# Short-Term Mortality Fluctuations and Longevity Risk-Adjusted Age: Learning the Resilience of a Country to a Health Shock

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## Abstract

**:**

## 1. Introduction

## 2. The Gompertz–Makeham Mortality Estimation from Weekly Data

#### L-RaG Age Indicator

## 3. Results

#### 3.1. Description of the Data

#### 3.2. Estimation of LRaG Age and dLRaG Indicator

#### 3.3. Does the Dynamic Evolution of the dLRaG Indicator Capture Quality of Life?

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

L-RaG | longevity-risk-adjusted global |

GM | Gompertz–Makeham |

TMR | total mortality rate |

wTMR | weekly total mortality rate |

INMR | initial natural mortality rate |

ME 100k | mortality excess per 100k |

MEAB | mortality excess as a percentage of the annual baseline |

HE | health expenditure |

## Appendix A

Country Code | Country Name | Country Code | Country Name |
---|---|---|---|

AUS | Australia | LVA | Latvia |

AUT | Austria | LTU | Lithuania |

BLR | Belarus | LUX | Luxembourg |

BEL | Belgium | NLD | Netherlands |

BGR | Bulgaria | NZL_NP | New Zealand Total population |

CAN | Canada | NZL_MA | New Zealand Maori |

CHL | Chile | NZL_NM | New Zealand Non-Maori |

HRV | Croatia | NOR | Norway |

CZE | Czechia | POL | Poland |

DNK | Denmark | PRT | Portugal |

EST | Estonia | KOR | Republic of Korea |

FIN | Finland | RUS | Russia |

FRATNP | France Total population | SVK | Slovakia |

FRACNP | France Civilian population | SVN | Slovenia |

DEUTNP | Germany Total population | ESP | Spain |

DEUTE | Germany East Germany | SWE | Sweden |

DEUTW | Germany West Germany | CHE | Switzerland |

GRC | Greece | TWN | Taiwan |

HKG | Hong Kong | GBR_NP | U.K. United Kingdom Total Population |

HUN | Hungary | GBRTENW | U.K. England and Wales Total Population |

ISL | Iceland | GBRCENW | U.K. England and Wales Civilian Population |

IRL | Ireland | GBR_SCO | U.K. Scotland |

ISR | Israel | GBR_NIR | U.K. Northern Ireland |

ITA | Italy | USA | U.S.A. |

Country | ME 100k | MEAB | HE 2019 | D2017 | D2018 | D2019 | D2020 |
---|---|---|---|---|---|---|---|

