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Estimating the impact of the COVID-19
pandemic using granular mortality data
Frank van Berkum∗
, Bertrand Melenberg†& Michel Vellekoop‡
September 14, 2022
Abstract
We present an extension of the Li and Lee model to quantify mortality in five European
countries during the COVID-19 pandemic. The first two factors are used to model the pre-
COVID mortality, with the first layer modelling the common trend and the second layer
the country-specific deviation from the common trend. We add a third layer to capture the
country-specific impact of COVID-19 in 2020 and 2021 in excess of the pre-COVID trend.
We use weekly mortality data from the Short Term Mortality Fluctuations Database to
calibrate this third factor, and we use a more granular dataset for deaths in the Netherlands
to assess the added value of more detailed data. We use our framework to define mortality
forecasts based on different possible scenarios for the future course of the pandemic.
1 Introduction and motivation
In this paper we use a three-layer Li and Lee model [14] to quantify mortality in five European
countries during the COVID-19 pandemic, namely, Belgium, France, Germany, Great Britain,
and the Netherlands. The first two layers model the pre-COVID mortality using annual data,
where the first layer quantifies the common trend in the five countries and the second layer the
country-specific deviation from the common trend. Our model adds a third layer to capture the
country-specific impact of the new cause of death COVID-19 in the years 2020 and 2021 in excess
of the pre-COVID trend. To model and quantify this third factor, we make use of weekly data
that have become available in the Short Term Mortality Fluctuations (STMF) dataset, which
forms a part of the Human Mortality Database (HMDB) [19]. We annualize the weekly outcomes
to generate annual mortality forecasts. We supplement our model with possible scenarios for
mortality rate predictions taking into account the impact of COVID-19.
The Li-Lee model is an extension of the Lee-Carter model [13]. In the Lee-Carter model
a factor which determines the impact of mortality changes per age is estimated, and a time-
dependent factor that describes the development over time when averaged among all ages, which
therefore takes the form of a time series. Next to the Li-Lee model [14], which extends the Lee-
Carter model by adding extra layers, many other modifications have been proposed, introducing,
for example, additional factors [5,20], cohort effects [21], or a more appropriate model for
measurement noise [4]. What these and alternative models have in common (see [6] for a good
overview) is the goal of an improved description of the development of survival probabilities over
time for different age groups, since this allows better forecasts for all sorts of statistics for human
survival and better pricing and risk management models for financial products that depend on
it.
∗Corresponding author. University of Amsterdam, Dutch Actuarial Association’s Working Group for Mortality
Research & PwC the Netherlands, f.vanberkum@uva.nl .
†Tilburg University & Dutch Actuarial Association’s Committee for Mortality Research,
b.melenberg@tilburguniversity.edu.
‡University of Amsterdam & Dutch Actuarial Association’s Committee for Mortality Research,
m.h.vellekoop@uva.nl .
1
Estimating the impact of the COVID-19 pandemic using granular mortality data 2
The calibration methods for parameters in statistical models such as the Lee-Carter model
rely on historical data and these are usually available in the form of yearly observations of deaths
and population numbers or related statistics. We use annual data from 1970 up to and including
2019, retrieved from the HMDB, to calibrate the first two factors of the Li-Lee model, i.e., the
pre-COVID mortality. However, since COVID-19 only plays a role in the years 2020 and 2021,
we use weekly data to model the third layer of the Li-Lee model. The transition from yearly to
weekly data means that some adjustments in the model need to be made.
Looking at more granular data necessitates that seasonal fluctuations in deaths during the
year are taken into account since the mere assumption that deaths will be distributed over the
different weeks of the year in a uniform manner can already be refuted after a first casual glance
at the data. But if the actual distribution of mortality over weeks can be reliably deduced from
historical data, and a calibrated mortality model is available for observations before the start
of the pandemic, then weekly observations of deaths and exposures during the pandemic should
allow us to assess its impact in terms of a time series describing its severity and a factor per
age-group which determines how much its members will be affected.
We perform our analysis for the two genders separately, since it is well-known by now that
gender is an important risk factor for death due to COVID-19. The risk for men is higher than
for women, while at the same time women may be more prone to what has become known as
the “long COVID” form of the disease, which can cause debilitating symptoms for a very long
time period but does not seem to be fatal. Such a difference between the genders is partially
explained by the distinct reactions of the immune systems which have been observed for the
body’s response to infection with the SARS-CoV-2 virus [25].
Apart from age and gender, we will not consider other characteristics which may influence
an individual’s change in survival probabilities as a result of the SARS-CoV-2 virus.1We also
will not make use of data that try to measure the number of deaths with known cause of death
COVID-19 either. The actual number of deaths that are directly related to getting the disease
will far outnumber those that have been confirmed by administrative and other data [1]. But
more importantly, our aim is also to include indirect effects on mortality as a result of the
pandemic as well.2
The two-layer Li-Lee model that we use to quantify the pre-COVID mortality shows in
the first layer a clear common trend. The country-specific deviations of the common trend in
the second layer turn out not to be stationary. Instead, we model these deviations, like the
first factor, using random walks with drifts. These drift terms are not statistically significantly
different from zero. Therefore, we set these drift terms equal to zero in the mortality projections.
To quantify the factors which extend the traditional Li-Lee model, we do not only need
death counts, but also exposures at a weekly frequency. These exposures are not available, so
we have to determine these ourselves. Given the available data, we can only approximate their
values. Therefore, as a comparison, we also briefly introduce the impact of COVID-19 using a
compositional data (CoDa) analysis. Such a CoDa analysis only requires death counts, but not
the corresponding exposures. We conduct this CoDa analysis using Dutch weekly death counts
per individual age in the years 2020 and 2021, which were provided by Statistics Netherlands.
Focusing on the years 2020 and 2021, the outcomes of the CoDa analysis turn out to be quite
similar to the outcomes based on the three-layer Li-Lee model.
1Other risk factors, which can be medical, socioeconomic or pertaining to lifestyle choices, have been shown
to increase the risk of a fatal outcome after an infection. In particular, cardiovascular disease [24], autoimmune
diseases [9], diabetes [16], and obesity and increased blood pressure [10] appear to have a significant impact [8]. In
addition, a study of COVID-19 deaths in British hospitals found a sharp increase in the risk for people with high
scores on a measurement scale for deprivation, a variable that characterizes negative socioeconomic factors such
as poverty, lack of social contacts and a lower level of education. Only part of this effect could be explained by the
more common occurrence of existing medical conditions that are known to increase the risk to become severely ill
as a result of COVID-19 [26]. Data from the UK also shows that the risk is not the same for people with different
ethnicities: in a study that explicitly corrected for age, gender, preexisting conditions and socioeconomic factors,
people with a white skin were found to have a lower risk of COVID-19 mortality than those with a different skin
colour. A relatively recent large cohort study that took many of these different factors into account found that
among the non-medical indicators, old age, male sex and black skin colour are the most severe ones [9].
2Such as the deferred care mentioned in footnote 1for other conditions and the reduction in the number of
deaths due to the flu or traffic accidents.
Estimating the impact of the COVID-19 pandemic using granular mortality data 3
The weekly data in the Short Term Mortality Fluctuations (STMF) dataset only includes
the death counts over five year age ranges. We use the Dutch weekly death counts per individual
age in combination with the corresponding approximated exposures to investigate the impact
of using five year age groups compared to individual ages. Comparing the resulting outcomes
shows that the time trends are quite similar, but the age effects, after aggregating to five year
intervals, might be quite sensitive to possible cohort effects. This applies in particular to ages
that correspond to the baby boom generation, borne after the second world war.
