after its birthday as
x
year old. For them we use the smoothed mortality probability
qx1,t +qx,t
2.
The other half of the
x
year old deceased belongs to the population of
x
year 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 [6]. Hence for
x
= 0
,...,
101 the random number
Dx,t
of deaths of
341
age xin year tis binomial distributed and satisfies 342
EDx,t =1
2lx1,t
qx1,t +qx,t
2+lx,t
qx,t +qx+1,t
2(2)
where lx1,t and lx,t are taken from the population table of the Federal Statistical 343
Office of Germany [17]. For
x
= 0 we set
l1,t
=
l0,t+1
if available,
l1,t
=
l0,t
else, and
344
q1,t =q0,t. The same considerations lead to EDy ,t.345
The 2017/2019 life table by the Federal Statistical Office of Germany contains the 346
mortality probabilities
qx,t
and
qy,t
, and the underlying population table the population
347
August 10, 2022 10/42
size
lx,t
and
ly,t
for the age
x
= 0
,...,
100. In principle it would be more precise to use
348
life tables and population tables up to age 113 but these data are not available. The 349
excess mortality is obtained by comparing the expected values EDx,t +EDy,t to the 350
observed data dx,t +dy,t for t= 2020,2021 and 2022 . 351
Some remarks are in order to contextualize the method. 352
Modelling the longevity factors is a challenging task. For example, the Actuarial 353
Association of Austria uses factors involving arctan t
100 20.01which has 354
serious advantages. The need for longevity factors depends heavily on the country,
355
it seems for example that in Japan and in England the mortality trend has 356
already vanished and the mortality probabilities are more or less constant. 357
The mortality probability heavily depends on gender and differs for the male and
358
female population. But the resulting excess mortality is nearly the same for the 359
male and female population. Hence in the following we calculate the expected 360
number of deaths separately and show only the total number of deaths. On the 361
other hand, huge differences occur for the excess mortality in different age groups
362
and therefore we present our results for each age group separately. 363
The mortality probability not only depends on age and gender, but also 364
significantly on social status, profession, health condition, region, etc. As is 365
common, the German life tables give average mortality probabilities. Also, it is 366
unclear - at least to the authors - whether the SARS-CoV-2-infection rate and 367
mortality depends on these factors, too. For a deeper investigation of COVID-19 368
mortality increase this should be taken into account, but at the moment 369
appropriate data are not available. 370
2.2 Results 371
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 (raw) number of deaths in
2021 only in certain age groups [18]. Therefore, 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
.
August 10, 2022 11/42
Table 1: Expected deaths and yearly excess mortality. 372
t= 2020 t= 2021 t= 2022
expected expected expected
age range observed rel.diff. observed rel.diff. observed
0-14 3.531 3.513 3.517
3.306 -6,38% 3.490 -0,67%
15-29 3.944 3.817 3.755
3.844 -2,53% 3.951 3,52%
30-39 6.626 6.585 6.546
6.668 0,64% 6.938 5,35%
40-49 15.345 14.877 14.601
15.507 1,06% 16.256 9,27%
50-59 58.641 57.705 56.471
57.331 -2,23% 59.387 2,91%
60-69 117.432 118.456 119.983
118.460 0,88% 126.477 6,77%
70-79 198.389 190.335 186.303
201.957 1,80% 204.089 7,23%
80-89 378.459 392.535 404.994
378.406 -0,01% 396.990 1,13%
90-199.191 201.884 202.375
200.093 0,45% 203.852 0,97%
total 981.557 989.707 998.545
985.572 0,41% 1.021.430 3,21%
373
Clearly, for the year 2022 we can only present the expected number of deaths. 374
The deviation in 2020 and 2021 must be compared to the deviation inherent in the 375
parameter choice of our model, and the empirical standard deviation which has occurred
376
in the years before. This will be done in Section 3 and Section 4. It will turn out, that
377
in year 2020 the observed number of deaths is extremely close to the expected number 378
with respect to the empirical standard deviation, whereas in 2021 the difference is of 379
order twice the empirical standard deviation. 380
The following graph illustrates, that the deviation of the observed mortality from the
381
expected mortality is not uniform over the different age groups, and, in particular, the 382
structure changes from 2020 to 2021. A closer look reveals that the excess mortality 383
observed in 2021 is almost entirely due to an above-average increase in deaths in the age
384
groups between 15 and 79. The highest values are reached in the age group 40-49, 385
where an increase in the number of deaths is observed that is nine percent higher than 386
the expected values. 387
August 10, 2022 12/42
388
Fig. 5: Yearly excess mortality. The red bars show the excess mortality in 2020 (left 389
panel) and 2021 (right panel) in different age groups, the grey bars the total excess mortality. 390
Some remarks are in order, to contextualize the results. 391
Our results show that there is some kink around the age of 50. We do not have an
392
explanation for this fact. 393
One has to take into account that the year 2020 is a leap year. Therefore we have
394
“added” an additional day by multiplying the result of the computations described
395
above by 366
365 .396
For infants something unexplained happens. In the beginning of 2020 there were 397
774.870 people of age 0, during the year 2.373 children of age 0 died, yet at the 398
end of 2020 there were 783.593 (!) people of age 1. This is maybe due to 399
migration effects, but we do not have sufficient precise data to model this effect. 400
And for our investigations concerning COVID-19 excess mortality, the infant 401
mortality can be ignored. 402
3 The model uncertainty 403
There are several parameters for modelling mortality probabilities which 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 and 2021, 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 404
longevity trend. 405
August 10, 2022 13/42
Table 2: Expected deaths for different life tables. 406
longevity
trend life table ED2020 ED2021
2015/17 1.010.478 1.025.768
none 2016/18 999.592 1.014.802
2017/19 988.288 1.003.270
2015/17 989.964 998.213
half 2016/18 986.021 994.294
2017/19 981.557 989.707
2015/17 969.896 971.451
full 2016/18 972.649 974.230
2017/19 974.875 976.341
observed 985.572 1.021.430
407
It turns out that the life tables have a significant effect on the question whether an 408
excess mortality exists. For example, the use of the life table 2015/17 of the Federal 409
Statistical Office of Germany without the longevity trend yields for both Corona-years 410
2020 and 2021 a mortality deficit. And even when keeping half the longevity trend, in 411
2021 the excess mortality of 31.723 deaths for the life table 2017/19 should be compared
412
to the smaller excess mortality of 23.217 deaths when using the life table 2015/17, the 413
total difference being 8.506 deaths. In other words, the life tables of the Federal 414
Statistical Office of Germany have a serious fluctuation over the years which should be
415
taken into account as the model uncertainty. 416
For a more convenient view we present the excess mortality using the relative 417
difference. 418
Table 3: Excess mortality for different life tables. 419
longevity
trend life table 2020 2021
2015/17 -2,46% -0,42%
no 2016/18 -1,40% 0,65%
2017/19 -0,27% 1,81%
2015/17 -0,44% 2,33%
half 2016/18 -0,05% 2,73%
2017/19 0,41% 3,21%
2015/17 1,62% 5,14%
full 2016/18 1,33% 4,84%
2017/19 1,10% 4,62%
420
August 10, 2022 14/42
421
Fig. 6: The model sensitivity.
