Article

ISSN 0006-2979, Biochemistry (Moscow), 2023, Vol. 88, No. 11, pp. 1778-1785 © Pleiades Publishing, Ltd., 2023.

Published in Russian in Biokhimiya, 2023, Vol. 88, No. 11, pp. 2156-2165.

1778

Actuarial Aging Rates in Human Cohorts

L e o n i d A . G a v r i l o v

1,2,a

* and Natalia S. Gavrilova

1,2

NORC at the University of Chicago, 60637 Chicago, IL, USA

Institute for Demographic Research, Federal Center of Theoretical and Applied Sociology, Russian Academy of Sciences,

109028 Moscow, Russia

e-mail: lagavril@yahoo.com

Received August 1, 2023

Revised October 8, 2023

Accepted October 9, 2023

Abstract— Aging rate is an important characteristic of human aging. Attempts to measure aging rates through the Gompertz

slope parameter lead to a conclusion that actuarial aging rates were stable during the most of the 20th century, but recently

demonstrate an increase over time in the majority of studied populations. These findings were made using cross-sectional

mortality data rather than by the analysis of mortality of real birth cohorts. In this study we analyzed historical changes of

actuarial aging rates in human cohorts. The Gompertz parameters were estimated in the age interval 50-80 years using data

on one-year cohort age-specific death rates from the Human Mortality Database (HMD). Totally, data for 2,294 cohorts

of men and women from 76 populations were analyzed. Changes of the Gompertz slope parameter in the studied cohorts

revealed two distinct patterns for actuarial aging rate. In higher mortality Eastern European countries actuarial aging rates

showed continuous decline from the 1910 to 1940 birth cohort. In lower mortality Western European countries, Australia,

Canada, Japan, New Zealand, and USA actuarial aging rates declined from the 1910th to approximately 1930th cohort

and then increased. Overall, in 50 out of 76 populations (68%) actuarial aging rate demonstrated decreasing pattern of

change over time. Compensation effect of mortality(CEM) was tested for the first time in human cohorts and the cohort

species-specific lifespan was estimated. CEM was confirmed using cohort data and human cohort species-specific lifespan

estimates were similar to the estimates obtained for the cross-sectional data published earlier.

DOI: 10.1134/S0006297923110093

Keywords: aging, mortality, compensation effect of mortality, species-specific lifespan, aging rate

Abbreviations: CEM,compensation effect of mortality.

* To whom correspondence should be addressed.

INTRODUCTION

Studies of age-specific mortality patterns and ac-

tuarial aging rate in particular are important for under-

standing the fundamental biology of aging and to devel-

op genuine anti-aging interventions [1]. The aging rate

is often estimated as a slope coefficient of the Gompertz

law describing exponential increase of mortality rate

with age (also known as the Gompertz slope or actuar-

ial aging rate) [2, 3]. This approach looks reasonable,

because hypothetical non-aging populations have slope

coefficient equal to zero, and because the slope param-

eter characterizes the rate of mortality increase with age.

Earlier studies of actuarial aging rate in humans

showed increasing trend during the last 30 years in most

countries [2, 3]. These results of increasing aging rates

in contemporary populations look counterintuitive given

continuous decline of mortality among adults. Math-

ematical reliability theory of aging indicates that the

slope coefficient is determined not only by the rate of

functional loss with age (“true aging rate”), but also by

the initial redundancy levels (initial reserve capacity)

[1, 4]. Thus, actuarial aging rate is determined not only

by intrinsic deterioration of organs and tissues, but also

by other factors including environmental effects.

Actuarial aging rates are different across different

populations and are organized in such a way that low-

er initial mortality is compensated for by its more rapid

growth with age. This means that high mortality rates in

disadvantaged populations (within a given species) are

compensated for by a low actuarial aging rate (longer

mortality doubling period). As a result of this compen-

sation, mortality rates tend to converge at older ages [1].

This phenomenon is called compensation effect of mor-

tality (CEM) [2].

ACTUARIAL AGING RATES IN HUMAN COHORTS 1779

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

We have previously examined the historical chang-

es of the actuarial aging rates and the compensation ef-

fect of mortality using cross-sectional mortality data [2].

However, this approach was subjected to a constructive

criticism that the CEM should be studied with cohort

data [5]. This paper takes this criticism into account and

investigates historical changes in the actuarial aging rate

and the compensation effect of mortality using cohort

data.

