Assessing the Impact of the COVID19 Shock on a Stochastic MultiPopulation Mortality Model
Abstract
:1. Introduction
2. Data and Notation
3. A Stochastic MultiPopulation Mortality Standard of Type Li and Lee
3.1. The Li and Lee Mortality Model
 1.
 In a first step, we calibrate the parameters ${A}_{x},\phantom{\rule{0.222222em}{0ex}}{B}_{x}$, and ${K}_{t}$ in the common mortality trend by assuming that the total number of deaths random variable ${D}_{x,t}^{T}$ follows a Poisson distribution with mean ${\mu}_{x,t}^{T}\xb7{E}_{x,t}^{T}$. Hereto, we maximize the following Poisson loglikelihood:$$\begin{array}{c}\hfill \underset{{A}_{x},{B}_{x},{K}_{t}}{max}\sum _{x\in \mathcal{X}}\sum _{t\in \mathcal{T}}\left({d}_{x,t}^{\mathrm{T}}log\left({\mu}_{x,t}^{\mathrm{T}}\right){E}_{x,t}^{\mathrm{T}}{\mu}_{x,t}^{\mathrm{T}}\right),\end{array}$$
 2.
 In a second step, we calibrate the countryspecific parameters ${\alpha}_{x},\phantom{\rule{0.222222em}{0ex}}{\beta}_{x}$ and ${\kappa}_{t}$ by assuming that the number of deaths random variable ${D}_{x,t}^{c}$, in the country of interest c, follows a Poisson distribution with mean ${\mu}_{x,t}^{c}\xb7{E}_{x,t}^{c}$. Hereto, we maximize the Poisson loglikelihood, conditional on the estimates obtained in step 1:$$\begin{array}{c}\hfill \underset{{\alpha}_{x},{\beta}_{x},{\kappa}_{t}}{max}\sum _{x\in \mathcal{X}}\sum _{t\in \mathcal{T}}\left({d}_{x,t}^{\mathrm{c}}log\left({\mu}_{x,t}^{\mathrm{c}}\right){E}_{x,t}^{\mathrm{c}}{\mu}_{x,t}^{\mathrm{c}}\right),\end{array}$$
3.2. The Time Dynamics
3.3. Generating Future Paths of Mortality Rates and Life Expectancies
3.4. The Li and Lee Mortality Model for the Belgian Population
4. Transforming Weekly Mortality Data in Age Buckets to Annual Mortality Data at Individual Ages
4.1. From Weekly to Annual Mortality Data Registered in Age Buckets
4.2. Ungrouping Data from Age Buckets to Individual Ages
4.3. Applying the Protocol to the MultiPopulation Data Set to Recalibrate the Li and Lee Mortality Model for the Belgian Population
5. Assessing the Impact of a Pandemic Shock on the MultiPopulation Mortality Model
5.1. Limiting the Time Series Likelihood Contribution of the Pandemic Data Point
5.2. Mitigating the Impact of the Pandemic Data Point with a Lee and Miller Inspired Mortality Model
6. Addressing a Mortality Shock in a Mortality Prediction Model: A Literature Review
7. Conclusions and Outlook
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Data Sources
Country  Exposures  Deaths  

2017  2018  2019  2020  2017  2018  2019  2020  
AUS  HMD  HMD  HMD  STMF  HMD  HMD  HMD  EURO.W 
BEL  HMD  HMD  EURO  STMF  HMD  HMD  EURO  STATBEL 
DNK  HMD  HMD  HMD  HMD  HMD  HMD  HMD  HMD 
FIN  HMD  HMD  HMD  STMF  HMD  HMD  HMD  EURO.W 
FRA  HMD  HMD  EURO  STMF  HMD  HMD  EURO  STMF 
GER  HMD  EURO  EURO  STMF  HMD  EURO  EURO  STMF 
ICE  HMD  HMD  EURO  STMF  HMD  HMD  EURO  EURO.W 
LUX  HMD  HMD  HMD  STMF  HMD  HMD  HMD  EURO.W 
NED  HMD  HMD  HMD  STMF  HMD  HMD  HMD  EURO.W 
NOR  HMD  HMD  EURO  STMF  HMD  HMD  EURO  EURO.W 
SWE  HMD  HMD  HMD  STMF  HMD  HMD  HMD  EURO.W 
SWI  HMD  HMD  EURO  STMF  HMD  HMD  EURO  EURO.W 
UNK  HMD  HMD  STMF  STMF  HMD  HMD  STMF  STMF 
Appendix B. Constructing Virtual Exposure Points
Age Bucket  Male Exp.  Female Exp. 