The bars show the mortality deficit, respectively the excess
422
mortality in 2020 (left panel) and 2021 (right panel) for different life tables and longevity 423
trends. 424
In the light of these results, we have decided to choose a model which avoids the 425
extremes and includes half of the longevity factor in Section 2.1. In this case, the range
426
between the three models which is an indicator for the model uncertainty is in both
427
years approximately 8.500 deaths per year. 428
Yet in all these results obtained by life tables of recent years of the Federal 429
Statistical Office of Germany, and in most other models [4
8], the main point coincides
430
with our results: for 2020 the number of deaths is close to the expected value, whereas 431
for 2021 there is a noticeable excess mortality. 432
A more detailed analysis of all the age groups introduced in Section2.2 shows that 433
independently of the model used the increase of the excess mortality from 2020 to 2021
434
is about approximately 6% for the age groups 0-79, except for the age group 40-49 435
where it is 8%. These more detailed results are given in the supplement, Section 8.2. 436
4 The empirical standard deviation 437
As remarked in Section 2.2, to contextualize the deviation in 2020 and 2021 it must be
438
compared to the model uncertainty, and to the empirical standard deviation occurred in
439
the years before. Since the precise value of the empirical standard deviation like the 440
expectation heavily depends on the underlying mathematical model, and since we are
441
only interested in a rough approximation of the empirical standard deviation we use an
442
extremely simple model: we approximate the expected number of deaths using a linear
443
regression model and calculate the empirical standard deviation in this model. 444
August 10, 2022 15/42
445
Fig. 7: The empirical standard deviation.
The red squares show the number of all-cause
446
deaths in Germany from 2010 to 2019. The blue line shows the regression line. 447
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 448
Germany. Observe that we have taken into account that the years 2012 and 2016 have 449
been leap years and the number of deaths has been normalized to 365 days per year. 450
Table 4: Linear regression of the observed deaths. 451
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
452
Calculating in this simple model the empirical standard deviation gives 453
ˆσ(dt) = 14.162 .(3)
We do not claim that this is the precise value of the standard deviation σ(Dt), yet we 454
are convinced that this at least reflects the order of magnitude. To check whether this 455
order of magnitude is plausible we also computed the empirical standard deviation for 456
the years 2000-2009 using again the linear regression model. For these years the 457
empirical standard deviation is approximately 12.600 which is the same order as (3). 458
At first sight this empirical standard deviation is somehow surprising and seems to 459
be in contrast to the model used for modelling Dx,t described in Section 2.1. As is 460
August 10, 2022 16/42
common, we assumed that the number of deaths follows a simple binomial distribution.
461
This is the most natural assumption. It would imply that the variance 462
VDx,t =lx,t(1 qx,t )qx,t is approximately the number of deaths lx,tqx,t, since for the 463
large majority of
x
the mortality probabilities are close to zero. Hence in Germany this
464
assumption and the independence property of the binomial model would lead to a total
465
variance is of order one million, and a standard deviation of approximately 1.000. Thus
466
in actuarial science we introduce a further randomization of qx,t which keeps the 467
expectation unchanged and thus our results in Sections 2.13 are still valid but 468
increases the variance to the observed 14.000. 469
We compare the excess mortality of 4.000 deaths in 2020 and 31.700 deaths in 2021
to the empirical standard deviation ˆσ(Dt). In 2020 this leads to
d2020 ED2020 0,28ˆσ(D2020 )
and for 2021
d2021 ED2021 2,24ˆσ(D2021 ).
In many applications an observed deviation beyond twice the standard deviation is 470
called significant because for normal distributed random variables the 5% confidence 471
interval leads to this bound. A bound of 2.24 times the standard deviation leads to a 472
2.5% confidence interval, which roughly speaking means that this event occurs 2-3 times
473
every hundred years. 474
On the other hand one could also take into account half of the model uncertainty of
approximately 4.250 deaths. This leads to
d2021 ED2021 4.250 + 1,94ˆσ(D2021 )
and thus the deviation in 2021 would be in a 5% confidence interval. 475
5 Monthly expected mortality 476
5.1 Methods 477
In the following two sections, we present a more detailed analysis of the number of 478
deaths during the years 2020 to 2022. It is well known that the mortality probabilities 479
are not constant but differ from month to month with peaks at the beginning and the 480
end of the year and also sometimes in summer when the weather is too hot (and 481
depending on many other circumstances). 482
Unfortunately, the data basis for such investigations provided by the Federal 483
Statistical Office of Germany is somehow weak. Therefore, again several approximation
484
steps have to be applied. We denote by
dx,t,m
, respectively
dy,t,m
, the number of deaths
485
of xyear old male and yyear old female in year tin month m. The Federal Statistical 486
Office of Germany offers tables for d¯x,t,m and d¯y ,t,m in the age groups 487
¯x, ¯y {[0,14],[15,29],[30,34],[35,39],...,[90,94],[95,)}which we use for the years 488
t= 2016,...,2021, see [18]. We ignore again migration issues. 489
Denote by fmthe estimated proportion of deaths in month m,m= 1,...,12. I.e.,
we distribute d¯x,t onto the monthly number of deaths d¯x,t,m via
f¯x,m =1
4
2019
X
t=2016
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 2016 was a leap year.
We list the obtained estimates in the supplement, Section 8.3. Then we distribute the
August 10, 2022 17/42
expected number of deaths for year t= 2020,2021,2022 according to the factors f¯x,m
and f¯y,m and obtain the approximation
ED¯x,t,m =f¯x,mED¯x,t ,ED¯y,t,m =f¯y,m ED¯y ,t,
for 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
490
February. These expected values should be compared to the observed data d¯a,t,m for 491
m= 1,...,12. The remarks made at the end of Section 2.1 apply similarly to the 492
computations made in this section. 493
5.2 Results 494
Following the computations described in the previous section, we calculate the expected
495
number of deaths ED¯a,2021,m for all months m= 1,...,12 in the years 496
t
= 2020
,
2021
,
2022. We emphasize that the observed number of deaths is the currently
497
available data set of the Federal Statistical Office of Germany and for the years 2021 498
and 2022 it is still preliminary. We concentrate in this section on four age ranges 499
¯a= [0,14], [15,59], [60,) and [80,). 500
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
and show our results in Fig. 8. 501
502
Fig. 8: Development of the monthly excess mortality. For four age groups the red 503
squares show the monthly excess mortality from January 2020 to June 2022. 504
August 10, 2022 18/42
5.2.1 Children 505
In the the age group [0
,
14] the number of deaths is (luckily) very 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
506
Fig. 8 and the table in the supplement, Section 8.4, with the calculated values show in 507
fact oscillations of this size. Hence we think that any conclusion relying on these 508
numbers has to be taken with great care. The maybe only notable results are first the 509
well accepted fact that children are extremely robust with respect to 510
SARS-CoV-2-infections and the curve seems to be independent of the usual 511
SARS-CoV-2-infection waves. Second, the presumably different social behavior during 512
the Corona crises seems to lead to a mortality deficit in the younger age groups which is
513
visible here. An exception are the months May to November 2021 with a visible positive
514
excess mortality for half a year. 515
5.2.2 Adults 516
The age group [15
,
59] is the largest group we discuss in this section with approximately
517
7.000 expected deaths per month. For this age group we list the results in detail. Using
518
the binomial model, deviations from the expected value in the range of twice the 519
standard deviation are to be expected, which is of order 170 deaths or 2,4% relative 520
difference. Some of the deviations listed in the following table are clearly beyond this 521
threshold, yet the remarks made in Section 4, that the standard deviation is 522
underestimated by the binomial model, should be kept in mind. 523
August 10, 2022 19/42
Table 5: Expected deaths and monthly excess mortality. 524
t= 2020 t= 2021 t= 2022
expected expected expected
observed rel.diff. observed rel.diff. observed rel.diff.