MATERIALS AND METHODS

One of the goals of this study was to analyze histor-

ical changes of actuarial aging rates and test the com-

pensation effect of mortality in human cohorts. These

analyses were not conducted before with cohort data

and all studies of actuarial aging rates and CEM so far

were made using cross-sectional data [1-3, 6].

Methods. In the first step of the analyses, we have

calculated parameters R

and α of the Gompertz equa-

tion(1):

μ(x) = R

exp(αx), (1)

where μ(x) is a cohort mortality (age-specific cohort

death rate) at age x and R

and α are the Gompertz

parameters. For historical cross-sectional data, actuar-

ial aging rate is usually estimated using the Gompertz–

Makeham equation with additional age-independent

Makeham term. In the case of cohort data (unlike

cross-sectional data) the Makeham term (or background

mortality) is not stable over age, because it is changing

with calendar time. It was also found that in contempo-

rary populations background mortality is very low [1, 7]

and hence has no noticeable effect on the estimates of

the Gompertz parameters. In this case we use the fact

that after the 1960s background mortality decreased to

very low levels close to zero [2,8]. Thus, the Makeham

parameter can be ignored for parameter estimates at ages

over 50 years and 1910 and later birth cohorts. Bongaarts

estimated background and senescent (age-dependent)

mortality by age using cause-of-death data and found

that after age 50-60 years mortality is determined almost

entirely by the senescent mortality [7].

Parameters of the Gompertz model were estimated

using method of non-linear regression in the age inter-

val 50-80 years (nlin procedure in Stata package, ver-

sion 14) as it was suggested before for cross-sectional

data [6]. Age interval 50-80 years is characterized by in-

creasing life table aging rate for females (mortality accel-

eration), whereas for male cohorts this regularity is not

so clear [9, 10]. It is believed that this age interval shows

pattern of age-specific mortality change for senescent

mortality [10]. Some researchers use logistic model to

study historical changes in period mortality in order to

capture mortality deceleration after age 85 years [6, 7].

In our study we analyze mortality below age 85years,

so that applying the Gompertz model is justified.

There is concern that the least squares fit often

leads to an ill-defined non-linear optimization problem,

which is extremely sensitive to sampling errors and the

smallest systematic demographic variations [11]. This

problem was discussed in our earlier publication [2].

Historical trends of actuarial aging rates for each

population of men and women (from 1910 to 1940 co-

hort) were estimated using linear regression model and

analyzing the sign and statistical significance of the re-

gression slope parameter.

CEM can be quantified using the inverse relation-

ship of the Gompertz parameters(2):

lnR

= lnM– Bα. (2)

Species-specific lifespan (parameter B in equation 2)

was obtained by running linear regressions between the

Gompertz parameters (lnR

and α) of the form present-

ed in equation 2. Thus, the species-specific lifespan

(slope parameter, B) and the intercept parameter(lnM)

have been estimated.

Data. Human Mortality Database (HMD) was used

as a source of mortality data for this study [12]. This da-

tabase contains mortality data for 42 countries with rea-

sonably good quality of demographic statistics. Totally

we used age-specific cohort death rates for 3304 cohorts

available in HMD covering data for 1900-1940 birth

cohorts. Study of historical changes in aging rates was

focused on more recent trends for 1910 to 1940 cohorts

and used 2294 cohorts. The age-specific cohort death

rates of men and women are available in the database

from ages 0 to 110 and older. However, cohort mortality

data are often not available for the whole age range, be-

cause many cohorts are not extinct. Data are available in

one-year age and time increments denoted asMx, where

x indicates single year of age.

Historical changes of actuarial aging rates in human

cohorts. Historical changes of actuarial aging rates in

demographic cohorts were not studied before. Cohort

mortality data are not as numerous as cross-sectional

data and require availability of long time series of mor-

tality for cohort mortality reconstruction. For this study

we selected single-year cohorts born from 1900 to 1940

in order to cover maximal number of countries.

Historical trends of cohort actuarial aging rate

were analyzed for time interval from 1910 to 1940 (birth

years of corresponding cohorts). For each country/sex

we run linear regression of the Gompertz slope param-

eter on year of birth for corresponding cohort in order

to estimate the general trend for actuarial aging rate.