After annualizing the weekly outcomes, we use our results to generate mortality forecasts.
Since the impact of COVID-19 on future mortality is quite uncertain at this stage, we present
these mortality forecasts for different scenarios, where each scenario represents a possible future
evolution of COVID-19.
Our approach differs from related research. For example, Robben et al. [22] and Schn¨urch et
al. [23] mention that the age profile of the impact may be different from the age profile for other
causes of death, but this profile is not estimated. In the first paper, the maximum likelihood
estimation problem for the time series of mortality dynamics from [11] is simply modified by
giving less weight in the likelihood function to the observation years during the pandemic. By
varying the corresponding weighting parameter and by adjusting the starting values of mortality
projections using a modified version of the approach of Lee and Miller [12], different projections
for the future forces of mortality in Belgium are generated and the effect on life expectancies
is determined. Schn¨urch et al. [23] provide a comparative analysis of the extra deaths in 2020
using Lee-Carter models for different countries, and Cairns-Blake Dowd models for robustness
checks. The authors do not introduce a new age-dependent factor but focus on the change in
the time series for mortality; their approach can therefore be interpreted as an extension of the
approach by Chen & Cox [7], where transitory jumps in mortality time series are assumed.
Two papers in which a distinctive age pattern is addressed explicitly are Liu and Li [15] and
Zhou and Li [27]. In the first paper, the consequences of an age effect of a sudden mortality shock
are analyzed, but under the assumption that the shock has been observed in the past and that
it only affects the year in which it occurred. In [27] the Lee-Carter model is extended by a new
age-and-time effect for COVID-19 and a new time series representing the overall impact over all
ages. Parameters are estimated using a penalized quasi-likelihood maximization. The estimated
age distribution of the effect of COVID-19 follows the shape of the pre-COVID distribution for
mortality changes. This is even true for higher ages, possibly because it is hard to separate the
effect of the pandemic from ordinary mortality changes in this approach.
The calibration results of the three-layer Li-Lee model for the five countries that we include in
our study show a different pattern. Different age groups are affected differently in the different
countries, but in all cases we find and increasing trend in age and negligible effects for the
youngest ages. This is in contrast to the results reported in [27] but in line with studies in
the epidemiological literature, such as the infection fatality ratios reported in the metastudy
of Driscoll et al. [17]. We believe that this shows the merit of the adjustments to existing
estimation methods which we propose in this paper: using a Li-Lee model for different countries
to identify a common pre-pandemic trend, using weekly data during the pandemic to estimate
the age distribution of its impact, and using robustness checks based a more granular dataset
for individual ages and a Compositional Data analysis to verify that our extrapolation method
for unknown exposures is sound.
The structure of the remainder of this paper is as follows. Section 2introduces the model
that will be used to analyse weekly mortality observations. This section describes how pre-
COVID mortality assumptions are derived, the estimation of seasonal effects in mortality, and
how COVID-19 age and week effects are calibrated. Section 3first presents results using Com-
positional Data analysis for which only death counts are needed. Then, using granular Dutch
mortality death and exposure information, results are shown for the COVID-19 age and week
effects (including various sensitivities), and a comparison is made between the COVID-19 effects
in the Netherlands, Belgium, France, Germany and Great Britain. Section 4illustrates how
estimated COVID-19 age and week effects can be used to construct scenarios for mortality rate
predictions that are adjusted for the impact of COVID-19. Finally, Section 5concludes.
Estimating the impact of the COVID-19 pandemic using granular mortality data 4
2 Model specification and calibration
In this section, we describe the framework for estimating the impact of COVID-19 on the level
of mortality. We start with introducing our COVID-19 mortality model as an extension of the
Li-Lee model for weekly observations to which an additional term is added to capture the impact
of COVID-19. Then, we describe how the baseline level of mortality is calibrated using the usual
Li-Lee model, and we investigate seasonal patterns in recent weekly mortality observations.
Finally, we describe how weekly exposures and death counts are obtained and how the full
model is calibrated.
2.1 A COVID-extension for the Li-Lee model
To assess the impact of the pandemic, we propose a model in which we distinguish a baseline
specification for mortality in a country prior to the pandemic, an adjustment for seasonal effects
when we make the transition from yearly to weekly data at the start of 2020, and a new age-
dependent factor and time series to capture the effect of the pandemic. We thus assume that
the logarithm of the force of mortality in country c for age-group3xand gender g in week wof
year thas the following structure:
ln µc,g
x, t,w = ln µc,g
x,t + ln φc,g
x,w +Bc,g
xKc,g
t,w.(1)
The first term, µc,g
x,t , is the force of mortality for the year tin a (two-layer) Li-Lee model [14]
ln µc,g
x, t =Bg
xKg
t+αc,g
x+βc,g
xκc,g
t.(2)
This force of mortality combines a Lee-Carter specification in the first term, which forces of
mortality for all countries in our chosen peer group have in common, with the last two terms that
define a country-specific deviation from the common dynamics. The age-dependent parameters
Bg
x,αc,g
xand βc,g
xand the time series Kg
tand κc,g
tare calibrated using yearly historical data for
time periods before the pandemic; the precise procedure will be discussed in the next subsection.
The forecast values for t= 2020 and t= 2021 based on these parameters determine the baseline
values for pre-pandemic mortality µc,g
x,t in those years since these are then based on historical
data before COVID-19 had any impact.
In Equation (1) for the force of mortality per week, we add two more terms. The first, ln φc,g
x,w,
is introduced because mortality is not evenly distributed over the different weeks of the year:
there is usually more mortality in the cold winter months and less mortality during the milder
months. The quantity φc,g
x,w describes this fluctuation of mortality over the different weeks and
will be called the seasonal effect. The last term in (1), the product Bc,g
xKc,g
t,w, is the third layer
of our Li-Lee model. It combines a new age effect Bc,g
xwith a new time effect Kc,g
t,w which is 0 for
t≤2019. Our specification thus preserves the model structure of a Lee-Carter or Li-Lee model,
while making it possible to work with a finer dataset of weekly instead of yearly data.
The age effect Bc,g
xis expected to be very different from the values that are found for Bg
xand
βc,g
xbecause we know that excess mortality in 2020 and 2021 was largest among the highest age-
groups. For this reason, it is less accurate to describe the effect of COVID-19 by only making an
adjustment in the time series Kg
tand κc,g
tfor t= 2020 and t= 2021 while retaining the existing
pre-pandemic model structure.
Note that we do not include a term that depends only on the age and the week (similar to
the terms Ag
xand αc,g
x), because this would mean that we make the a priori assumption that
there could be a lasting effect of the virus, even if values of the corresponding time series Kc,g
t,w
would converge to zero in the future.
When analyzing mortality on an annual basis it is common to assume the force of mortality
µx,t to remain constant during the year. Since we consider mortality on a weekly basis, it
may seem more appropriate to assume the force of mortality µx,t,w to gradually move from
µx,t,1=µx,t to µx,t,wt=µx,t+1 where wtequals the last week in year t. However, in Section 4
we show how the impact of COVID-19 on the level of mortality can be incorporated in mortality
3Note that xmay refer to an individual age x={x}or a group of ages x={x1, x2, ..., xn}.