m=1 7.575 7.455 7.309
7.395 -2,38% 7.717 3,52% 7.270 -0,53%
m=2 7.257 6.896 6.761
6.811 -6,14% 6.615 -4,07% 6.523 -3,52%
m=3 7.703 7.583 7.435
7.280 -5,49% 7.175 -5,38% 7.039 -5,33%
m=4 6.969 6.860 6.727
6.960 -0,13% 7.429 8,30% 6.823 1,43%
m=5 7.025 6.915 6.781
6.898 -1,81% 7.385 6,80% 6.613 -2,47%
m=6 6.792 6.684 6.554
6.664 -1,89% 6.987 4,53% 6.444 -1,67%
m=7 6.946 6.838 6.706
6.816 -1,88% 6.912 1,08%
m=8 6.887 6.779 6.647
6.807 -1,16% 6.819 0,60%
m=9 6.593 6.489 6.363
6.528 -0,99% 6.820 5,09%
m=10 6.941 6.833 6.700
6.841 -1,45% 7.258 6,22%
m=11 6.833 6.728 6.598
6.818 -0,22% 7.351 9,27%
m=12 7.034 6.926 6.793
7.532 7,08% 8.064 16,43%
525
Whereas the numbers in year 2020 are mostly unremarkable, and reflect the minimal
526
number of deaths by the first COVID-19 wave in April 2020 and a serious excess 527
mortality in December 2020 in this age range, something unexpected is happening in 528
2021 and 2022. 529
The significant excess mortality in December 2020 continues slightly in January 2021
530
and then is mostly compensated until March 2021. That is, by the end of March, the 531
cumulative excess mortality was close to zero. In April and May 2021, a significant 532
increase in excess mortality is observed, followed by a decrease up to August. However,
533
other than at the beginning of the year, excess mortality remains above zero so that the
534
increase in excess mortality in April and May is not compensated for. In September 535
there is again a significant excess mortality, which increases in November and is more 536
than doubled in December 2021. 537
In 2022 the results are inconspicuous with a maybe notable mortality deficit in 538
March. Note that the numbers in 2022 are from the most current data set of the 539
Federal Statistical Office of Germany, this is still preliminary and in particular in these
540
months there will still be changes within the next weeks and months. 541
August 10, 2022 20/42
5.2.3 Old ages 542
The last group consists of the ages
60, where large parts of the vulnerable population
543
belong to, and a SARS-CoV-2-infection can be particularly dangerous. We list the 544
expected deaths, the observed deaths and the relative difference in the supplement, 545
Section 8.6. 546
That this age group largely belongs to the vulnerable population is clearly visible in
547
the data for 2020. Fig. 8 shows for the age group [60
,
79] and [80
,
) a decent peak for
548
April 2020, and a significant peak around December 2020. The peak of December 2020
549
continues in January 2021, but then turns into a mortality deficit until April 2021, 550
where suddenly the downwards trend stops. In September and October we see a decent,
551
and in November and December 2021 again a serious excess mortality. The year 2022 552
starts with a mortality deficit which again in April turns into an excess mortality. 553
Although the trend in both age groups looks parallel, it is interesting to split this age 554
group into the two groups [60,79] and [80,) and point our the differences. 555
The curve for the age group [80,) in Fig. 8 is below and somehow parallel to the 556
curve for the age group [60,79]. The main difference is the deviation of the age group 557
[60
,
79] in April and May 2021 where a jump in the mortality behaviour for the this age
558
group is visible. The expected number of deaths, the observed deaths and the difference
559
are given in the supplement, Section 8.7 and Section 8.8. 560
The age group [80,) beyond the expected life time in Germany (which is 561
approximately at the age of 80), seems to be somehow resistant to all mortality causes 562
at a larger scale. At certain moments some people die some months before or after the
563
‘expected‘ time of death, but the curve for the excess mortality always oscillates around
564
the 0% axes. A visible mortality deficit at the beginning of 2020 is nearly compensated
565
in April 2020, the huge peak at the turn of the year 2020/2021 is more or less 566
compensated by the mortality deficit in January to July 2020 and then in February to 567
August 2021, the peak around November and December 2021 is nearly compensated in
568
February to March 2022. 569
It is interesting to make this shift visible by calculating the cumulative excess 570
mortality since January 2020 in absolute numbers. Maybe due to the non-existing flue 571
in 2019 and 2020 we start with a negative value. In July 2020 up to 20
.
000 people more
572
than expected are still alive, which is compensated in December 2020 to February 2021,
573
where the curve is 10.000 above the expectation, and then the curve fluctuates to 574
10.000, to +10.000 and back to 0 which is the expectation. This shows that a 575
mortality deficit or a excess mortality in the age group [80,) just shifts the time of 576
death by some months. 577
This is in extreme contrast to the situation for the age group [60,79]. The 578
cumulative excess mortality is increasing up to 30.000 deaths at the end of year 2022, a
579
result we have not seen before in any publication and we find worth noting! 580
The difference between both age groups is demonstrated in Fig. 9. 581
August 10, 2022 21/42
582
Fig. 9: The cumulative excess mortality. The green areas show the regions of a 583
cumulative mortality deficit, the red areas of a cumulative excess mortality from January 2020
584
to June 2022. The age group [80,) is oscillating, the age group [60,79] nearly monotone 585
increasing. 586
5.3 Stillbirths in the years 2019 to 2022 in Germany 587
In all previous studies on excess mortality during the COVID-19 pandemic, only the 588
increase in mortality for the age groups 0 and above has been examined. In the 589
following, it is examined whether similar increases in mortality than that found for the
590
age groups 0 and above are also found at the level of stillbirths. 591
One problem with analyzing excess mortality at the level of stillbirths in Germany is
592
that the definition of a ‘stillbirth’ has been changed at the end of 2018. Until then, a 593
stillborn child was considered a stillbirth if a birth weight of at least 500 grams was 594
reached. Since the end of 2018, a stillborn child is considered a stillbirth if at least 500
595
grams or the 24th week of pregnancy was reached, which led to a diagnostically related
596
increase in stillbirths. This means that the figures on stillbirths are only validly 597
comparable from 2019 onwards. 598
Furthermore, when analyzing the number of stillbirths [19], it is important to note 599
that they must be placed in relation to the number of total births [20], because an 600
increase or decrease in the number of total births is automatically accompanied by an 601
increase or decrease in stillbirths. Fig. 10 shows the number of live births per quarter 602
since 2019 in the upper panel, the number of stillbirths in the middle panel, and the 603
number of stillbirths per 1000 births in the lower panel. 604
August 10, 2022 22/42
605
Fig. 10: Monthly stillbirths in the years 2020 to 2022 in Germany.