Table1 presents slope coefficients of this linear regres-

sion together with corresponding p-values. Note that

in contrast to cross-sectional data, actuarial aging rates

GAVRILOV, GAVRILOVA178 0

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

Table 1. Historical changes of actuarial aging rates from 1910 to 1940 human birth cohorts*

Country/Region

Men Women

Slope coefficient

of linear regression ×10

p-value

Slope coefficient

of linear regression ×10

p-value

Australia 0.2194 0.001 0.1387 0.019

Austria –0.3734 <0.001 –0.1755 0.004

Belgium –0.0585 0.031 –0.0137 0.721

Bulgaria –1.1383 <0.001 –0.7642 <0.001

Belarus –1.3228 <0.001 –1.3422 <0.001

Canada –0.0321 0.415 –0.0740 0.076

Switzerland –0.1837 0.001 –0.0882 0.033

Czech Republic –0.0871 <0.001 –0.6300 <0.001

Germany –0.2687 <0.001 0.0352 0.347

Denmark –0.7425 <0.001 –0.8647 <0.001

Spain –0.3475 <0.001 0.0445 0.346

Estonia –1.4662 <0.001 –1.5405 <0.001

Finland –0.0140 0.750 –0.2237 0.008

France, total population –0.3126 <0.001 –0.1574 <0.001

England and Wales, total population –0.1505 0.032 –0.1443 0.030

Northern Ireland –0.000006 0.892 0.0960 0.283

United Kingdom –0.1622 0.020 –0.1272 0.053

Scotland –0.2686 <0.001 0.0398 0.541

Hungary –1.2918 <0.001 –0.7143 <0.001

Ireland –0.5714 <0.001 –0.2981 0.001

Iceland –0.1458 0.435 0.3480 0.195

Italy 0.0291 0.614 0.0944 0.039

Japan –0.2001 <0.001 0.0812 0.003

Lithuania –1.1191 <0.001 –1.0182 <0.001

Luxemburg –0.3288 0.001 –0.0594 0.668

Latvia –1.2239 <0.001 –1.2308 <0.001

Netherlands –0.4269 <0.001 –0.5814 <0.001

Norway –0.4545 <0.001 –0.5405 <0.001

New Zealand 0.0900 0.335 0.0643 0.401

Poland –1.3398 <0.001 –1.1047 <0.001

Portugal –0.5002 <0.001 –0.5316 <0.001

ACTUARIAL AGING RATES IN HUMAN COHORTS 1781

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

Table 1 (cont.)

Country/Region

Men Women

Slope coefficient

of linear regression ×10

p-value

Slope coefficient

of linear regression ×10

p-value

Russia –1.0891 <0.001 –1.3581 <0.001

Slovakia –1.0837 <0.001 –0.6272 <0.001

Slovenia 2.1571 <0.001 2.9576 <0.001

Sweden –0.2394 <0.001 –0.2537 <0.001

Taiwan –0.1583 <0.001 –0.4012 <0.001

Ukraine –1.0314 <0.001 –1.1605 <0.001

USA 0.1323 0.003 0.0343 0.400

Note. Slope coefficients of linear regression for dependencies of actuarial aging rate on birth cohort.

* Gompertz parameters were estimated in age interval 50-80 years. Linear regression of the Gompertz slope coefficient on time

(cohort) was run over 31 cohorts for each region/sex. Cases with statistically significant (p<0.05) changes of actuarial aging rate

are highlighted in bold.

forcohort data tend to decrease over time in the majority

of studied populations. Actuarial aging rates decreased in

22 cases for both sexes, in 7 cases for men only and one

case for women only. In 10 cases for men and 15 cases

for women actuarial aging rates showed no statistically

significant change. Thus, 68% of populations demon-

strated decreasing trend of actuarial aging rate.

While analyzing individual trends for each country

we found two distinct patterns of change for actuarial ag-

ing rate. In lower mortality Western European countries

(Austria, Germany, Denmark, Spain, Finland, France,

England and Wales, Northern Ireland, Italy, Scotland,

Great Britain, Ireland, Netherlands, Norway, Portugal,

Sweden), Australia, Canada, Japan, New Zealand, and

USA actuarial aging rates decreased until approximately

the 1930th cohort and then increased (Fig.1a).

In higher mortality Baltic countries and countries of

the Eastern Europe (Bulgaria, Belarus, Czech Republic,

Estonia, Hungary, Lithuania, Latvia, Poland, Russia,

Slovakia, Ukraine) actuarial aging rates always declined

(Fig.1b). Figure1 is only an illustration, but the same two

patterns are observed for other countries as well. There

are a few exceptions. In Belgium actuarial aging rates

remained flat for both men and women. In Switzerland

actuarial aging rate increased for later birth cohorts only

for women and remained flat for men. In Taiwan actuar-

ial aging rate increased for men and continued its decline

for women.