$[0,14]$  988,713.02  944,379.40 
$[15,64]$  3,699,434.72  3,638,808.41 
$[65,74]$  568,101.96  618,244.99 
$[75,84]$  305,175.72  399,015.96 
$85+$  112,577.56  223,565.55 
Appendix C. Constructing Virtual Death Counts
Age Bucket  Male Deaths  Female Deaths 

$[0,4]$  410  325 
$[5,9]$  27  27 
$[10,14]$  41  38 
$[15,19]$  115  68 
⋯  ⋯  ⋯ 
$[75,79]$  12,730  9202 
$[80,84]$  15,125  12,899 
$[85,89]$  14,737  17,246 
$90+$  12,231  24,974 
Appendix D. Validation of the Constructed Virtual Death Counts and Exposure Points
Notes
1  Numbers are retrieved from https://www.statista.com/statistics/1102209/coronaviruscasesdevelopmenteurope/ (accessed on 13 December 2021) and https://www.statista.com/statistics/1102288/coronavirusdeathsdevelopmenteurope/ (accessed on 13 December 2021) and represent the situation at 5 December 2021. 
2  These numbers of COVID19 deaths come from the COVID19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU), see https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 (accessed on 13 December 2021). 
3  See https://ec.europa.eu/info/liveworktraveleu/coronavirusresponse/safecovid19vaccineseuropeans_en (accessed on 21 October 2021) for an overview of the approved, European COVID19 vaccines and those currently under development, as well as corresponding references. 
4  See https://www.actuaries.org.uk/learnanddevelop/continuousmortalityinvestigation/cmiworkingpapers/mortalityprojections (accessed on 9 November 2021). 
5  This database is our primary database and can be consulted at https://www.mortality.org/ (accessed on 13 April 2021). 
6  Eurostat is the statistical office of the European Union, see https://ec.europa.eu/eurostat (accessed on 13 April 2021). 
7  Statbel is the Belgian statistical office, see https://statbel.fgov.be/en (accessed on 13 April 2021). 
8  This information can be explored using the visualization toolkit on https://mpidr.shinyapps.io/stmortality/ (accessed on 30 July 2021). 
9  Eurostat provides weekly death statistics at https://ec.europa.eu/eurostat/web/COVID19/data (accessed on 30 July 2021). 
10  The years 1992, 1998, 2004, 2009, 2015 and 2020 contain 53 weeks instead of the usual 52 weeks (ISO 8601 standard). 
11  The HMD team uses a Lee and Carter model to extrapolate recent exposures. The documentation can be consulted via https://www.mortality.org/Public/STMF_DOC/STMFNote.pdf (accessed on 30 July 2021). 
12  See https://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=demo_r_mwk_05&lang=en (accessed on 30 July 2021). 
13  Eurostat only provides weekly death counts for Germany for age buckets of length 10. 
14  We use the nlminbfunction in the statspackage of R. 
15  See https://data.worldbank.org/indicator/NY.GDP.PCAP.CD (accessed on 16 June 2021). 
16  See https://www.iso.org/iso8601dateandtimeformat.html (accessed on 3 November 2021). 
17  The years 1992, 1998, 2004, 2009, 2015 and 2020 contain 53 weeks. 
18  
19  The case ${\lambda}_{2020}=1$ corresponds to the Lee and Miller mortality model as discussed in Lee and Miller (2001). 
20  For all countries except the United Kingdom, we do have the annual exposures ${E}_{x,t}$ at an individual age level in 2019 from either HMD or Eurostat (see Table A1). For Denmark, we already have the annual exposures ${E}_{x,t}$ at an individual age level in the year 2020 available from the HMD. 
21  We use the exposure points $(1,{E}_{1,t})$ and $(2,{E}_{2,t})$ to linearly extrapolate to age 0:
$${E}_{0,t}={E}_{1,t}+{\displaystyle \frac{{E}_{2,t}{E}_{1,t}}{21}}\xb7(01).$$

22  For all countries except the United Kingdom, we do have the annual death counts ${d}_{x,t}$ at individual ages in 2019 from either HMD or Eurostat (see Table A1). For Belgium and Denmark, we even have the annual death counts ${d}_{x,t}$ at an individual age level in 2020 available from HMD and Statbel, respectively. For these two countries, there is no need to create virtual death counts. 
23  For Belgium and Denmark, we already have the death counts at individual ages in 2020 from Statbel and HMD, respectively. We take the average of both ratios. 
24  We only use the weekly death counts collected in age buckets from Eurostat when they match the reported death counts in the larger age buckets from the STMF data series. We do this for safety reasons because some deviations between the weekly death counts on Eurostat and the STMF data series may occur due to for example territorial differences, e.g., France with or without overseas regions. 
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Weight 2020  ${\mathit{\theta}}^{\mathit{M}}$  ${\mathit{\theta}}^{\mathit{F}}$  ${\mathit{c}}^{\mathit{M}}$  ${\mathit{c}}^{\mathit{F}}$  ${\mathit{\varphi}}^{\mathit{M}}$  ${\mathit{\varphi}}^{\mathit{F}}$ 

0  −0.2315  −0.1938  0.0071  −0.0091  0.9027  0.9033 
0.25  −0.2236  −0.1872  0.0025  −0.0142  0.9306  0.9600 
0.50  −0.2159  −0.1807  −0.0028  −0.0180  0.9424  0.9730 
0.75  −0.2083  −0.1743  −0.0083  −0.0216  0.9487  0.9790 
1  −0.2009  −0.1681  −0.0138  −0.0250  0.9527  0.9825 
IA∣BE 2020  −0.2285  −0.1882  0.0140  −0.0240  0.9682  0.9226 
Cohort Life Expectancy in 2020  Males  Females  