AUS2 | −14.40 | −2.50 | 4919.24 | −2.30 | −1.80 | −2.29 | −1.23 |

AUT | 108.90 | 11.70 | 5705.10 | −0.31 | −0.52 | −0.24 | −0.16 |

BEL | 138.70 | 14.50 | 5458.40 | 0.06 | 0.12 | 0.10 | −0.15 |

BGR | 457.50 | 29.00 | 1842.05 | 0.72 | 0.64 | 0.62 | 0.21 |

CAN | 40.00 | 5.10 | 5370.44 | 1.19 | 0.99 | 1.14 | 1.26 |

CHE | 99.70 | 12.60 | 7138.06 | 0.14 | 0.20 | −0.00 | 0.23 |

CHL | 158.50 | 26.90 | 2291.46 | 1.94 | 1.61 | 2.00 | 0.98 |

CZE | 323.60 | 30.10 | 3417.49 | −0.99 | −0.81 | −0.87 | −0.85 |

DEUTNP | 47.10 | 4.00 | 6518.00 | 0.07 | 0.04 | 0.12 | 0.21 |

DNK | −10.80 | −1.10 | 5477.57 | −0.57 | −0.44 | −0.69 | −0.26 |

ESP | 186.20 | 20.30 | 3600.28 | −0.48 | −0.36 | −0.30 | −0.30 |

EST | 137.90 | 11.50 | 2507.07 | 0.48 | 0.50 | 0.17 | 0.52 |

FIN | 7.40 | 0.70 | 4558.54 | −1.18 | −1.31 | −1.42 | −0.74 |

FRATNP | 109.60 | 11.60 | 5274.26 | 0.84 | 0.75 | 0.95 | 0.51 |

GBR | 160.70 | 17.70 | 4500.14 | 0.22 | 0.18 | 0.35 | 0.11 |

GBR_NL | 28.50 | 4.50 | 2318.96 | −0.21 | −0.07 | −0.14 | −0.08 |

GRC | 72.40 | 5.90 | 2014.20 | −0.47 | −0.37 | −0.53 | −0.23 |

HUN | 243.60 | 17.80 | 2169.77 | −0.26 | −0.53 | −0.10 | −0.21 |

ISL | −4.50 | −0.70 | 4540.76 | 0.14 | −0.21 | −1.60 | 0.08 |

ISR | 56.00 | 10.40 | 2903.41 | 0.14 | 0.05 | 0.07 | −0.19 |

ITA | 206.30 | 19.10 | 3653.40 | −0.42 | −0.14 | −0.60 | −0.45 |

LTU | 350.70 | 24.90 | 3406.26 | −0.30 | −0.23 | −0.22 | −0.14 |

LUX | 31.00 | 4.30 | 2727.19 | −0.02 | 0.16 | 0.31 | −0.06 |

LVA | 158.10 | 10.40 | 5414.48 | 0.67 | 0.89 | 0.72 | 0.35 |

NOR | −28.20 | −3.70 | 2039.22 | 0.38 | −0.00 | 0.18 | 0.38 |

NZL_NP | −40.00 | −5.40 | 5739.20 | −0.29 | −0.35 | −0.20 | 0.02 |

POL | 309.30 | 27.60 | 6744.62 | −1.42 | −1.26 | −1.18 | −0.88 |

PRT | 184.90 | 16.30 | 4211.85 | 0.58 | 0.58 | 1.23 | 0.78 |

RUS | 339.80 | 28.20 | 2289.31 | 0.48 | 0.31 | 0.30 | −0.08 |

SVK | 305.40 | 30.50 | 3347.43 | −0.52 | −0.06 | −0.15 | −0.27 |

SVN | 178.70 | 17.40 | 1850.26 | 1.28 | 0.73 | 0.59 | 0.04 |

SWE | 88.10 | 9.70 | 2189.05 | 0.35 | 0.53 | 0.87 | 0.47 |

**Table A3.**Quartile ranking with respect to excess mortality per 100k people in 2020–2021, the dLRaG indicator in 2019, and excess mortality per annual baseline.

Country | Rank EM100k 2020–2021 | Rank dLRaG 2019 | Rank EMAB 2020–2021 |
---|---|---|---|