Figure 2 shows mortality for 1920th, 1930th, and

1940th cohorts of Swedish and Polish men. Note that

mortality trajectories diverge for all three cohorts in

Poland, which corresponds to declining actuarial aging

rates over subsequent birth cohorts. On the other hand,

only 1920th and 1930th cohorts show mortality diver-

gence in the case of Sweden corresponding to the pat-

tern presented in Fig. 1 for the lower mortality Western

European country.

Mortality data for human cohorts provided us an

opportunity to study time trends of cohort actuarial ag-

ing rate in different countries. Overall, cohort actuarial

aging rate demonstrates a declining trend. On the other

hand, lower mortality countries show a V-shaped pat-

tern of the actuarial aging rate changes.

These results are different from results obtained for

cross-sectional mortality data. Our earlier studies demon-

strated that the period actuarial aging rates are relatively

stable over time at least until the 1960s [1, 8]. Studies

by Bongaarts confirmed this initial finding [6, 7]. With

longer time series it became clear that actuarial aging

rates have more complex trajectories after the 1960s and

demonstrate an increasing pattern after the 1980s [2,3].

Compensation effect of mortality for human cohorts.

Compensation effect of mortality (CEM) refers to mor-

tality convergence, when higher values for the slope

parameter (in the Gompertz function) are compensat-

ed by lower values of the intercept parameterR

in dif-

ferent populations of a given biological species [1, 13].

CEM can be quantified using inverse relationship of the

Gompertz parameters of the Gompertz–Makeham equa-

tion presented in equation (2). Study of CEM was in-

spired by the pioneer publication of Strehler and Mildvan

who found an inverse correlation between the Gompertz

parameters [14]. However, these authors did not take

into account the Makeham parameter when using rather

old mortality data and hence obtained a spurious cor-

relation. The term “compensation effect of mortality”

GAVRILOV, GAVRILOVA1782

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

Fig. 1. Historical changes of actuarial aging rate in the lower mortality Western European (Italy)(a) and higher mortality Baltic (Lithuania)(b)

countries.

Fig. 2. Mortality (common log scale) as a function of age for three cohorts of Swedish and Polish men.

ACTUARIAL AGING RATES IN HUMAN COHORTS 1783

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

Table 2. Characteristics of compensation law of mortality for three birth cohorts based on the Gompertz model*.

Human Mortality Database

Population

Regression coefficients

Correlation coefficient

between lnR

and α

Number

of observations

lnM ±σ B ±σ, years

All single-year birth cohorts from 1920 to 1940

Men –2.39 ± 0.10 84.02 ± 1.33 –0.9085 850

Women –4.36 ± 0.11 66.13 ± 1.28 –0.8716 850

Both sexes –3.29 ± 0.09 75.35 ± 1.23 –0.9029 850

All single-year birth cohorts from 1930 to 1940

Men –1.57 ± 0.09 97.81 ± 1.29 –0.9624 460

Women –3.36 ± 0.15 79.18 ± 1.77 –0.9025 460

Both sexes –2.38 ± 0.10 89.59 ± 1.32 –0.9536 460

1940 birth cohort

Men –1.68 ± 0.16 97.33 ± 2.23 –0.9894 43

Women –2.87 ± 0.39 86.24 ± 4.53 –0.9479 43

Both sexes –2.45 ± 0.19 89.89 ± 2.53 –0.9842 43

* Gompertz parameters were estimated in age interval 50-80 years.

wasintroduced in 1978 when account for the Makeham

parameter resulted in totally different parameter estimates

of the Strehler–Mildvan correlation [15]. Compensation

effect of mortality is defined as a convergence of age-spe-

cific senescent (age-dependent) mortality dependencies

at advanced ages [1,15].

The coordinate corresponding to the age at which

all the mortality trajectories intersect(B) has been called

the species-specific life span [1]. It was found that for

humans its value is equal to 95 ± 2 years when using

cross-sectional data [1, 2]. It should be noted that the

compensation effect of mortality can be observed by a

simple visual inspection of mortality trajectories without

calculation of the Gompertz parameters [2]. Also, CEM

can be observed not only for humans, but for some oth-

er biological species [1,13].

Data on cohort mortality for different countries al-

lows us to test the existence of the compensation law of

mortality among cohorts. Compensation law of mortal-

ity for cohort data was never tested before. We estimat-

ed parameters of linear regression presented by equa-

tion(2) using cohort data.