The upper panel
606
shows the number of live births per quarter since 2019, the middle panel the number of 607
stillbirths per quarter since 2019, and the lower panel the number of stillbirths per 1000 births
608
per quarter since 2019. 609
As can be seen, regarding the number of live births, a relatively large decrease of 10.1
610
percent is observed in the first quarter of 2022 compared to the mean across the first 611
quarters in the years 2019 to 2021. Regarding the number of stillbirths per 1000 births,
612
the year 2020 is comparable to the year 2019. However, in the year 2021, a sudden 613
increase of 10.7 percents is observed in the second quarter of the year 2021 compared to
614
the mean across the years 2019 and 2020. The number of stillbirts remains increased in
615
the following quarters, reaching an increase of 9.9 percent in the first quarter of 2022. 616
Taken together, these findings indicate that a simiar increases in mortality than that
617
found for the age groups 0 and above are also found at the level of stillbirths. Whereas
618
in the year 2020 no noticable excess mortality at the level of stillbirths is observed, in 619
the year 2021, starting in April, a striking excess mortality is observed. 620
6 Discussion 621
In the previous sections we estimated the expected number of all-cause deaths and the 622
increase in all-cause mortality for the pandemic years 2020 to 2022 in Germany. The 623
results revealed several previously unknown mortality dynamics that require a 624
reassessment of the mortality burden brought about by the COVID-19 pandemic. 625
The analysis of the yearly excess mortality showed a marked difference between the
626
pandemic years 2020 and 2021. Whereas in the year 2020 the observed number of deaths
627
was extremely close to the expected number with respect to the empirical standard 628
deviation, in 2021, the observed number of deaths was far above the expected number in
629
the order of twice the empirical standard deviation. An age-dependent analysis showed
630
that the strong excess mortality observed in 2021 was almost entirely due to an 631
above-average increase in deaths in the age groups between 15 and 79, reaching more 632
than nine percent in the age group 40-49. A detailed analysis of the monthly excess 633
August 10, 2022 23/42
mortality showed that the high excess mortality observed in the age groups between 15
634
and 79 in the year 2021 started to accumulate only from April 2021 onwards. 635
An analysis of the number of stillbirths revealed a similar mortality pattern than 636
observed for the age group between 15 and 79 years. Whereas in 2020 the number of 637
stillbirths per 1,000 births was similar than in the year before, an increase of about 11 638
percent was observed in the second quarter of the year 2021 compared to the previous 639
years. 640
Taken together, these findings raise the question what happened in 2021 that led to
641
a sudden and sustained increase in mortality in April in the age groups below 80 years
642
and to a sudden and sustained increase in the number of stillbirths, although no such 643
effects on mortality had been observed during the COVID pandemic so far. In the 644
following sections, possible explanatory factors are explored. 645
6.1 Possible factors influencing mortality 646
As already mentioned, apart from the population structure the number of deaths in a 647
year depends on several different factors, the most important being maybe the severity
648
of the flue, and the number of extremely hot weeks. The fluctuations between different
649
years, and thus the approximation of the empirical standard deviation
ˆσ
(
Dt
) in Section
650
4, includes all these factors. It is unclear, rather subjective, and most probably 651
impossible to precisely define ‘extreme events’, to calculate the influence of such 652
extreme events and to adjust mortality to ‘entirely normal’ years. Thus our calculations
653
gives the expected number of deaths taking into account all these extreme and 654
non-extreme effects which are contained in the different life tables. We tried to quantify
655
the sensitivity of our approach in Section 3 and Section 4 against the background of 656
extreme events in the last years. 657
For the pandemic years 2020 and 2021, it is clear that the number of deaths has been
658
influenced directly and indirectly by COVID-19. First, clearly there has been a serious
659
number of COVID-19 deaths, either as the only reason for death or in combination with
660
several other causes, which also might have caused death independently of COVID-19, 661
see e.g. the discussion in Section 5.2.3 and in the forthcoming Section 6.2. Second, the
662
vaccination campaign which started in 2021 should be visible in a reduced excess 663
mortality, or even better as a mortality deficit. An attempt to compare our results to 664
the number of vaccinations is the content of Section 6.3. 665
Third, the indirect effects on mortality due to the COVID-19 measures are extremely
666
harder to quantify. Several aspects may contribute to an excess mortality or a mortality
667
deficit. In Germany strict control measures since 2020 limited personal freedom, schools
668
were partially closed, there were severe lockdowns. This substantially influenced the risk
669
of road accidents and other outdoors casualties. On the other hand many clinical 670
services have been delayed or avoided in 2020 and 2021. All these and many more 671
factors influenced mortality in different directions and on different time scales, but most
672
of them are hard to measure, many effects are highly correlated, and it seems to be 673
impossible to quantify the overall impact of the control measures on the number of 674
deaths. Hence we decided not to discuss this issue. 675
6.2 COVID-19 deaths and mortality 676
In this section we compare the excess mortality since March 2020 to the observed and 677
reported number of COVID-19 deaths by the German Robert Koch Institute. The 678
Robert Koch Institute provides the number of COVID-19 deaths [21] for the age groups
679
[0
,
9], [10
,
19], . . . . Because the Federal Statistical Office of Germany uses different age
680
groups, direct comparisons are made impossible, e.g. for the age group [20,59]. 681
Therefore we compare the number of COVID-19 deaths of the age group [20
,
59] to the
682
August 10, 2022 24/42
mortality of the age group [15
,
59] and assume that the number of COVID-19 deaths in
683
the group [15
,
19] can be neglected. Even when the reporting system in Germany seems
684
to be imprecise and partially insufficient, there should be a serious correlation between
685
the reported number of deaths and the excess mortality. We show the development of 686
the number of reported COVID-19 deaths and the excess mortality in Fig. 11. 687
688
Fig. 11: COVID-19 deaths versus excess mortality.
The blue squares show the number
689
of reported COVID-19 deaths, the red squares the mortality deficit, respectively the excess 690
mortality from January 2020 to June 2022 in three different age groups. 691
The age group [20
,
59] starts with a mortality deficit until November 2020, although
692
the number of COVID-19 deaths is positive. In December 2020 both numbers coincide.
693
After that, the number of COVID-19 deaths stays on a high level while mortality 694
decreases and a noticeable mortality deficit occurs until summer 2021. In April, a 695
marked increase in excess mortality is observed that is not accomponied by a 696
comparable increase in COVID-19 deaths. From June 2021 onwards, the number of 697
excess deaths stays above the number of COVID-19 deaths, until both curves decouple
698
in January 2022. 699
A similar picture occurs for the age group [60,79] with more distinct deviations 700
between both curves. One difference to the age group [80,) is that in the older age 701
group the peak in April 2021 is nearly invisible: there is no increase in COVID-19 702
deaths despite a marked increase in the excess mortality curve. 703
It is elucidating to compare the cumulative number of COVID-19 deaths to the 704
cumulative number of excess deaths in Fig. 12. In all age groups, the cumulative 705
number of reported COVID-19 deaths is increasingly higher than the cumulative 706
number of excess deaths. In the age group between 20 and 59 years, of the 707
approximately 7.500 COVID-19 deaths reported until the end of June 2022, 708
approximately 5.500, did not show up as excess deaths. In the age group between 60 709
August 10, 2022 25/42
and 79 years, of the approximately 42.000 COVID-19 deaths reported until the end of 710
June 2022, approximately 10.000 did not show up as excess deaths. The strongest 711
divergence is found in the age group over 80 years, where of the approximately 90.000 712
COVID-19 deaths reported until the end of June 2022 approximately 78.000 did not 713
show up as excess deaths. Taken together, of the 140.000 reported COVID-19 deaths in
714
the agre groups over 20 years, more than 93.000 did not show up as excess deaths and 715
are thus contained in the ‘expected’ number of deaths. 716
717
Fig. 12: Cumulative COVID-19 deaths versus cumulative excess mortality. For 718
three different age groups the blue squares show the cumulative number of reported COVID-19
719
deaths, the blue squares the cumulative mortality deficit, respectively the excess mortality in 720
three different age groups from March 2020 to June 2022. 721
It thus is obvious that the number of reported COVID-19 deaths contains a large 722
number of ‘expected’ deaths and it seems to be misleading to measure the risk of the 723
COVID-19 pandemic just using the reported deaths. One should rather use the excess 724
mortality curve than the number of reported COVID-19 deaths, or a combination of 725
both, to carve out the moments of high risk and to evaluate the total risk of a pandemic.