Estimating the impact of the COVID-19 pandemic using granular mortality data 5
Male
Female
NLD
DEU
FRA
BEL
GBR
0 20 40 60 80
0.05
0.10
0.15
0.20 Peer group Bx
Age
1970 1980 1990 2000 2010
−6
−4
−2
0
2
4
Peer group Kt
Year
0 20 40 60 80
−8
−6
−4
−2
Male αx by country
Age
0 20 40 60 80
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Male βx by country
Age
1970 1980 1990 2000 2010
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
Male κt by country
Year
0 20 40 60 80
−8
−6
−4
−2
Female αx by country
Age
0 20 40 60 80
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Female βx by country
Age
1970 1980 1990 2000 2010
−1.5
−1.0
−0.5
0.0
0.5
1.0 Female κt by country
Year
Figure 1: Estimated parameter values for the baseline model in (2). The left column shows Bg
xand
Kg
ton the middle and the bottom row respectively. The middle column shows αc,g
x,βc,g
xand κc,g
ton the
top, middle and bottom row respectively when the gender g is equal to male, the third column shows
the same information as the second column but for females.
forecasts. For this purpose, it turns out to be more convenient to assume that the baseline level
of mortality remains constant during the year.
2.2 Estimation of the baseline
The baseline mortality model in (2) contains common age-dependent parameters Bg
x, country-
specific age-dependent parameters αc,g
xand βc,g
x, and the time series for the common trend Kg
t
and country-specific deviations from the trend κc,g
t. We use maximum likelihood methods to cal-
ibrate these parameters for individual ages, i.e., with x={x}, for the time period before the pan-
demic started. The Human Mortality Database contains historical data in terms of deaths (Dc,g
x,t )
and exposures (Ec,g
x,t ) for individual ages for our peer-group of countries, which consist of the
Netherlands, Belgium, Germany, France, and the United Kingdom of Great Britain and Northern
Ireland, which we will abbreviate in the country set C={NLD,BEL,DEU,FRA,GBR}.
Data from the years t∈TEU ={1970, ..., 2019}and ages x∈X={0, ..., 90}are used
for the calibration and we determine the relevant parameters in two separate steps. We first
estimate the parameters which are common to all countries, Bg
xand Kg
t, using the aggregated
deaths and exposures
Dg
x,t =X
c∈C
Dc,g
x,t , Eg
x,t =X
c∈C
Ec,g
x,t .
Under the distributional assumption that deaths conditioned on exposures follow the Poisson
distribution
Dg
x,t ∼Poisson Eg
x,t µcomm,g
x,t (3)
ln µcomm,g
x,t =Ag
x+Bg
xKg
t(4)
Estimating the impact of the COVID-19 pandemic using granular mortality data 6
we can find parameter estimates (Ag
x, Bg
x, Kg
t) by maximizing the log-likelihood
ln Lcomm =X
g∈{m,f}X
x∈XX
t∈TDg
x,t(Ag
x+Bg
xKg
t)−Eg
x,t eAg
x+Bg
xKg
t+C(5)
with Ca constant which does not affect the optimization of the likelihood. Once the estimates
Bg
xand Kg
thave been determined (as well as Ag
x, which will no longer be needed in the sequel),
we can start the second stage of the calibration based on the specification
Dc,g
x,t ∼Poisson Ec,g
x,t µc,g
x,t,
with µc,g
x,t as defined in (2) with x={x}. This implies that the remaining parameters αc,g
x,βc,g
x
and κc,g
tcan be determined by maximization of the log-likelihood
ln L=X
g∈{m,f}X
c∈CX
x∈XX
t∈TDc,g
x,t (αc,g
x+βc,g
xκc,g
t)−Ec,g
x,t eBg
xKg
t+αc,g
x+βc,g
xκc,g
t+˜
C. (6)
We impose the parameter restrictions kBgk=kβc,gk= 1 (with k · k the Euclidian norm) and
Pt∈TKg
t=Pt∈Tκc,g
t= 0 when we maximize (5)-(6) for all countries c and genders g to ensure
that no identification issues can arise.
Estimated parameters for the Li-Lee baseline model. The estimated parameters are
shown in Figure 1. The Bg
xparameters show that the highest improvements in mortality in
the observed period occurred at ages below 20 years. For higher ages the sensitivity to the Kg
t
parameter is relatively stable, though ages 70-80 seem to benefit a bit more than average. The
general trend in mortality across the countries, as represented by the slope of the Kg
tparameter,
is stable, indicating that mortality (averaged over the countries) improved at a stable pace. The
αc,g
xparameters exhibit the well-known structure: the lowest level of mortality at around 10
years, the accident hump for young adults, and mortality linearly increasing by age for higher
ages. Differences between the countries are small except for ages close to the accident hump,
and French female at higher ages seem to have structurally lower levels of mortality than the
other countries considered.
The country-specific improvements, captured by the βc,g
xand κc,g
tparameters, are less straight-
forward to interpret. The κc,g
tparameters indicate that countries may experience periods with
higher and periods with lower improvement rates than the peer group. For example, the Ger-
man male κtparameter decreases from 1970 to 1985, then increases until 2010, and remains
steady afterwards. Such periods of higher and lower improvement rates can alternate from one
year to the other, as indicated by the abrupt change in slope in those period effects. The βc,g
x
parameters, which indicate how sensitive ages are towards changes in the κc,g
tparameter, are not
consistently above or below zero. Therefore, we cannot say in general that for a specific dataset
mortality increased less or more severely for all ages compared to the peer group.
Forecasting period effects in Li-Lee baseline model. We have used data until 2019 to
calibrate the Li-Lee baseline model. To calibrate the new age-dependent factor and time series
for the effect of the pandemic, we need to predict the baseline mortality rates for the years 2020
and 2021. In Section 4we investigate the impact of the pandemic on forecasts of cohort life
expectancies, for which we need to forecast mortality far into the future. Hence, we need to
specify a time series model for the calibrated period effects Kg
tand κc,g
t.
The common approach to forecast the period effects in the peer group of countries is using a
random walk with drift, see, for example, [14] and [11]. The assumption of a random walk with
drift implies that the shared mortality trend continues in the future. For the country-specific
period effects, an autoregressive model of order one, including a constant, is often used. We
aim to calibrate one large time series model for all period effects simultaneously. Preliminary
analyses highlighted that for various country-specific period effects the mean reversion parameter
is larger than one, indicating that the time series is not stationary. We therefore impose a random
walk with drift for the country-specific period effects. For projection purposes, we neglect the
Estimating the impact of the COVID-19 pandemic using granular mortality data 7
drift term for the country-specific period effects to ensure mortality in the different countries
does not diverge. This approach is justified by the fact that the estimated drift terms for most
country-specific period effects are not statistically different from zero.
We define the assumed time series for the period effect of the peer group of countries and for
the period effect of the country-specific deviations as:
Kg
t=Kg
t−1+θg+εg
t
κc,g
t=κc,g
t−1+δc,g+ξc,g
t.
We assume that the error terms εg
tand ξc,g
tfor g ∈ {m,f}and c ∈Cfollow a multivariate
normal distribution with mean vector 012 and covariance matrix Σ. The parameters θg,δc,g
and all elements of the covariance matrix Σ are estimated using maximum likelihood estimation
techniques.
2.3 Incorporation of the seasonal effect
The specification for the weekly force of mortality in (1) contains a parameter for seasonal effect.
We consider two approaches to incorporate a seasonal effect. If no seasonal effect is estimated
in advance, which means ln φc,g
x,w is taken to be zero, it will be included in the time series Kc,g
t,w.
An alternative approach is to include a pre-determined estimator based on historical data which
ensures that the product Bc,g
xKc,g
t,w only represents the observed deviation from the baseline
predictions that are based on statistical information before the virus struck. We will designate
the first approach with the term “time series for seasonal effect plus COVID-19” and the second
approach with “time series for COVID-19”. By comparing these two methods, we can investigate
whether the two choices can lead to different conclusions about the impact of the pandemic.