0  65  0  65  
Recalibration 2020 weight = 0  Best. Est.  89.36  19.70  91.42  22.78 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[87.56;89.34;90.90]$  $[18.95;19.70;20.44]$  $[89.28;91.41;93.18]$  $[21.82;22.78;23.73]$  
Recalibration 2020 weight = 0.25  Best. Est.  89.07  19.65  91.11  22.65 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[86.88;89.06;90.86]$  $[18.79;19.65;20.52]$  $[88.62;91.09;93.04]$  $[21.60;22.65;23.69]$  
Recalibration 2020 weight = 0.50  Best. Est.  88.77  19.58  90.78  22.56 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[86.23;88.74;90.81]$  $[18.61;19.58;20.54]$  $[88.08;90.76;92.87]$  $[21.42;22.55;23.65]$  
Recalibration 2020 weight = 0.75  Best. Est.  88.45  19.52  90.46  22.48 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[85.62;88.44;90.68]$  $[18.49;19.52;20.58]$  $[87.49;90.43;92.74]$  $[21.29;22.49;23.68]$  
Recalibration 2020 weight = 1  Best. Est.  88.13  19.45  90.13  22.41 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[84.98;88.10;90.56]$  $[18.31;19.44;20.58]$  $[86.79;90.10;92.66]$  $[21.12;22.40;23.69]$  
IA∣BE 2020  Best. Est.  89.91  20.38  91.54  23.14 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[88.11;89.89;91.46]$  $[19.57;20.37;21.17]$  $[89.46;91.53;93.25]$  $[22.15;23.14;24.07]$ 
Weight 2020  ${\mathit{\theta}}^{\mathit{M}}$  ${\mathit{\theta}}^{\mathit{F}}$  ${\mathit{c}}^{\mathit{M}}$  ${\mathit{c}}^{\mathit{F}}$  ${\mathit{\varphi}}^{\mathit{M}}$  ${\mathit{\varphi}}^{\mathit{F}}$ 

0  −0.2177  −0.1881  −0.0474  −0.0300  0.9588  0.9124 
0.25  −0.2139  −0.1836  −0.0305  −0.0277  0.9545  0.9607 
0.50  −0.2094  −0.1809  −0.0319  −0.0303  0.9243  0.8567 
0.75  −0.2061  −0.1791  −0.0424  −0.0337  0.8539  0.7110 
1  −0.2037  −0.1774  −0.0587  −0.0324  0.8248  0.5297 
Cohort Life Expectancy in 2020  Males  Females  

0  65  0  65  
Recalibration 2020 weight = 0  Best. Est.  89.95  20.31  92.15  23.32 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[88.09;89.95;91.47]$  $[19.56;20.30;21.02]$  $[89.42;92.14;94.20]$  $[22.12;23.32;24.45]$  
Recalibration 2020 weight = 0.25  Best. Est.  89.25  19.89  91.58  22.99 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[87.08;89.24;91.06]$  $[19.03;19.89;20.78]$  $[88.76;91.59;93.79]$  $[21.72;22.98;24.18]$  
Recalibration 2020 weight = 0.50  Best. Est.  88.60  19.57  91.12  22.82 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[85.83;88.61;90.75]$  $[18.45;19.57;20.69]$  $[88.35;91.10;93.31]$  $[21.54;22.82;24.05]$  
Recalibration 2020 weight = 0.75  Best. Est.  88.05  19.37  90.71  22.71 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[84.98;88.02;90.33]$  $[18.11;19.36;20.53]$  $[87.94;90.69;92.84]$  $[21.47;22.70;23.87]$  
Recalibration 2020 weight = 1  Best. Est.  87.57  19.19  90.32  22.58 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[84.18;87.53;89.94]$  $[17.82;19.18;20.46]$  $[87.39;90.30;92.36]$  $[21.28;22.57;23.74]$  
IA∣BE 2020  Best. Est.  89.91  20.38  91.54  23.14 
$[{q}_{0.5};{q}_{50};{q}_{99.5}]$  $[88.11;89.89;91.46]$  $[19.57;20.37;21.17]$  $[89.46;91.53;93.25]$  $[22.15;23.14;24.07]$ 
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Robben, J.; Antonio, K.; Devriendt, S. Assessing the Impact of the COVID19 Shock on a Stochastic MultiPopulation Mortality Model. Risks 2022, 10, 26. https://doi.org/10.3390/risks10020026
Robben J, Antonio K, Devriendt S. Assessing the Impact of the COVID19 Shock on a Stochastic MultiPopulation Mortality Model. Risks. 2022; 10(2):26. https://doi.org/10.3390/risks10020026
Chicago/Turabian StyleRobben, Jens, Katrien Antonio, and Sander Devriendt. 2022. "Assessing the Impact of the COVID19 Shock on a Stochastic MultiPopulation Mortality Model" Risks 10, no. 2: 26. https://doi.org/10.3390/risks10020026