726
Beyond the problem that the number of reported COVID-19 deaths cannot be 727
validly used to assess the effects of the COVID-19 pandemic on mortality, it seems also
728
unlikely that the high excess mortality in 2021 in the age groups under 80 years can be
729
explained by COVID-19 deaths, because the marked increases in excess mortality in 730
April to June 2021 - the mortality increases abruptly by 13% from March to April 2021
731
in the age group between 15 and 59 - and also in October to December 2021 were not 732
accomponied by comparable increased in the number of COVID-19 deaths. 733
Furthermore, it seems also very unlikely that the abrupt increase of the mortality is due
734
to delayed or avoided clinical services, which should lead to much smoother changes, or
735
August 10, 2022 26/42
due to side effects of COVID-19 measures. Thus, it remains to investigate the factors 736
which could lead to the surprising jumps in excess mortality in April to June 2021 and
737
also in October and November 2021. 738
6.3 COVID-19 vaccination and mortality 739
In April 2021 an extensive vaccination campaign started in Germany. Comparing the 740
number of vaccinations [22] to the excess mortality should show the sum of two effects 741
of the vaccination: on the one hand a decreasing excess mortality because of successful
742
immunisation, and on the other hand an increasing mortality if the vaccinations would
743
cause side effects in the form of deaths. 744
The following Fig. 13 shows on the left scale the number of excess deaths, 745
respectively death deficit, and on the right scale the number of vaccinations. 746
747
Fig. 13: Number of vaccinations versus excess mortality. The red line shows the 748
death deficit, respectively the excess deaths, the four dashed lines the number of vaccinations 749
from January 2021 to June 2022. 750
The figure shows that the strong increase in mortality in April 2021 and the further
751
development of the excess deaths covaries with the strong increase of the number of 752
vaccinations. Furthermore, the peaks of the excess mortality nearly coincides with the 753
peaks of the vaccination campaign. Such a strong covariation suggests that the increase
754
in excess mortality might be related to the increase in vaccinations. Since covariation 755
does not neccessarily imply causation, further studies are needed to investigate this 756
assumption. However, a further hint that vaccinations may indeed have increased 757
mortality in the negative is the fact that the age group [0,29] has a peak in the excess 758
mortality in June 2021 instead of April 2021, see the table and the graph in the 759
supplement, Section 8.5. Data of the Robert Koch Institute show that for this age 760
group the peak in the vaccination campaign is in fact only in June 2021. 761
Because we computed excess mortality in monthly intervals, it is possible to compare
762
the excess mortality in the months April 2020 to March 2021, where vaccination for the
763
population was not available, to the second year of Corona April 2021 to March 2022, 764
where large groups of the population have been vaccinated. We show in the following 765
table the excess mortality in this time periods for four age groups. 766
August 10, 2022 27/42
Table 6: Expected deaths and excess mortality 04/20–03/22. 767
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
3.238 -281 -7,98% 3.492 -22 -0,64%
15-59 83.954 82.556
83.371 -583 -0,69% 85.857 3.301 4,00%
60-79 313.125 308.066
323.349 10.224 3,27% 327.576 19.510 6,33%
80-581.040 598.103
593.678 12.638 2,18% 599.021 918 0,15%
0-981.638 992.239
1.003.636 21.998 2,24% 1.015.946 23.707 2,39%
768
The maybe most surprising fact is that the second year produces in all age groups a
769
significant mortality increase, which is in sharp contrast to the expectation that the 770
vaccination should decrease the number of COVID-19 deaths. The only exception is the
771
last age group [80,), where in the first year a large number of excess deaths was 772
observed. However, when interpreting this finding, it has to be taken into account that
773
there was a huge mortality deficit from 2019 and until October 2020 which was 774
compensated in November, December 2020 and January 2021. This effect could not 775
occur a second time within one year. Even if for the age group [80,) the strong 776
mortality decrease would be a direct effect of the vaccination, we doubt whether this 777
would justify the vaccination of the whole population independent of age. In total, the 778
decrease of the mortality of people of age 80 and the increase of mortality of 779
comparatively young people, yields a negative net effect. 780
The German Paul-Ehrlich-Institut, which is responsible for judging the known and 781
unknown risks with respect to new vaccinations, uses a method for risk-evaluation called
782
“expected vs. observed”, but both authors of this contribution have not been able to 783
understand the strange way the method is used by the Paul-Ehrlich-Institut. In the 784
analysis of the Paul-Ehrlich-Institut, the expected number of all-cause deaths in the 785
vaccinated group is compared to the observed number of suspected vaccine-related 786
deaths, instead of the observed number of all-cause deaths. Therefore, since the number
787
of the expected all-cause deaths includes all other (vaccine-independent) causes of 788
deaths such as, for instance, heart diseases, strokes, accidents, such an analysis would 789
only reveal a safety signal if the number of reported suspected vaccine-related deaths is
790
larger than the total number of all-cause deaths.791
The table above represents how an “expected vs. observed” analysis should be done:
792
the first column gives the number of deaths and the excess mortality in a Corona year 793
without vaccination, the second column shows the number of deaths and the excess 794
mortality in a Corona year with vaccination. One would expect that a vaccination 795
reduces excess mortality and possible negative side-effects are overcompensated by the 796
positive effect of the immunisation. Obviously the contrary happened according to the 797
table above. The numbers are totally surprising and should lead to several more detailed
798
investigations from different scientific fields, to find the sources for this alarming signal.
799
August 10, 2022 28/42
7 Conclusion 800
The present study used the state-of-the-art method of actuarial science to estimate the
801
expected number of all-cause deaths and the increase in all-cause mortality for the 802
pandemic years 2020 to 2022 in Germany. 803
In 2020 the observed number of deaths was extremely close to the expected number,
804
but in 2021 the observed number of deaths was far above the expected number in the 805
order of twice the empirical standard deviation. The analysis of the age-dependent 806
monthly excess mortality showed, that a high excess mortality observed in the age 807
groups between 15 and 79 starting from April 2021 is responsible for the excess 808
mortality in 2021. An analysis of the number of stillbirths revealed a similar mortality 809
pattern than observed for the age group between 15 and 79 years. 810
As a starting point for further investigations explaining this mortality patterns, we 811
compared the excess mortality to the number of reported COVID-19 deaths and the 812
number of COVID-19 vaccinations. This leads to several open questions, the most 813
important beeing the covariation between the excess mortality and the COVID-19 814
vaccinations. 815
References 816
1. von Stillfried S, ulow RD, ohrig R, Boor P, for the German Registry of 817
COVID-19 Autopsies (DeRegCOVID). First report from the German COVID-19
818
autopsy registry. The Lancet Regional Health - Europe (2022); 15:100330. 819
doi.org/10.1016/j.lanepe.2022.100330 820
2. Saragih ID, Advani S, Saragih IS, Suarilah I, Susanto I, Lin CJ. Frailty as a 821
mortality predictor in older adults with COVID-19: A systematic review and 822
meta-analysis of cohort studies. Geriatric Nurs. (2021); 42(5):983–992. 823
doi.org/10.1016/j.gerinurse.2021.06.003 824
3.