To illustrate the second approach, Figure 2shows how mortality spreads over different weeks
of the year for the different countries. Values are aggregated over the two sexes and over all ages
x∈X. The gray lines for the years 2010 to 2019 show how mortality fluctuates over the weeks
and we see a clear variation in time: for example, the severe flu wave around the tenth week of
2018 in the Netherlands is clearly visible in one of the gray lines. A value of 100% in the figure
corresponds to the situation in which mortality is uniformly distributed in a year. The green line
is the average over all gray lines, and thus equals the observed average effect per week. Mortality
during the year is not evenly distributed: as expected, more people die in the winter months
than during the summer months. Using cyclic cubic splines, we have estimated a smooth effect
based on the annual historical observations shown here, and the result is represented by the
orange line. The values at the beginning and at the end of the year ensure a smooth transition
from week 52 to week 1.
2.4 Estimation of the impact of COVID-19
Once the Li-Lee model for the baseline and the seasonal effects have been calibrated, we can
estimate the remaining parameters Bc,g
xand Kc,g
t,w that describe the effect of COVID-19. We
make the assumption that during the pandemic the number of deaths for given exposures still
follows a Poisson distribution which now includes the additional factors. This implies that for
t= 2020 and t= 2021 we impose
Dc,g
x,t,w ∼Poisson Ec,g
x,t,w µc,g
x,t φc,g
x,w exp(Bc,g
xKc,g
t,w)
(where µc,g
x,t and φc,g
x,w are set to the earlier calibrated values), and the required dataset for
calibration is
{(Ec,g
x,t,w, Dc,g
x,t,w),x∈X,g∈ {m,f},c∈C, t ∈T, w ∈Wt}.
We choose a subset Xof ages and the collections T={2020,2021}and W2020 ={1, .., 53}and
W2021 ={1, .., 52}for weeks4after January 1st, 2020. The parameters that describe the impact
4The year 2020 counted among the used calendar convention NEN 2772/ISO 8601 an (incomplete) 53rd week
and 2021 an (incomplete) 0th week; a correction has been made in the datasets by merging the two.
Estimating the impact of the COVID-19 pandemic using granular mortality data 8
0 10 20 30 40 50
0.8
0.9
1.0
1.1
1.2
1.3
Week
The Netherlands
0 10 20 30 40 50
0.8
0.9
1.0
1.1
1.2
1.3
Week
Germany
0 10 20 30 40 50
0.8
0.9
1.0
1.1
1.2
1.3
Week
France
0 10 20 30 40 50
0.8
0.9
1.0
1.1
1.2
1.3
Week
Belgium
0 10 20 30 40 50
0.8
0.9
1.0
1.1
1.2
1.3
Week
Great Britain
Average 2010−2019
Estimated smooth effect
Individual years
Figure 2: Observed fraction of annual mortality per week in the years 2010-2019 for the five countries
considered, and estimated seasonal effect, aggregated across ages and sexes.
of the pandemic thus follow by determining for g ∈ {m,f}and c ∈C:
(b
Bc,g
x,b
Kc,g
t,w) = arg min
(Bx,Kt,w)X
x∈XX
t∈TX
w∈WtDc,g
x,t,wBxKt,w −Dpred,g,c
x,t,w exp(BxKt,w),(7)
where we define
Dpred,c,g
x,t,w =Ec,g
x,t,wµc,g
x,t φc,g
x,w (8)
for the expected number of deaths in a certain week based on given exposures during the week
and the baseline mortality calibration, while possibly taking into account a seasonal effect.
Estimation of weekly deaths and exposures by age. The weekly exposures during the
pandemic in (8), i.e., the values of Ec,g
x,t,w, cannot be based on observed data so we must generate
estimates based on the population data that we have at our disposal.5Various population
statistics are available in the Eurostat database. Unfortunately, only yearly population estimates
are given, and the most recent population estimate at the time that this paper was written refers
to the measurements on 1 January 2020 for c∈ {NLD,BEL,DEU,FRA}and 1 January 2019
for c= GBR.
We determine the population on January 1 of year t+ 1 as Px,t+1 =Px−1,t −Cx,t , where Cx,t
denotes the number of people that died in year tthat would have had age xat 31 December
of that year; these data can be obtained from the Eurostat database6. Some adjustments are
necessary to estimate weekly exposures. We combine the population sizes from Eurostat with
available weekly mortality observations per age-group, Dc,g
x,t,w, from the Short Term Mortality
Fluctuations dataset, which works with age-groups of five years, starting from {0,1,2,3,4}until
5To generate estimates of weekly exposures, we will make the assumption that each year consists of 52 weeks,
the month February consists of 28 days, and the month December consists of 30 days. This results in a year of
364 days, which equals exactly 52 weeks. This approach has some clear drawbacks. For example, week 1 does
not start on 1 January in each year, and December obviously has 31 days. However, this approach does take into
account the development in the population estimates through the year. In case a year has 53 weeks, we assume
that the exposure in week 53 is the same as it is in week 52 of that year. The exposure for a certain week is
estimated as the average population during that week multiplied by 7/365, so in the sequel we can focus on the
determination of the population numbers per day during 2020 and 2021.
6We used a backtest over the years 2015 up to, and including, 2019 to investigate whether this is a good
method to predict the change in population figures for an individual age over the course of a year, and found
satisfactory results. Including migration information improves the accuracy of this method, but weekly migration
information is not available and is therefore not used when estimating the weekly population sizes.
Estimating the impact of the COVID-19 pandemic using granular mortality data 9
{85,86,87,88,89}, and an open-ended final age-group {90,91,92, ...}. We transform the values
Dc,g
x,t,w for age-groups xto values Dc,g
x,t,w for individual ages xin t= 2020 and t= 2021 by
assuming that the proportional distribution of deaths over ages in an age-group in a certain
week wequals the historical average of that distribution for age-group over the years 2015 up to
(and including) 2019:
Dc,g
x,t,w =Dc,g
x,t,w ·P2019
t=2015 Dc,g
x,t
Px∈xP2019
t=2015 Dc,g
x,t
.(9)
Define Cc,g
x,t,w as the number of people that died during week win year tthat would have had
age xat 31 December of year t. We construct Cc,g
x,t,w using the approximation7:
Cc,g
x,t,w =1−w
wt·Dc,g
x−1,t,w +w
wt
·Dc,g
x,t,w,
where wtrepresents the number of weeks in year t, so w2019 = 52, w2020 = 53, and w2021 = 52.
We could estimate the population size at the first day of week w+ 1 using
Pc,g
x,t,w+1 =Pc,g
x,t,w −w
wt·Cc,g
x+1,t,w + (1 −w
wt)·Cc,g
x,t,w.
However, we then ignore the fact that people may have their birthday during a week, and we
therefore replace the above formula by
Pc,g
x,t,w+1 =1−w
wt
Pc,g
x,t,1−X
i≤w
Cc,g
x+1,t,i
+w
wt
Pc,g
x−1,t,1−X
i≤w
Cc,g
x,t,i
,(10)
which accounts for this effect under the assumption that births are distributed uniformly during
the year. We initialize this procedure using the last known value of Px,t and we apply this
formula for w= 0, ..., wt, such that Px,t,1=Px,t and Px,t,wt+1 =Px,t+1,1≈Px,t+1.
Once we have a value for the population at the beginning and the end of the week, we can
take the average to find the average population during the week, and we then have the required
exposure Ec,g
x,t,w for that week after we have multiplied the result by 7
365 .
