#
Implications of the SARS-Cov-2 Pandemic for Mortality Forecasting: Case Study for the Czech Republic and Spain^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methodology

#### 2.1. Data

_{x}

_{,t}). We took the population state on 1 January in year t and t + 1, summed it, and divided it by 2 (i.e., simple arithmetic mean). The central mortality rate in time t (m

_{x}

_{,t}) was then calculated as (1):

_{x}

_{,t}is the number of deaths in time t. Data about mortality rates for 2019 in the Czech Republic comes from the Czech Statistical Office (Prague, Czech Republic) database, and for Spain for 2019 are taken from Instituto Nacional de Estadística (Madrid, Spain). Data for 2020 are available in The Human Mortality Database, but they are not age-specific (there are only age categories 0–14, 15–64, 65–74, 75–84, 85+), so the projection cannot include this year directly.

#### 2.2. Methods

**k**$\left(\frac{d\mathrm{log}{m}_{x,t}}{dt}=\frac{{b}_{x}dk}{dt}\right)$” (Lee and Carter [14]). The coefficient can be negative for some ages, which indicates that mortality at those ages tends to rise when mortality in other ages is falling. Error term

**e**

_{x,t}≈ N(0,σ

^{2}) captures age-specific historical influences not captured by the model and is supposed to be homoscedastic. “As the model written in this way is over parametrized, the two additional constraints are introduced in order to identify the model.” (Danesi, Haberman, Millossovich [16]): $\sum}_{x=1}^{N}{b}_{x}=1$ and $\sum}_{t=1}^{T}{k}_{t}=0$. Using these constrains, the least squares estimator for ${a}_{x}$ can be obtained by (4):

_{t}), while

**a**

_{x}and

**b**

_{x}stayed unchanged. This adjustment gives greater weight to the ages at which the numbers of deaths are large. Booth, Maindonald, and Smith [15] modified the method to adjust the time component to reproduce the age distribution of deaths, rather than total deaths, and to determine the optimal fitting period to address non-linearity in the time component.

**R**, and other calculations were done in MS Excel.

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Empirical values of age-specific mortality rates of males (

**left**), females (

**middle**), and total (

**right**)—Czech Republic; x—years, y—ages, z—rates; Source: Human mortality database, own elaboration.

**Figure 2.**Empirical values of age-specific mortality rates of males (

**left**), females (

**middle**), and total (

**right**)—Spain; x—years, y—ages, z—rates; Source: Human mortality database, own elaboration.

**Figure 3.**Empirical values of the number of deaths and five-year moving average—Czech Republic; Source: Human mortality database, own elaboration.

**Figure 4.**Empirical values of the number of deaths and five-year moving average—Spain; Source: Human mortality database, own elaboration.

**Figure 5.**Results of forecasts A and B for the Czech Republic and Spain—example of mortality in higher ages. Source: own elaboration.

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**MDPI and ACS Style**

Šimpach, O.; Pechrová, M.Š. Implications of the SARS-Cov-2 Pandemic for Mortality Forecasting: Case Study for the Czech Republic and Spain. *Eng. Proc.* **2021**, *5*, 58.
https://doi.org/10.3390/engproc2021005058

**AMA Style**

Šimpach O, Pechrová MŠ. Implications of the SARS-Cov-2 Pandemic for Mortality Forecasting: Case Study for the Czech Republic and Spain. *Engineering Proceedings*. 2021; 5(1):58.
https://doi.org/10.3390/engproc2021005058

**Chicago/Turabian Style**

Šimpach, Ondřej, and Marie Šimpachová Pechrová. 2021. "Implications of the SARS-Cov-2 Pandemic for Mortality Forecasting: Case Study for the Czech Republic and Spain" *Engineering Proceedings* 5, no. 1: 58.
https://doi.org/10.3390/engproc2021005058