Hung IFN, Zhang AJ, To KKW, Chan JFW, Zhu SHS, Zhang R, Chan T, Chan
825
K, Yuen K: Unexpectedly Higher Morbidity and Mortality of Hospitalized Elderly
826
Patients Associated with Rhinovirus Compared with Influenza Virus Respiratory
827
Tract Infection. Int. J. Mol. Sci. (2017); 18:259. doi.org/10.3390/ijms18020259 828
4. Baum K. Considerations on excess mortality in Germany in the year 2020 and 829
2021. Dtsch. Med. Wochenschr. (2022); 147(7):430–434. 830
5. COVID-19 Excess Mortality Collaborators: Estimating excess mortality due to 831
the COVID-19 pandemic: a systematic analysis of COVID-19-related mortality, 832
2020–21. The Lancet (2022); 399:1513–1536. 833
www.thelancet.com/action/showPdf?pii=S0140-6736%2821%2902796-3 834
6.
De Nicola G, Kauermann G, ohle G. On assessing excess mortality in Germany
835
during the COVID-19 pandemic (Zur Berechnung der ¨
Ubersterblichkeit in 836
Deutschland ahrend der COVID-19-Pandemie). AStA Wirtsch Sozialstat Arch 837
(2022); 16:5–20. link.springer.com/article/10.1007/s11943-021-00297-w 838
7.
De Nicola G, Kauermann G: An update on excess mortality in the second year of
839
the COVID-19 pandemic in Germany (Ein Update zur ¨
Ubersterblichkeit im 840
zweiten Jahr der COVID-19 Pandemie in Deutschland). AStA Wirtsch Sozialstat
841
Arch (2022); 16:21–24. doi.org/10.1007/s11943-022-00303-9 842
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8. Kowall B, Standl F, Oesterling F, Brune B, Brinkmann M, Dudda M, Pflaumer 843
P, ockel K, Stang A. Excess mortality due to Covid-19? A comparison of total 844
mortality in 2020 with total mortality in 2016 to 2019 in Germany, Sweden and 845
Spain. PLoS ONE (2021); 16(8):e0255540. doi.org/10.1371/journal.pone.0255540
846
9. Levitt M, Zonta F, Ioannidis JPA. Comparison of pandemic excess mortality in 847
2020–2021 across different empirical calculations. Environ Res. (2022); 848
213:113754. doi.org/10.1016/j.envres.2022.113754 849
10. World Health Organization: Global Excess Deaths Associated with COVID-19, 850
January 2020–December 2021. https://www.who.int/data/stories/global-excess- 851
deaths-associated-with-covid-19-january-2020-december-2021 (Accessed on May 6,
852
2022) 853
11. Federal Statistical Office of Germany: Number of deaths and excess mortality 854
(Sterbefallzahlen und ¨
Ubersterblichkeit). 855
www.destatis.de/DE/Themen/Querschnitt/Corona/Gesellschaft/bevoelkerung- 856
sterbefaelle.html (Accessed on February 21, 857
2022) 858
12.
Mølbak K, Mazick A. European monitoring of excess mortality for public health
859
action (EuroMOMO). European Journal of Public Health (2013); 23 suppl. 860
1:ckt126.113. doi.org/10.1093/eurpub/ckt126.113 861
13. Keiding N, Clayton D. Standardization and control for confounding in 862
observational studies: a historical perspective. Stat. Sci. (2014); 29:529–558. 863
doi.org/10.1214/13-STS453 864
14.
Staub K, Panczak R, Matthes KL, Floris J, Berlin C, Junker C, et al. Historically
865
High Excess Mortality During the COVID-19 Pandemic in Switzerland, Sweden, 866
and Spain. Annals of Internal Medicine (2022); 175(4):523–532. 867
www.acpjournals.org/doi/10.7326/M21-3824 868
15. Federal Statistical Office of Germany: Life tables 2015/17, 2016/18, 2017/2019. 869
www-genesis.destatis.de/genesis//online?operation=table&code=12411-0006 870
(Accessed on February 21, 2022) 871
16.
German Association of Actuaries (DAV): Life table DAV 2004R. www.aktuar.de
872
17. Federal Statistical Office of Germany: Population statistics. 873
www-genesis.destatis.de/genesis//online?operation=table&code=12411-0005 874
(Accessed on June 20, 2022) 875
18. Federal Statistical Office of Germany: Death statistics. 876
www.destatis.de/DE/Themen/Gesellschaft-Umwelt/Bevoelkerung/Sterbefaelle- 877
Lebenserwartung/Tabellen/sonderauswertung-sterbefaelle.html (Accessed on July
878
20, 2022) 879
19. Federal Statistical Office of Germany: Number of stillbirths. Available upon 880
request from the FSOG. (Accessed on July, 2022) 881
20. Federal Statistical Office of Germany: Number of births. 882
www-genesis.destatis.de/genesis//online?operation=table&code=12612-0002 883
(Accessed on July 25, 2022) 884
21. Robert Koch Institut: COVID-19 Todesf¨alle nach Sterbedatum. 885
www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Projekte RKI/ 886
COVID-19 Todesfaelle.html (retrieved on July 28, 2022) 887
August 10, 2022 30/42
22. Robert Koch Institut: Digitales Impfquoten-Monitoring COVID-19. 888
www.rki.de/DE/Content/InfAZ/N/Neuartiges Coronavirus/Daten/ 889
Impfquotenmonitoring.xlsx? blob=publicationFile (retrieved on Juli 18, 2022) 890
Christof Kuhbandner Matthias Reitzner
Universit¨at Regensburg Universit¨at Osnabr¨uck
Institut ur Experimentelle Psychologie Institut ur Mathematik
93040 Regensburg 49069 Osnabr¨uck
Germany Germany
891
August 10, 2022 31/42
8 Supplementary Material 892
8.1 Yearly Mortality Excess 893
In Section 2.2 we have stated the total expected number of deaths EDtin 2020–2022 894
only for certain age groups, in the following table we list the detailed expected number
895
of deaths EDx,t for males and EDy,t for females and for each age x, y separately. 896