For the Netherlands, we have more granular data at our disposal. Population sizes are
available for the first day of each month until 1 January 2022. For the other days in the
month, we determine the population sizes through linear interpolation. We do not need to
project population sizes as in (10), since linear interpolation over monthly estimates is more
accurate then projecting monthly estimates over a two-year period. The weekly exposures are
then determined by multiplying the average population during the week by 7
365 . The mortality
observations Dc,g
x,t,w are available for individual ages and can be used directly.
3 Empirical Results
In this section, we present the empirical results when the model introduced in the previous section
is applied to actual data. First, we analyze the impact of COVID-19 on mortality observations
without using exposure information. This approach can be applied relatively easily, since during
pandemics mortality observations may quickly become available, whereas exposure observations
are often estimated and published on an annual basis only. We calibrate the COVID-19 model
to Dutch mortality data for which we have mortality data available for individual ages. We use
this granular dataset to investigate different ways to include a seasonal effect, analyze which ages
are most affected by COVID-19, and examine the importance of having granular data. Finally,
we compare the impact of COVID-19 on the level of mortality over time for the collection of five
countries: NLD, DEU, FRA, BEL, and GBR.
7This approximation is based on the assumption that at all times mortality is uniformly spread over all people
in a Lexis-parallelogram who have the same age at the end of the year, and uniformly spread during the year
over all people with a common (rounded) age.
Estimating the impact of the COVID-19 pandemic using granular mortality data 10
3.1 Analysis based on Dutch death counts only
In this section we show, as comparison, the impact of COVID-19 using only the number of
deaths during the weeks of 2020 and 2021. This avoids the use of exposures which may be hard
to estimate. We follow [18] in which the Compositional Data (CoDa) analysis is introduced,
referring to [2], as an alternative to the Lee-Carter way of modeling mortality. In [18], the CoDa
analysis is presented in a number of steps which make use of so-called CoDa operators. These
operators are summarized in an appendix of [3]. We follow this approach and apply it to Dutch
mortality data for the years 2010 to 2021 and ages 0 to 98. The CoDa analysis cannot deal
with zero observations. Since the weekly death counts by age contain many zero observations,
we increase all death counts by 1. This adjustment has negligible impact on the shape of the
distribution of the deaths over the ages and over the weeks.
0 20 40 60 80 100
0
20
40
60
80
Age
Female αx
Years 2010−2019
Average over 2010−2019
Year 2020
Year 2021
0 20 40 60 80 100
−0.4
−0.2
0.0
0.2
0.4
Age
Female βx
0 10 20 30 40 50
−2
−1
0
1
2
3
4
Standardized deaths in 2020
Standardized deaths in 2021
Week
Female κt,w
0 20 40 60 80 100
0
10
20
30
40
50
60
70
Age
Male αx
0 20 40 60 80 100
−0.4
−0.2
0.0
0.2
0.4
Age
Male βx
0 10 20 30 40 50
−2
−1
0
1
2
3
4
Week
Male κt,w
Figure 3: Results from CoDa analysis using Dutch weekly number of deaths. The gray lines show the
estimated CoDa parameters for the years 2010 to 2019 and the black line represents the average over
these estimates. The estimated CoDa parameters for the years 2020 and 2021 are shown in red and blue
respectively. For the years 2020 and 2021, the graphs on the right also show the standardized weekly
death counts (normalized using the mean and variance in that year).
Figure 3shows the CoDa-parameter estimates. In the CoDa analysis applied to mortality
data, the αxparameters show the (average) age distribution of the death counts.8These age
distributions in 2020 and 2021 are close to the average over the years 2010 to 2019 for lower ages.
For higher ages, however, the age distributions in 2020 and 2021 are higher than the average
over the years 2010-2019 and in particular also the age distribution in 2019. This confirms that
COVID-19 mainly increased the number of deaths of the elderly. The age distribution of 2021
is close to that of 2020, although there seems to be a slight shift to younger ages. In particular,
there seems to be a small increase for the ages between 50 and 70.
The κt,w process in the CoDa analysis shows shifts in the age distribution over the weeks.
By construction, both the κwprocess (summed over time) and the βxparameters (summed over
ages) add up to zero. A positive value of κwimplies a shift in the age distribution of number
of deaths from the ages with a negative βxto the ages with a positive βx. The κwprocess
shows a clear peak between weeks 10 and 20 of the year 2020, the first wave of the COVID-19
pandemic in the Netherlands. It also shows increasing positive values near the end of 2020,
followed by positive values at the start of 2021, the second wave of COVID-19. Finally, near
the end of 2021 the κwprocess shows the third wave of COVID-19. For comparison we also
8This age distribution shows the absolute number of deaths per age, aggregating to the total number of deaths
per week.
Estimating the impact of the COVID-19 pandemic using granular mortality data 11
show the standardized weekly death counts, i.e., the death counts normalized using the mean
and variance in that year.
These peaks imply a shift in the age distribution from the young to the old, since the βxs
of the older people (above age 60) are positive, while the βxvalues of the younger ages are
mostly negative (although there are some exceptions as far as the younger ages are concerned).
However, the βxs for the older males are substantially higher in 2020 than in 2021. The shifts
in the age distribution (corresponding to an increase in the κwprocess) of the males in 2021 are
far less dramatic than the shifts in 2020. For females, the βxcoefficients of 2021 are close to
those in 2020 for ages above 80, but for ages between 60 and 80 the βxvalues in 2021 are lower
than those in 2020. Thus, the shifts in the age distribution for females in 2021 is particularly in
the direction of the very old, above age 80.
3.2 Calibration results based on a Dutch dataset with high granularity
In this section we continue our analysis using the Dutch mortality data as obtained from Statistics
Netherlands. First, we investigate the two different approaches for incorporating the seasonal
effect, then we compare the results when using different age ranges for calibration, and finally
we analyze the importance of using granular data.
Incorporation of seasonal effect. In Section 2.3 we described two approaches for including
the seasonal effect:
Method 1 Time series for seasonal effect plus COVID-19 : the seasonal effect φc,g
χ,t in (1) is set
equal to 1, which results in any seasonal effect being captured by the COVID-19 term Kc,g
t,w.
Method 2 Time series for COVID-19 : the seasonal effect φc,g
χ,t is set equal to the estimated
smooth seasonal effect as illustrated in Figure 2, which results in the COVID-19 term Kc,g
t,w
reflecting the impact of COVID-19 corrected for this seasonal effect.
Figure 4shows the calibrated parameters. We observe that the differences in the COVID-19 age
effects, as shown in the top row, are hardly visible. The COVID-19 age effect is erratic and close
to zero for both females and males up to age 60. This indicates that for those ages there was
hardly any impact on the level of mortality due to COVID-19 in the years 2020 and 2021. For
higher ages, the age effect for females increases from age 60 to 70 and remains constant for ages
70 to 98, and for males the age effect increases steadily from age 60 to 98.
The COVID-19 week effects (bottom row) show substantial differences between the two
methods. In the Netherlands, the corona virus was first identified in February 2020, which
means that in the first four weeks of 2020 we would expect the COVID-19 week effect to be
close to zero. The week effect represented by the dashed lines clearly starts above 0 for males
and females. Further, in the summer of 2020 (around week 26) the dashed lines for the week
effect are below zero, indicating mortality levels were below expectation if seasonality was not
taken into account. The COVID-19 week effect of Method 2 (the solid line) starts close to zero
and remains above zero for nearly all weeks included in the dataset; Method 2 therefore seems
to capture only the impact of COVID-19 on observed mortality.
The observed weekly mortality death counts as shown in Figure 5are very volatile for the ages
45 and 55 and do not exhibit the peaks as observed in the COVID-19 week effects in Figure 4.