AUS2 | 1 | 1 | 1 |

AUT | 2 | 2 | 3 |

BEL | 3 | 3 | 3 |

BGR | 4 | 4 | 4 |

CAN | 2 | 4 | 2 |

CHE | 2 | 2 | 3 |

CHL | 3 | 4 | 4 |

CZE | 4 | 1 | 4 |

DEUTNP | 2 | 3 | 1 |

DNK | 1 | 1 | 1 |

ESP | 3 | 2 | 4 |

EST | 3 | 3 | 2 |

FIN | 1 | 1 | 1 |

FRATNP | 2 | 4 | 2 |

GBR | 3 | 3 | 3 |

GBR_NL | 1 | 2 | 2 |

GRC | 2 | 1 | 2 |

HUN | 4 | 2 | 3 |

ISL | 1 | 1 | 1 |

ISR | 2 | 3 | 2 |

ITA | 4 | 1 | 3 |

LTU | 4 | 2 | 4 |

LUX | 1 | 3 | 1 |

LVA | 3 | 4 | 2 |

NOR | 1 | 3 | 1 |

NZL_NP | 1 | 2 | 1 |

POL | 4 | 1 | 4 |

PRT | 3 | 4 | 3 |

RUS | 4 | 3 | 4 |

SVK | 4 | 2 | 4 |

SVN | 3 | 4 | 3 |

SWE | 2 | 4 | 2 |

**Figure A1.**Country ranking by excess mortality per 100k people (i.e., EM100k) in 2020–2021 and LRaG gap in 2019 (upper panel); by health expenditure in 2019 and LRaG gap in 2017 (middle panel); by health expenditure in 2019 and LRaG gap in 2020 (lower panel). Countries on the horizontal axis are sorted alphabetically.

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**Figure 1.**dLRaG for three age classes by country. Each panel shows the dLRaG across the countries for a given year and age class.

**Figure 2.**dLRaG for age class 15–64 per country and years 2017 and 2020. Each panel shows the dLRaG across the countries.

**Figure 4.**Country ranking by health expenditure in 2019 (i.e., HE 2019) and by LRaG gap in 2017. Countries on the horizontal axis are sorted alphabetically.

**Figure 5.**Country ranking by LRaG gap in 2017 and 2020 and the variation of health expenditure in 2018–2019 as a percentage. Countries on the horizontal axis are sorted alphabetically.

**Figure 6.**Country ranking by excess mortality per 100k people (i.e., EM100k) in 2020–2021 and by LRaG gap in 2019.

**Table 1.**Gompertzian regime limit age, ${x}^{\ast}$, non-country-specific part, ${\lambda}^{\ast}$, of the total hazard rate and p-value of the slope in the linear regression (11) for different years.

Year | ${\mathit{x}}^{\ast}$ | ${\mathit{\lambda}}^{\ast}$ | ${\mathit{p}}_{\mathbf{value}}$ |
---|---|---|---|

2016 | 91 | 0.32 | $4.01\times {10}^{-13}$ |

2017 | 88 | 0.27 | $7.42\times {10}^{-11}$ |

2018 | 84 | 0.20 | $8.93\times {10}^{-11}$ |

2019 | 87 | 0.25 | $1.95\times {10}^{-11}$ |

2020 | 81 | 0.16 | $7.86\times {10}^{-9}$ |

2021 (42 weeks) | 72 | 0.08 | $5.05\times {10}^{-8}$ |

**Table 2.**Normality test for residuals of the multivariate linear regressions (20).

Test | Statistic | p-Value |
---|---|---|

Shapiro–Wilk | 0.9835 | 0.8921 |

Kolmogorov–Smirnov | 0.0822 | 0.9696 |

Anderson–Darling | 0.2509 | 0.7205 |

**Table 3.**Multivariate linear regression (20). Signif. codes: 0 “***” 0.001 “**”, 0.01 “*”, 0.05 “.”, 0.1 “ ”, 1.

Parameter | Estimate | Std. Error | t Value | Pr(>$\left|\mathit{t}\right|$) |
---|---|---|---|---|

(intercept) | −0.2061 | 0.1095 | −1.88 | 0.0706 |

$dLRa{G}_{2017}$ | 0.3638 | 0.1860 | 1.96 | 0.0609 $(\xb7)$ |

$dLRa{G}_{2018}$ | 0.1232 | 0.2534 | 0.49 | 0.6307 |

$dLRa{G}_{2019}$ | 0.1426 | 0.1186 | 1.20 | 0.2394 |

$H{E}_{2019}$ | 0.0001 | 0.0000 | 2.03 | 0.0526 $(\xb7)$ |

**Table 4.**Multivariate linear regression (21). Signif. codes: 0 “***”, 0.001 “**”, 0.01 “*”, 0.05 “.”, 0.1 “ ”, 1.

Parameter | Estimate | Std. Error | t Value | Pr(>$\left|\mathit{t}\right|$) |
---|---|---|---|---|