This linear regression was run for more recent birth

cohorts of 1920-1940 using estimates of the Gompertz

parameters available for 76 populations of men and

women obtained in the age interval 50-80 years. Table2

presents the results of these estimations. These results

Fig. 3. Compensation effect of mortality for human cohorts. Mortality

(common log scale) of 1930 birth cohorts in six populations. Designa-

tions: M,male; F,female.

GAVRILOV, GAVRILOVA178 4

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

confirm the existence of compensation effect of mor-

tality for human cohorts, although values of species-

specific life span [parameter B of linear regression in

equation (2)] are somewhat lower compared to the cross-

sectional data [1,2]. These results also show lower val-

ues of the species-specific life span for historically older

birth cohorts, which is in agreement with findings ob-

tained for cross-sectional data [2].

Overall, we can conclude that estimates of the spe-

cies-specific lifespan based on cohort data (for later

birth cohorts) demonstrate a good agreement with ear-

lier publications based on cross-sectional data [1, 2].

These results mean that quantitative measures of CEM

for humans are rather stable.

Figure 3 demonstrates mortality convergence at old-

er ages (compensation effect of mortality) for 1930 birth

cohorts of several populations. Note that men of Poland

have lower actuarial aging rates compared to men of Swe-

den and women have higher actuarial aging rates com-

pared to men. Also, note that age trajectories for women

show slightly accelerated trend confirming earlier studies

of life table aging rate [10].

Figure 3 is another illustration of CEM existence,

which does not require statistical estimation of the

Gompertz parameters.

DISCUSSION AND CONCLUSIONS

It was found that the compensation effect of mor-

tality does exist for cohort data. Study of the quanti-

tative measures of CEM using cohort data confirmed

that the inverse correlation between the Gompertz in-

tercept parameter, ln(R

), and the Gompertz slope pa-

rameter(α) of the Gompertz equation does exist and is

highly statistically significant when we compare differ-

ent human populations. Estimates of the species-spe-

cific lifespan (parameter B) for later birth cohorts are

close to estimates obtained for cross-sectional data [1,2]

although still are somewhat lower. It was also found

that the estimates of the species-specific lifespan are

lower for historically older birth cohorts. Thus, com-

pensation effect of mortality is confirmed for human

birth cohorts.

In this study we also analyzed historical changes

of the actuarial aging rates (Gompertz slope parame-

ter) for human cohorts. In our study we analyzed time

trends in cohort actuarial aging rates for each popula-

tion separately from 1910 to 1940 birth cohort. It turns

out that the majority of populations (52 out of 76

or 68%) demonstrate declining pattern of actuarial ag-

ing rate. Actuarial aging rates remain stable for 23% of

human cohorts. This result obtained for the cohort ac-

tuarial aging rates is different from the result obtained

for cross-sectional data. It was found that period actu-

arial aging rates demonstrate an increasing pattern after

the 1980s for the majority of studied populations [2, 3].

Before the 1980s period actuarial aging rates showed

stability over time [1, 2, 8]. On the other hand, actuarial

aging rates measured for birth cohorts of the same coun-

try and sex has a tendency to decrease over time.

When studying cohort mortality, it is important to

realize that over the life course, a cohort is affected both

by aging, which increases mortality, and by improvement

in living conditions, which reduces mortality. It is theoret-

ically possible that the actuarial aging rate may eventually

decrease to very low levels close to zero, which can be

considered as an apparent aging arrest. For example, if

the increase in mortality with age is 8% per year, but mor-

tality decreases over time at a rate of 2% per year, then

the observed increase in mortality in the cohort would

end up being only 6% per year (8–2 = 6) [16]. If the

rate of historical decline in mortality is age-dependent,

this might look on the cohort data like a decrease in the

actuarial aging rate in more recent cohorts of people.

In any case, a decrease in the actuarial aging rate looks

like a slowing down of aging. It is still unclear what can

explain the increase in the actuarial aging rate in the

lower mortality countries among more recent birth co-

horts. One possible explanation is that these countries

already exhausted most resources for mortality decline

such as decline in cardiovascular mortality and smoking

habit. Indeed, mortality in many of these countries sta-

bilized during the last 10-15 years suggesting that the

main force of decreasing actuarial aging rate almost dis-

appeared [17]. Thus, now the actuarial aging rate in the

lower mortality countries is determined mostly by the

aging process and depends on the senescent mortality.

Higher mortality countries like countries of the Eastern

Europe still have some remaining room for mortality

decline and hence their actuarial aging rate continues

to decline.