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 28 30 28
13 37 31 37 31 37 30
14 49 35 48 35 49 35
15 65 43 64 42 64 42
16 85 49 83 48 82 47
17 112 53 110 52 108 51
18 146 62 140 60 138 59
19 171 68 162 65 158 64
20 185 72 175 69 168 66
21 191 73 185 70 178 69
22 198 73 190 70 186 69
23 207 76 202 74 197 71
24 215 80 216 80 214 79
25 218 82 217 82 221 83
26 221 86 215 84 217 84
27 232 98 226 95 222 93
28 251 113 243 109 239 107
29 281 136 263 127 257 124
897
August 10, 2022 32/42
age EDx,2020 EDy,2020 EDx,2021 EDy,2021 EDx,2022 EDy,2022
30 313 159 301 153 283 143
31 342 173 338 170 326 164
32 373 181 370 180 368 178
33 396 191 401 192 400 192
34 400 208 408 210 416 213
35 415 224 420 226 430 230
36 463 240 461 238 468 241
37 516 264 507 258 508 257
38 553 298 547 294 540 288
39 589 327 587 324 583 319
40 634 338 643 343 643 341
41 677 354 691 363 704 369
42 714 389 715 391 733 401
43 761 424 765 426 768 429
44 825 457 835 464 842 468
45 911 512 907 510 921 519
46 1020 577 1002 565 1000 566
47 1185 667 1119 633 1101 622
48 1432 815 1296 742 1226 706
49 1686 969 1565 902 1419 824
50 1981 1136 1851 1068 1722 997
51 2346 1334 2188 1250 2048 1178
52 2725 1519 2602 1452 2431 1363
53 3142 1719 3026 1657 2896 1587
54 3572 1932 3477 1887 3355 1823
55 4027 2174 3934 2126 3837 2080
56 4500 2426 4432 2385 4338 2336
57 4867 2614 4917 2631 4851 2592
58 5216 2799 5287 2821 5353 2847
59 5601 3009 5678 3039 5768 3070
60 5967 3229 6074 3287 6169 3327
61 6275 3395 6460 3496 6588 3566
62 6605 3551 6780 3639 6991 3756
63 6991 3784 7081 3818 7280 3921
64 7305 4009 7438 4053 7541 4097
65 7615 4256 7734 4271 7880 4325
66 7923 4519 8039 4539 8168 4564
898
August 10, 2022 33/42
age EDx,2020 EDy,2020 EDx,2021 EDy,2021 EDx,2022 EDy,2022
67 8321 4819 8324 4810 8449 4839
68 8767 5190 8694 5159 8700 5154
69 9254 5657 9147 5614 9079 5590
70 9731 6065 9673 6072 9571 6038
71 9780 6144 10155 6440 10098 6458
72 9640 6194 10183 6552 10582 6879
73 9276 6133 10055 6621 10635 7015
74 8621 5867 9697 6532 10528 7064
75 10089 7013 8972 6208 10108 6923
76 12388 8682 10482 7408 9333 6560
77 13162 9438 12930 9297 10965 7951
78 15407 11605 13810 10284 13590 10141
79 18457 14696 16206 12756 14551 11313
80 20017 16839 19386 16175 17062 14053
81 20467 18190 21048 18632 20434 17918
82 20225 19066 21459 20099 22110 20602
83 20114 20102 21052 20796 22377 21961
84 20104 21285 20790 21697 21806 22506
85 19414 21854 20524 22790 21299 23292
86 16879 20329 19502 23152 20686 24195
87 14702 19213 16687 21203 19333 24195
88 14305 20118 14286 19596 16274 21718
89 13939 21297 13524 20135 13539 19632
90 12811 21495 12710 20897 12381 19758
91 11210 20458 11421 20568 11357 20012
92 9394 18809 9797 19013 9982 19172
93 7378 16790 7937 16852 8243 17080
94 5678 14855 6000 14498 6427 14573
95 4077 12447 4461 12477 4696 12182
96 2841 9845 3118 10055 3380 10057
97 2086 7628 2116 7584 2284 7723
98 1509 5910 1502 5676 1495 5643
99 1057 4389 1062 4296 1046 4087
100 1085 4294 1236 4835 1368 5108
101 646 2497 805 2970 929 3393
total 488.440 493.117 494.269 495.439 500.190 498.355
899
August 10, 2022 34/42
8.2 Mortality prediction using different life tables 900
In the following table we list the expected number of deaths for certain age groups using
901
the life tables 2015/17, 2016/18, 2017/19 of the Federal Statistical Office of Germany 902
and half the longevity factors of the DAV. 903
age range expected expected expected observed
¯a2015/17 2016/18 2017/19 d¯a,2020
0-14 3.565 3.585 3.531 3.306
15-29 4.070 3.996 3.944 3.844
30-39 6.718 6.655 6.626 6.668
40-49 15.673 15.557 15.345 15.507
50-59 60.494 59.796 58.641 57.331
60-69 116.457 117.236 117.432 118.460
70-79 197.146 197.428 198.389 201.957
80-89 387.256 382.712 378.459 378.406
90-198.585 199.056 199.191 200.093
total
0-989.964 986.021 981.557 985.572
904
age range expected expected expected observed
¯a2015/17 2016/18 2017/19 d¯a,2021
0-14 3.538 3.559 3.513 3.490
15-29 3.938 3.866 3.817 3.951
30-39 6.677 6.614 6.585 6.938
40-49 15.190 15.081 14.877 16.256
50-59 59.526 58.844 57.705 59.387
60-69 117.481 118.264 118.456 126.477
70-79 188.917 189.319 190.335 204.089
80-89 401.711 396.993 392.535 396.990
90-201.236 201.753 201.884 203.852
total
0-998213 994294 989.707 1.021.430
905
August 10, 2022 35/42
8.3 Monthly expected mortality: allocation factors 906
We list the estimated proportion of deaths in month mand different age ranges ¯x, ¯y.907
The first table lists the results f¯x,m for the male population, the second f¯y,m for the 908
female population. 909
¯xm1 2 3 4 5 6 7 8 9 10 11 12
0-15 8,9 7,9 9,3 8,3 7,7 8,3 8,7 8,6 7,7 8,2 8,2 8,3
15-30 8,9 7,8 8,3 7,9 8,6 8,6 9,1 8,7 8,0 8,2 8,0 7,9
30-35 9,2 8,1 9,0 8,7 8,3 8,2 8,9 8,4 7,4 8,5 7,9 7,6
35-40 7,9 8,0 9,0 7,9 9,0 8,1 8,2 8,7 8,0 8,2 8,2 8,8
40-45 8,9 8,4 9,1 8,0 8,6 8,1 8,5 8,5 7,9 7,7 8,3 8,1
45-50 9,3 8,4 9,1 8,5 8,3 8,3 8,4 8,2 7,8 8,0 7,8 7,9
50-55 9,1 8,6 9,2 8,3 8,4 8,1 8,0 8,2 7,8 8,3 8,0 8,2
55-60 9,0 8,4 9,3 8,5 8,2 8,0 8,2 8,0 7,7 8,3 8,1 8,3
60-65 9,1 8,5 9,1 8,2 8,2 8,0 8,2 8,2 7,5 8,2 8,2 8,6
65-70 8,9 8,4 9,2 8,1 8,3 7,7 8,2 8,3 7,7 8,3 8,2 8,7