(intercept) | −0.2113 | 0.1116 | −1.89 | 0.0682 |

$dLRa{G}_{2017}$ | 0.6002 | 0.0516 | 11.64 | 0.0000 (***) |

$H{E}_{2019}$ | 0.0001 | 0.0000 | 2.00 | 0.0544 $(\xb7)$ |

**Table 5.**Normality test for residuals of the multivariate linear regression (20).

Test | Statistic | p-Value |
---|---|---|

Shapiro–Wilk | 0.9649 | 0.3711 |

Kolmogorov–Smirnov | 0.1046 | 0.8393 |

Anderson–Darling | 0.4189 | 0.3086 |

Linear Regression (22) | ||||
---|---|---|---|---|

Parameter | Estimate | Std. Error | t Value | Pr(>$\left|\mathbf{t}\right|$) |

(intercept) | 138.8954 | 18.7357 | 7.41 | 0.0000 |

$dLRa{G}_{2017}$ | 107.5452 | 91.3934 | 1.18 | 0.2496 |

$dLRa{G}_{2018}$ | 45.9393 | 122.1673 | 0.38 | 0.7098 |

$dLRa{G}_{2019}$ | 77.6427 | 55.0293 | 1.41 | 0.1697 |

$dLRa{G}_{2020}$ | −309.4199 | 84.0705 | −3.68 | 0.0010 (**) |

Linear Regression (23) | ||||

Parameter | Estimate | Std. Error | t Value | Pr(>$\left|\mathbf{t}\right|$) |

(intercept) | 12.8720 | 1.4451 | 8.91 | 0.0000 |

$dLRa{G}_{2017}$ | 8.1244 | 7.0490 | 1.15 | 0.2592 |

$dLRa{G}_{2018}$ | 6.0949 | 9.4225 | 0.65 | 0.5232 |

$dLRa{G}_{2019}$ | 7.4316 | 4.2443 | 1.75 | 0.0913 $(\xb7)$ |

$dLRa{G}_{2020}$ | −27.4200 | 6.4842 | −4.23 | 0.0002 (***) |

Linear Regression (22) | ||
---|---|---|

Test | Statistic | p-Value |

Shapiro–Wilk | 0.9814 | 0.8391 |

Kolmogorov–Smirnov | 0.0630 | 0.9987 |

Anderson–Darling | 0.1601 | 0.9427 |

Linear Regression (23) | ||

Test | Statistic | p-Value |

Shapiro–Wilk | 0.9815 | 0.8429 |

Kolmogorov–Smirnov | 0.0925 | 0.9236 |

Anderson–Darling | 0.2581 | 0.6955 |

Linear Regression (24) | ||||
---|---|---|---|---|

Parameter | Estimate | Std. Error | t Value | Pr(>$\left|\mathbf{t}\right|$) |

(intercept) | 141.11 | 21.77 | 6.483 | 3.65 × 10${}^{-7}$ *** |

$dLRa{G}_{2019}$ | 44.92 | 24.60 | 1.826 | 0.0778 (·) |

Linear Regression (25) | ||||

Parameter | Estimate | Std. Error | t Value | Pr(>$\left|\mathbf{t}\right|$) |

(intercept) | 13.077 | 1.766 | 7.406 | 2.98 × 10${}^{-8}$ |

$dLRa{G}_{2019}$ | 4.857 | 1.996 | 2.434 | 0.0211 (*) |

**Table 9.**Bivariate absolute frequency distribution of the quartile ranking of the LRaG gap in 2019 and the excess of mortality per 100 k in 2020–21 (top panel) and excess mortality as a percentage of the annual baseline in 2020–2021 (bottom panel).

dLRaG_r1 | dLRaG_r2 | dLRaG_r3 | dLRaG_r4 | Row Tot. | |
---|---|---|---|---|---|