The fitted death counts are close to the seasonally adjusted death counts, and the impact of
COVID-19 on mortality at these ages is therefore negligible. For ages 65 and 85, the seasonally
adjusted expected deaths clearly exhibit a seasonal pattern, but this pattern is not sufficient to
capture the peaks that coincide with the COVID-19 waves. The fitted deaths using Method 1
and Method 2 follow the wave pattern in the observed deaths more closely.
We would like to be able to distinguish between effects induced by a typical seasonal effect
and effects due to a pandemic. Method 2, in which we correct for historically observed seasonal
effects, allows us to assess the impact of only the pandemic on the level of mortality. Further,
Method 2 is more flexible than Method 1, since the seasonal effect is not enforced to affect
mortality the same way as the pandemic. Therefore, in the remainder of the analyses we will
only use Method 2 when estimating the impact of COVID-19 on the level of mortality.
Estimating the impact of the COVID-19 pandemic using granular mortality data 12
0 20 40 60 80 100
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Age
NLD female COVID age effect
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
NLD female COVID week effect
0 20 40 60 80 100
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Age
NLD male COVID age effect
Method 1
Method 2
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
NLD male COVID week effect
Figure 4: COVID-19 parameters estimated with and without predetermined seasonal effect (solid
respectively dashed line) for females and males using Dutch data.
0 20 40 60 80 100
0
2
4
6
8
10
12
14
Week since 1 Jan 2020
NLD male deaths at age 45
Observations
Expected deaths (constant mu)
Expected deaths (seasonally adj. mu)
0 20 40 60 80 100
0
5
10
15
20
Week since 1 Jan 2020
NLD male deaths at age 55
Fitted deaths (COVID, method 1)
Fitted deaths (COVID, method 2)
0 20 40 60 80 100
0
10
20
30
40
50
Week since 1 Jan 2020
NLD male deaths at age 65
0 20 40 60 80 100
0
20
40
60
80
100
120
Week since 1 Jan 2020
NLD male deaths at age 85
Figure 5: The black dots represent observed weekly deaths for Dutch males at ages 45, 55, 65 and 85
during the years 2020 and 2021. The black line shows the expected deaths assuming a constant force of
mortality, and the blue line represents expected death counts taking into account the historical seasonal
effect. The yellow and red lines show the fitted deaths using Method 1 respectively Method 2.
Selection of ages. The COVID-19 age effects for ages below 40 in Figure 4are volatile and
seem to be centered around zero. In Appendix Awe show estimates using ages 0-98 and ages
40-98. From the parameter estimates in Figure 12, we conclude that the COVID-19 week param-
eters and age parameters for higher ages are hardly affected when including information from
Estimating the impact of the COVID-19 pandemic using granular mortality data 13
younger ages. For interpretation of parameters and reliability of estimates and projections, it is
desirable that only structural effects are analyzed; noise due to low exposures should preferably
be excluded. Since the age effects at the younger ages do not seem to capture a systematic
COVID-19 effect, we exclude data for ages below 40 from calibration.
Importance of granular data. For the Netherlands we have weekly mortality data available
for both genders and individual ages. The Short Term Mortality Fluctuations database (STMF)
contains weekly mortality data for many countries, but only for specific age groups. For some
countries the age groups are small (age groups of five years), whereas for other countries there
are only five age groups.
In this section, we use the dataset from Statistics Netherlands to investigate the importance
of having granular data in analyzing the impact of COVID-19 on mortality. We consider three
levels of granularity:
•Level 1, individual ages with x∈ {0,1,2, ..., 99}; this is the most granular type of data;
•Level 2, age groups of five years with x∈ {{0, ..., 4},{5, ..., 9}, ..., {90, ..., 94},{95, ...}};
•Level 3, five age groups with x∈ {{0, ..., 14},{15, ..., 64},{65, ..., 74},{75...84},{85, ...}};
this is the least granular type of data.
We use the original dataset with observations by individual ages (Level 1) to artificially construct
the datasets based on Level 2 and Level 3 granularity.
Figure 6shows the parameter estimates when calibrating the COVID-19 model to the three
datasets. The top row shows the estimated COVID-19 age effects, and we observe clear differ-
ences between the estimates from the datasets with different levels of granularity. The age effects
are volatile for all levels of granularity, but the Level 2 and Level 3 estimates show remarkable
peaks and dips at the ages 65-75. In the year 2020, these ages correspond with the years of
birth 1945-1955 (with a clear peak in 1946), which is the so-called baby boom generation after
the second World War. The baby boom generation appears as a cohort effect in the exposures,
which in turn results in cohort effects in the observed deaths. For most age groups (either Level
2 or Level 3), the distribution of deaths over the ages is relatively stable over time, except for
this baby boom generation. This cohort effect makes the allocation of deaths to individual ages
increasingly inaccurate if larger age groups and more historical years are used when applying
Equation (9).
In contrast, the COVID-19 week effects estimated using the datasets of different granularity
look remarkably similar. This is as expected since the observed deaths are not relocated over
different weeks, only over different ages. The impact of COVID-19 on total mortality will
therefore be similar, which results in stable estimates of the COVID-19 week effect, regardless
of the level of granularity of the dataset.
From this analysis we conclude that datasets with deaths by age group can be used to obtain
a first impression of the impact of a pandemic over time. However, if the impact on individual
ages must be assessed, one needs to use death counts for individual ages.
3.3 Calibration results based on the STMF dataset
In this section, we calibrate the COVID-19 model to the countries cin the set given by C=
{NLD,DEU,FRA,BEL,GBR}using mortality data as obtained from STMF. The deaths by age
group are allocated to individual ages using Equation (9), and we include data for the ages 40 to
95. We investigate to what extent similarities and differences between countries can be observed.
Figure 7shows the estimated COVID-19 parameters.9The COVID-19 age effect is close to
zero at lower ages for most countries. The exception to this is Great Britain, where the COVID-
19 age effect is substantial for all ages included. The volatile behavior between ages 65 and 80
is probably the result of cohort effects in the death counts, as described in the previous section.
9Results obtained using ages 0-98 are available in Appendix B.
Estimating the impact of the COVID-19 pandemic using granular mortality data 14
40 50 60 70 80 90 100
−0.1
0.0
0.1
0.2
0.3
Age
NLD female COVID age effect
0 20 40 60 80 100
−1
0
1
2
3
4
5
Week since 1 Jan 2020
NLD female COVID week effect
Level 1 (most granular)
Level 2
Level 3 (least granular)
40 50 60 70 80 90 100
−0.1
0.0
0.1
0.2
0.3
Age
NLD male COVID age effect
0 20 40 60 80 100
−1
0
1
2
3
4
5
Week since 1 Jan 2020
NLD male COVID week effect
Figure 6: COVID-19 parameter estimates using datasets with different levels of granularity.
40 50 60 70 80 90
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
Age
Female COVID age effect
40 50 60 70 80 90
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
Age
Male COVID age effect
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
Female COVID week effect
NLD DEU FRA
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
Male COVID week effect
BEL GBR
Figure 7: Estimated COVID-19 parameters for various countries using ages 40 to 95.
We observe that the different countries exhibit similar COVID-19 week effects, though there
are a few notable differences. Germany experienced a minor first COVID-19 wave compared to
the other four countries, whereas later COVID-19 waves were similar to those in other countries.
Around week 33, Belgium, the Netherlands and Germany experienced a temporary peak, which
may be the result of temporarily relaxing COVID-related restrictions.