70-75 9,2 8,9 9,5 8,3 8,1 7,7 8,1 8,0 7,5 8,1 8,0 8,7
75-80 9,3 8,9 9,7 8,3 8,1 7,7 7,9 7,8 7,4 8,0 8,1 8,7
80-85 9,2 8,8 9,5 8,1 8,1 7,5 7,8 7,8 7,5 8,2 8,4 9,1
85-90 9,4 9,2 9,7 8,1 8,0 7,4 7,8 7,7 7,3 8,1 8,2 9,1
90-95 9,6 9,1 9,7 8,1 7,8 7,3 7,6 7,5 7,3 8,2 8,5 9,4
95-9,7 9,0 9,9 8,0 7,8 7,3 7,5 7,4 7,1 8,2 8,6 9,5
910
¯ym1 2 3 4 5 6 7 8 9 10 11 12
0-15 8,7 8,7 9,7 8,0 8,2 8,3 8,1 7,8 8,0 8,2 7,3 9,0
15-30 8,8 8,6 8,7 7,9 8,6 7,6 8,1 8,7 8,5 8,1 8,1 8,4
30-35 8,5 7,6 8,9 8,3 8,4 8,5 7,9 8,8 8,8 7,8 7,7 8,8
35-40 8,1 7,7 8,6 8,5 8,9 8,3 8,0 8,0 7,8 8,6 8,9 8,6
40-45 9,1 8,7 9,2 8,0 8,3 8,0 8,0 8,0 7,8 8,6 8,0 8,5
45-50 9,2 8,4 9,4 8,2 8,2 7,9 8,1 8,0 8,1 8,3 7,9 8,3
50-55 8,9 8,4 9,0 8,1 8,3 8,1 8,3 8,1 7,9 8,3 8,1 8,5
55-60 8,9 8,4 9,0 8,1 8,2 7,9 8,3 8,1 7,8 8,2 8,3 8,8
60-65 8,9 8,5 9,5 8,2 8,3 7,9 8,1 8,1 7,6 8,1 8,1 8,7
65-70 8,9 8,6 9,3 8,0 8,2 7,6 8,2 8,1 7,8 8,1 8,2 8,8
70-75 9,2 9,0 9,6 8,4 8,1 7,5 8,0 7,8 7,6 8,1 8,0 8,7
75-80 9,3 9,0 9,7 8,3 8,0 7,6 7,9 7,9 7,5 8,0 8,1 8,7
80-85 9,1 8,9 9,7 8,0 8,0 7,5 7,9 8,0 7,5 8,1 8,3 9,0
85-90 9,6 9,2 10,0 8,2 7,9 7,3 7,8 7,8 7,4 7,9 8,1 8,8
90-95 9,7 9,5 10,0 8,1 7,9 7,3 7,8 7,7 7,2 7,9 8,1 8,9
95-9,5 9,3 10,0 8,0 7,8 7,2 7,7 7,8 7,3 8,1 8,3 9,1
911
August 10, 2022 36/42
8.4 Monthly development: age group 0-14 912
We list the total expected monthly number of deaths ED¯a,t,m for children, ¯a= [0,14], 913
the observed number of deaths and the relative difference. 914
t= 2020 t= 2021 t= 2022
expected expected expected
observed rel.diff. observed rel.diff. observed rel.diff.
m=1 311 311 311
272 -12,60% 296 -4,68% 267 -14,11%
m=2 298 287 287
291 -2,37% 222 -22,70% 267 -7,13%
m=3 334 333 333
313 -6,16% 290 -12,88% 276 -17,17%
m=4 288 287 287
289 0,50% 260 -9,39% 256 -10,88%
m=5 280 279 280
277 -1,09% 310 10,92% 257 -8,14%
m=6 292 291 292
275 -5,77% 303 4,05% 298 2,22%
m=7 297 297 297
278 -6,51% 290 -2,27%
m=8 290 289 289
273 -5,71% 308 6,60%
m=9 275 275 275
277 0,69% 299 8,92%
m=10 289 289 289
260 -10,14% 319 10,49%
m=11 275 275 275
240 -12,76% 306 11,46%
m=12 302 302 302
261 -13,69% 287 -4,89%
915
August 10, 2022 37/42
8.5 Monthly development: age groups 0-29 and 30-79 916
We list the total expected monthly number of deaths ED¯a,t,m for the younger 917
population
¯a
= [0
,
29] and the age group [30
,
79], the observed number of deaths and the
918
relative difference. 919
t= 2020 t= 2021 t= 2022
expected expected expected
¯a= [0,29] observed rel.diff. observed rel.diff. observed rel.diff.
m=1 659 648 643
601 -8,80% 587 -9,42% 611 -4,97%
m=2 623 592 587
621 -0,32% 498 -15,83% 564 -3,92%
m=3 666 656 651
633 -5,00% 585 -10,79% 637 -2,14%
m=4 598 589 584
577 -3,58% 593 0,74% 559 -4,30%
m=5 619 609 604
588 -5,05% 638 4,82% 595 -1,43%
m=6 618 608 603
604 -2,26% 690 13,53% 589 -2,32%
m=7 644 633 628
613 -4,80% 652 2,99%
m=8 631 620 615
630 -0,12% 620 -0,02%
m=9 596 586 581
582 -2,34% 642 9,58%
m=10 609 599 595
580 -4,81% 681 13,64%
m=11 592 582 577
549 -7,27% 625 7,36%
m=12 619 609 604
572 -7,60% 630 3,43%
920
August 10, 2022 38/42
t= 2020 t= 2021 t= 2022
expected expected expected
¯a= [30,79] observed rel.diff. observed rel.diff. observed rel.diff.
m=1 36.117 35.422 35.040
34.971 -3,17% 39.736 12,18% 35.608 1,62%
m=2 34.364 33.914 33.341
32.850 -4,41% 32.816 -3,24% 31.751 -4,77%
m=3 37.276 36.777 36.157
35.668 -4,31% 34.007 -7,53% 35.395 -2,11%
m=4 32.644 32.244 31.695
33.986 4,11% 34.852 8,09% 33.234 4,85%
m=5 32.327 31.963 31.403
31.788 -1,67% 34.533 8,04% 32.331 2,96%
m=6 30.641 30.302 29.756
30.295 -1,13% 31.825 5,03% 31.440 5,66%
m=7 31.959 31.628 31.068
31.164 -2,49% 31.943 1,00%
m=8 31.698 31.353 30.803
32.045 1,10% 31.405 0,17%
m=9 30.054 29.719 29.200
30.791 2,45% 31.556 6,18%
m=10 32.155 31.782 31.237
32.622 1,45% 33.920 6,73%
m=11 32.007 31.624 31.081
33.720 5,35% 36.366 14,99%
m=12 34.107 33.681 33.123
40.023 17,35% 40.188 19,32%
921
We visualize the different trends of the excess mortality, respectively mortality 922
deficit in Fig. 14 for the two age groups. 923
924
Fig. 14: Development of the monthly excess mortality. The blue squares show the 925
monthly excess mortality for the age group [0
,
29], the red squares the monthly excess mortality
926
for the age group [30,79] from January 2020 to June 2022. 927
August 10, 2022 39/42
8.6 Monthly development: age group 60+ 928
We list the total expected monthly number of deaths
ED¯a,t,m
for the elderly population,
929
¯a= [60,), the observed number of deaths and the relative difference. 930
t= 2020 t= 2021 t= 2022
expected expected expected
observed rel.diff. observed rel.diff. observed rel.diff.
m=1 83.255 84.439 85.435
77.313 -7,14% 98.566 16,73% 81.643 -4,44%
m=2 82.343 80.638 81.586
72.928 -11,43% 75.253 -6,68% 75.698 -7,22%
m=3 86.268 87.482 88.498
79.803 -7,49% 74.336 -15,03% 86.003 -2,82%
m=4 72.682 73.693 74.540
76.581 5,36% 74.028 0,46% 78.557 5,39%
m=5 71.408 72.405 73.236
68.660 -3,85% 73.123 0,99% 73.941 0,96%
m=6 66.626 67.539 68.300
65.220 -2,11% 69.441 2,82% 71.736 5,03%
m=7 70.238 71.222 72.040
66.701 -5,04% 69.411 -2,54%
m=8 69.932 70.907 71.717
71.662 2,47% 69.213 -2,39%
m=9 66.232 67.158 67.925
67.438 1,82% 70.655