EMk100_r1 | 4 | 2 | 2 | 0 | 8 |

EMk100_r2 | 1 | 2 | 2 | 3 | 8 |

EMk100_r3 | 0 | 1 | 3 | 4 | 8 |

EMk100_r4 | 3 | 3 | 1 | 1 | 8 |

Col. Tot. | 8 | 8 | 8 | 8 | 32 |

dLRaG_r1 | dLRaG_r2 | dLRaG_r3 | dLRaG_r4 | Row Tot. | |

EMAB_r1 | 4 | 1 | 3 | 0 | 8 |

EMAB_r2 | 1 | 1 | 2 | 4 | 8 |

EMAB_r3 | 1 | 3 | 2 | 2 | 8 |

EMAB_r4 | 2 | 3 | 1 | 2 | 8 |

Col. Tot. | 8 | 8 | 8 | 8 | 32 |

**Table 10.**Distribution of the absolute difference of the quartile ranking of the LRaG gap in 2019 and excess mortality per 100k people in 2020–2021 (top panel) and excess mortality as a percentage of the annual baseline in 2020–2021 (bottom panel).

$|\mathit{R}\mathit{a}\mathit{n}\mathit{k}\phantom{\rule{0.166667em}{0ex}}\mathit{E}\mathit{M}\mathit{k}{100}_{2020-21}-\mathit{R}\mathit{a}\mathit{n}\mathit{k}\phantom{\rule{0.166667em}{0ex}}\mathit{d}\mathit{L}\mathit{R}\mathit{a}{\mathit{G}}_{2019}|$ | 0 | 1 | 2 | 3 |

n. units | 10 | 11 | 8 | 3 |

$|Rank\phantom{\rule{0.166667em}{0ex}}EMA{B}_{2020-21}-Rank\phantom{\rule{0.166667em}{0ex}}dLRa{G}_{2019}|$ | 0 | 1 | 2 | 3 |

n. units | 9 | 10 | 11 | 2 |

**Table 11.**Distribution of the absolute difference of the quartile ranking of health expenditure in 2019 and the LRagG gap in 2017 (top panel) and in 2020 (bottom panel).

$|\mathit{R}\mathit{a}\mathit{n}\mathit{k}\phantom{\rule{0.166667em}{0ex}}\mathit{H}{\mathit{E}}_{2019}-\mathit{R}\mathit{a}\mathit{n}\mathit{k}\phantom{\rule{0.166667em}{0ex}}\mathit{d}\mathit{L}\mathit{R}\mathit{a}{\mathit{G}}_{2017}|$ | 0 | 1 | 2 | 3 |

n. units | 10 | 11 | 8 | 3 |

$|Rank\phantom{\rule{0.166667em}{0ex}}H{E}_{2019}-Rank\phantom{\rule{0.166667em}{0ex}}dLRa{G}_{2020}|$ | 0 | 1 | 2 | 3 |

n. units | 7 | 12 | 7 | 6 |

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**MDPI and ACS Style**

Polinesi, G.; Recchioni, M.C.; Rimondi, A.; Sysoev, A.
Short-Term Mortality Fluctuations and Longevity Risk-Adjusted Age: Learning the Resilience of a Country to a Health Shock. *Computation* **2022**, *10*, 47.
https://doi.org/10.3390/computation10040047

**AMA Style**

Polinesi G, Recchioni MC, Rimondi A, Sysoev A.
Short-Term Mortality Fluctuations and Longevity Risk-Adjusted Age: Learning the Resilience of a Country to a Health Shock. *Computation*. 2022; 10(4):47.
https://doi.org/10.3390/computation10040047

**Chicago/Turabian Style**

Polinesi, Gloria, Maria Cristina Recchioni, Andrea Rimondi, and Anton Sysoev.
2022. "Short-Term Mortality Fluctuations and Longevity Risk-Adjusted Age: Learning the Resilience of a Country to a Health Shock" *Computation* 10, no. 4: 47.
https://doi.org/10.3390/computation10040047