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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 [4–10]. 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,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,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(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 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,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,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 (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. [6]. 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

x

year old. For them we use the smoothed mortality probability

qx−1,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

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 343

Office of Germany [17]. For

x

= 0 we set

l−1,t

=

l0,t+1

if available,

l−1,t

=

l0,t

else, and

344

q−1,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 2.1–3 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, B¨ulow RD, R¨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, H¨ohle G. On assessing excess mortality in Germany

835

during the COVID-19 pandemic (Zur Berechnung der ¨

Ubersterblichkeit in 836

Deutschland w¨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, J¨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 f¨ur Experimentelle Psychologie Institut f¨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