Between week 40 and 60 (winter season 2020-2021) the second COVID-19 wave hit West-
ern Europe, but we observe substantial differences between countries. From week 50 onward,
Estimating the impact of the COVID-19 pandemic using granular mortality data 15
COVID-19 vaccines became available in various countries, though the availability and timing of
the vaccine shots for people of different ages differed between countries. Belgium had a high peak
at around week 45 after which excess mortality soon disappeared. The Netherlands, Germany
and France all experienced lower peaks, but in these countries it also took a few more weeks
before excess mortality had vanished. This decrease in excess mortality may have been the result
of vaccines being applied to parts of the population but also due to new COVID-19 restrictions.
Finally, Great Britain experienced a high peak which was similar to the one in Belgium, but a
few weeks later, and measures were taken resulting in excess mortality decreasing rapidly. Great
Britain was one of the countries in which vaccines were provided to the population earlier and
faster.
It will always remain challenging to assess the effectiveness of policy decisions during a
pandemic. But using the best possible data and using improved estimation methods can help to
improve such assessments.
4 Forecasting mortality adjusted for COVID-19
In the previous section we have analyzed the impact of COVID-19 on the level of mortality in
the years 2020 and 2021. For insurance companies and pension funds it is particularly important
to assess the impact of COVID-19 on mortality rates. At the time this paper was written, it was
too early to predict whether long term mortality improvement rates should be adjusted upward,
downward, or not at all. In this section, we therefore introduce a general framework in which
the impact of COVID-19 on the level of mortality in the years 2020 and 2021 can be used to
generate scenarios for future mortality.
Transforming weekly effects to annual effects. The impact of COVID-19 was calibrated
using weekly mortality observations. From these calibrations, we obtained weekly parameter
estimates Bxand Kt,w. To generate scenarios for future mortality, we first need to transform
these week effects into annual effects Vxand Xt. We temporarily impose that Px∈XVx= 1
such that analytical results can be used for the transformation. Afterwards, we again apply the
restriction kVk= 1. To transform week effects into annual effects, we make the two one-year
survival probabilities for the years 2020 and 2021 equal to the product of the weekly survival
probabilities in those years:
exp (−µx,t ·exp[VxXt]) =
wt
Y
w=1
exp −1
wt·µx,t ·φx,w ·exp [BxKt,w],(11)
for t= 2020 and t= 2021, and for all x∈X. After taking the natural logarithm on both sides,
dividing by µx,t, taking the natural logarithm again, and summing over all ages x∈X, we find
Xt=X
x∈X
ln (1
wt
wt
X
w=1
φx,w ·exp[BxKt,w]).
Next, we determine Vxby making survival over both the years 2020 and 2021 equal to surviving
over all weeks in those years (which follows directly from (11)):
2021
Y
t=2020
exp (−µx,t ·exp[VxXt]) =
2021
Y
t=2020
wt
Y
w=1
exp(−1
wt·µx,t ·φx,w ·exp[BxKt,w]).
Rewriting this equation results in
2021
X
t=2020
µx,t
wt
X
w=1
1
wt(exp[VxXt]−φx,w ·exp[BxKt,w]) = 0,
and this non-linear equation in Vxcan be solved numerically for each age x∈Xseparately.
Finally, we again renormalize the parameters Vxand Xtsuch that kVk= 1.
Estimating the impact of the COVID-19 pandemic using granular mortality data 16
Figure 8shows the weekly and annual effects calibrated using the Dutch mortality data from
Statistics Netherlands. The weekly and annual COVID-19 age effects are nearly identical, and
the COVID-19 year effects are close to the averages of the week effects in that year.
40 50 60 70 80 90
−0.1
0.0
0.1
0.2
0.3
Age
COVID age effect
Female
Male
0 20 40 60 80 100
−1
0
1
2
3
4
5
Week since 1 Jan 2020
COVID week effect
Week estimates
Year estimates
Figure 8: COVID-19 parameters from calibration on weekly data (solid line) and annualized COVID-
19 parameters (dashed line).
Future scenarios for the COVID-19 period effect. At the end of the observed period,
the week effect as shown in Figure 8is not far from zero. For prediction purposes, one might
be tempted to choose Xtequal to 0 for t > 2021. However, the week effect also shows that after
periods of (almost) no excess mortality, new periods of excess mortality may occur. We therefore
investigate how mortality rates could develop under varying assumptions for the future impact
of COVID-19 on mortality.
We choose scenarios that can be written as:
ln µx,2021+h= ln µpre-covid
x,2021+h+VxX2021+h
X2021+h=Xstartηh+ (1 −ηh)X∞,
for 0 ≤η≤1. The value of the COVID-19 period effect in the limit, X∞, is chosen to be a
multiple of X2021 where the multiplicative factor is scenario-dependent. The parameter ηdefines
how fast convergence to this limit value takes place. This generic approach for generating
COVID-19 scenarios is similar to that of [27] who estimate similar COVID-19 effects using a
penalized quasi-likelihood approach. We define the following scenarios:
1. Completely incidental:Xstart =X∞= 0.
The mortality forecast is completely based on the pre-COVID forecast.
2. Completely structural:Xstart =X∞=X2021.
It is assumed that COVID-19 is a new cause of death that will remain permanently with
an impact that does not change over time.
3. Decreasing impact: 0 < η < 1, Xstart =X2021 ,X∞= 0.
It is assumed that the effect of COVID-19 converges to zero after a certain period, for
example because herd immunity is (almost) reached in a population.
4. Growing impact: 0 < η < 1, Xstart =X2021 ,X∞= 1.25X2021 .
The impact grows, for example when new variants appear which are not impacted by
current and new vaccines, or when other mitigating policies imposed by the government
become less effective.
5. New normal: 0 < η < 1, Xstart =X2021 ,X∞= 0.25X2021 .
The impact decreases to a constant but does not completely vanish, which results in a
permanent effect that can be compared to other causes of death such as the flu.
Estimating the impact of the COVID-19 pandemic using granular mortality data 17
6. Increased resilience: 0 < η < 1, Xstart =X2021 ,X∞=−0.25X2021 .
The impact gradually converges to a value below zero, for example because the population
that survived the pandemic is stronger than the population before the pandemic.
At the time of writing this paper, no information is available to make an informed decision for
the value of η. We choose η= 0.5 for illustration purposes. Given the other choices made for
specifying the COVID-19 scenarios, the predicted period effects for Dutch males are shown in
Figure 9; the figure for females looks similar in case the same choices are made.
2020 2025 2030 2035 2040 2045 2050
−0.5
0.0
0.5
1.0
1.5
Year
Future scenarios for COVID period effect
Completely incidental
Completely structural
Decreasing impact
Growing impact
New normal
Increased resilience
Figure 9: Projected COVID-19 period effects for Dutch males under various assumptions for future
development of impact of COVID-19 on mortality.
We also need to specify how the COVID-19 age effect is defined for ages x /∈X. Define xmin
and xmax as the lowest respectively highest age included in the set of ages used for calibration,
X(i.e. xmin = 40 and xmax = 95 in the previous section). Figure 12 suggests that for lower
ages the impact of COVID-19 on the level of mortality was negligible, and therefore these ages
are excluded from calibration. In line with that approach, we define Vx= 0 for x < xmin . For
higher ages, we observe that the estimated parameter Vxseems relatively stable at higher ages.
Based on that observation, we define Vx=Vxmax for x>xmax.
Mortality predictions under various COVID-19 scenarios. For the various scenarios
illustrated in Figure 9, we have constructed the corresponding forecasts of mortality rates. Fig-
ure 10 shows the mortality forecasts for Dutch males at ages 55, 65 and 85. Though at age
55 mortality rates in 2020 and 2021 were not markedly different from the pre-COVID expec-
tation, at age 65 the level of mortality was elevated and at age 85 it was substantially higher
than expected. The pattern of the projected COVID-19 period effects from Figure 9are clearly
visible in the mortality forecasts. When constructing mortality rates, the COVID-19 period
effects are multiplied with the COVID-19 age effects from Figure 8which explains why the
forecast of mortality rates at age 55 are hardly affected by the COVID-19 period effect while
at age 85 mortality in all scenarios (except for Completely incidental) deviates greatly from
the pre-COVID forecast. For the scenarios Decreasing impact,New normal and Increased
resilience, the impact on mortality diminishes quickly after a few years (given our choice for
η), but for Completely structural and Growing impact the impact remains.
The impact on predicted period life expectancies and cohort life expectancies is shown in
the top row respectively bottom row of Figure 11. The period life expectancy at all ages is far
below the pre-COVID expectation. Though the COVID-19 age effects are calibrated for the ages
40-95, the period life expectancy at age 0 is also affected, since mortality rates at all ages are
used to compute this life expectancy. The gap between the most positive scenario (Increased
resilience) and the most negative scenario (Growing impact) increases to approximately one
year around 2040 for the ages shown. This impact remains constant further in the future once the
COVID-19 period effect has converged to its limit value, and all remaining future developments
are driven by the original Li-Lee model.
The pre-COVID forecast and the scenarios Completely incidental and Decreasing impact
result in comparable projected cohort life expectancies. While in the predicted period life ex-
Estimating the impact of the COVID-19 pandemic using granular mortality data 18
2000 2010 2020 2030 2040
0.0030
0.0035
0.0040
0.0045
0.0050
0.0055
Year
Male mortality rate at age 55
Observation (regular years)
Observation (COVID years)
Pre−COVID forecast
2000 2010 2020 2030 2040
0.008
0.010
0.012
0.014
0.016
Year
Male mortality rate at age 65
Completely incidental
Completely structural
Decreasing impact
2000 2010 2020 2030 2040
0.06
0.08
0.10
0.12
Year
Male mortality rate at age 85
Growing impact
New normal
Increased resilience
Figure 10: Projected mortality rates for Dutch males aged 55, 65 and 85 under various assumptions
for future development of impact of COVID-19 on mortality.
2000 2010 2020 2030 2040
76
78
80
82
84
Year
Male PLE at age 0
Observation (regular years)
Observation (COVID years)
Pre−COVID forecast
2000 2010 2020 2030 2040
16
17
18
19
20
21
Year
Male PLE at age 65
Completely incidental
Completely structural
Decreasing impact
2000 2010 2020 2030 2040
5.0
5.5
6.0
6.5
Year
Male PLE at age 85
Growing impact
New normal
Increased resilience
2020 2030 2040 2050 2060
89
90
91
92
93
Year
Male CLE at age 0
2020 2030 2040 2050 2060
20
21
22
23
24
25
Year
Male CLE at age 65
2020 2030 2040 2050 2060
5.5
6.0
6.5
7.0
7.5
Year
Male CLE at age 85
Figure 11: Projected period and cohort life expectancies for Dutch males at birth and at ages 65 and
85 under various assumptions for future development of impact of COVID-19 on mortality.
pectancies differences between scenarios grow over time, in predicted cohort life expectancies the
difference between scenarios at the start of the projection is close to the difference in the limit.
5 Conclusions
In this paper, we introduce a model to quantify weekly deviations from expected levels of mor-
tality during a pandemic. The model adds an extra layer to the two-layer Li-Lee model by
including an additional seasonal effect to capture regular seasonal patterns and an additional
age and week effect to measure deviations from pre-pandemic weekly mortality expectations.
We apply our model to data from Belgium, France, Germany and Great Britain using mor-
tality data from the Short Term Mortality Fluctuations database. There are differences between
countries in the extent to which mortality at different ages is affected by COVID-19 and how
COVID-19 affected mortality through time. Yet, there are also clear similarities since in all
countries periods of high excess mortality are followed by periods of lower or no excess mortal-
ity.
The application of the model requires the availability of exposures which often are not avail-
able on a weekly basis and these must therefore be approximated. Useful insights on the de-
Estimating the impact of the COVID-19 pandemic using granular mortality data 19
velopment of mortality through a year can be obtained through a Composition Data (CoDa)
analysis that can be performed using weekly death counts only.
Most sources of weekly mortality observations provide data by gender and by age groups.
Our sensitivity analysis shows that such data can be used to accurately monitor the development
of the pandemic through time, which is represented by the COVID-19 week effect. However, the
COVID-19 age effects are inaccurate in case of cohort effects exist in population sizes, and use
of weekly observations by individual ages is therefore recommended.
The future COVID-19 scenarios analyzed in this paper all assume convergence to pre-COVID
long term improvement rates. At this stage, there is insufficient data and information available
to determine whether, and if so how long term mortality improvement rates should be adjusted.
This paper provides no solution for this problem, and this is a problem that is likely to challenge
the imagination of demographers and actuaries for years to come.
Acknowledgment. Certain parts of the approach proposed in this paper have been used in a
model to generate a prognosis for future survival probabilities that was recently developed for
the Royal Dutch Actuarial Association. The authors gratefully acknowledge all comments from
members of the Association’s Committee and Working Group on Mortality Research: Wies de
Boer, Friso Cuijpers, Corn´e van Iersel, Marieke Klein, Hans de Mik, Erica Slagter, Janinke Tol,
Erik Tornij, Raymond Waucomont, Wouter van Wel, Menno van Wijk, Marco van der Winden
and Kim Wittekoek. We also thank Statistics Netherlands, and in particular Lenny Stoeldraijer,
for helping us to obtain the required datasets.
Estimating the impact of the COVID-19 pandemic using granular mortality data 20
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Estimating the impact of the COVID-19 pandemic using granular mortality data 22
A Selection of ages
In Figure 4we observed that for ages below 40 the COVID-19 age effects are erratic. Since the
age effects do not seem to capture a systematic COVID-19 effect at the younger ages, it might
be better to exclude those ages from calibration of the model. The COVID-19 age effects are
more stable from age 40 onward, though between ages 40 and 60 the estimated effect is close to
zero.
For interpretation of parameters and reliability of outcomes it is desirable that only structural
effects are analyzed; noise due to low exposures should preferably be excluded. Therefore, we
compare parameter estimates if we use different age ranges for calibration of the COVID-19
model. Figure 12 shows the estimated parameters for females and males using ages 0-98 (dashed
line) and ages 40-98 (solid line) for calibration, where the former are renormalized such that
the norms over the age range 40-98 are equal in both cases. This allows a comparison of the
parameter estimates.
0 20 40 60 80 100
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Age
NLD female COVID age effect
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
NLD female COVID week effect
0 20 40 60 80 100
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Age
NLD male COVID age effect
Ages 0−98
Ages 40−98
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
NLD male COVID week effect
Figure 12: COVID-19 parameters estimated using ages 0-98 (dashed line) and ages 40-98 (solid line).
From the parameter estimates in Figure 12 we conclude that the COVID-19 week parameters
and age parameters for higher ages are hardly affected when including information from younger
ages. However, the COVID-19 age effects for ages below 40 years are erratic, and therefore we
exclude these ages from further analyses.
Estimating the impact of the COVID-19 pandemic using granular mortality data 23
B Additional calibration results based on STMF data
0 20 40 60 80 100
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Age
Female COVID age effect
0 20 40 60 80 100
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Age
Male COVID age effect
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
Female COVID week effect
NLD DEU FRA
0 20 40 60 80 100
−2
0
2
4
6
Week since 1 Jan 2020
Male COVID week effect
BEL GBR
Figure 13: Estimated COVID-19 parameters for various countries using ages 0 up to and including
99.