It appears that aging should be measured not

by the rate of mortality increase with age, but by the

rate of loss of functional elements (mainly special-

ized cells) in the body. Such an approach to measur-

ing the true aging rate was proposed by the reliability

theory of aging [1, 4, 18]. According to this theory, a

rough estimate of the true aging rate can be obtained

by measuring one of the CEM parameters [inverse of

the species-specific lifespan in equation (2)]. Anoth-

er theory of aging (metronomic theory of aging) has

been proposed by Alexey M. Olovnikov [19]. Although

he considered the aging process as a program, he em-

phasized that this program refers mainly to the repro-

duction aspects of this process. Indeed, the time of the

beginning and the end of reproduction in women have

certain signs of a program [20]. However, the total lifes-

pan can be better described by reliability models, which

assume gradual destruction of the organism and loss of

functional elements including telomeres and special-

ized cells.

ACTUARIAL AGING RATES IN HUMAN COHORTS 1785

BIOCHEMISTRY (Moscow) Vol. 88 No. 11 2023

Acknowledgments. We are most grateful to the Rus-

sian theorist, Alexey Matveyevich Olovnikov (1936-

2022), for many hours of friendly scientific discussions

with him on many broad topics of the biology of aging.

This article is a part of a special issue of the journal de-

voted to his memory. We knew Dr.Olovnikov in person

since 1978, and greatly benefited from scientific conver-

sations with him. One example of his active participa-

tion in the discussion of our scientific studies and pre-

sentations is provided here: https://www.youtube.com/

watch?v=3cI5M88I-NU [in Russian].

Contributions. Leonid A. Gavrilov designed the

study, analyzed, and interpreted results, and edited the

manuscript. Natalia S. Gavrilova conducted statistical

analyses, and prepared the manuscript.

Funding. This work was partially support-

ed by the National Institutes of Health (project no.

NIH R21AG054849).

Ethics declarations. The authors declare no conflict

of interest in financial or any other sphere. This article

does not contain any studies with human participants or

animals performed by any of the authors.

REFERENCES

1. Gavrilov, L. A., and Gavrilova, N. S. (1991) The Biology

of Life Span: A Quantitative Approach, Harwood Academic

Publisher, New York.

2. Gavrilov, L. A., and Gavrilova, N. S. (2022) Trends in

human species-specific lifespan and actuarial aging rate,

Biochemistry (Moscow), 87, 1998-2011, doi: 10.1134/

S0006297922120173.

3. Tai, T. H., and Noymer, A. (2018) Models for estimating

empirical Gompertz mortality: with an application to evo-

lution of the Gompertzian slope, Popul. Ecol., 60, 171-184,

doi: 10.1007/s10144-018-0609-6.

4. Gavrilov, L. A., and Gavrilova, N. S. (2006) Reliability

Theory of Aging and Longevity. in Handbook of the Biology

of Aging (Masoro, E. J., and Austad, S. N., eds) 6Edn.,

Academic Press, San Diego, pp. 3-42, doi: 10.1016/

B978-012088387-5/50004-2.

5. Mikhalsky, A. I. (2023) On the Paper by Leonid A. Gavri-

lov and Natalia S. Gavrilova entitled “Trends in Human

Species-Specific Lifespan and Actuarial Aging Rate” Pub-

lished in Biochemistry (Moscow), Vol. 87, Nos. 12-13,

pp. 1622-1633 (2022), Biochemistry (Moscow), 88,

162-163, doi:10.1134/S0006297923010145.

6. Bongaarts, J. (2005) Long-range trends in adult mortality:

Models and projection methods, Demography, 42, 23-49,

doi:10.1353/dem.2005.0003.

7. Bongaarts, J. (2009) Trends in senescent life expec-

tancy, Popul. Stud. (Camb), 63, 203-213, doi: 10.1080/

00324720903165456.

8. Gavrilov, L. A., Gavrilova, N. S., and Nosov, V. N. (1983)

Human life span stopped increasing: why? Gerontology,

29, 176-180, doi:10.1159/000213111.

9. Horiuchi, S. (1997) Postmenopausal acceleration of

age-related mortality increase, J. Gerontol. A Biol. Sci.

Med. Sci., 52, B78-B92, doi: 10.1093/gerona/52A.1.B78.

10. Li, T., Yang, Y. C., and Anderson, J. J. (2013) Mor-

tality increase in late-middle and early-old age: het-

erogeneity in death processes as a new explanation,

Demography, 50, 1563-1591, doi: 10.1007/s13524-013-

0222-4.

11. Tarkhov, A. E., Menshikov, L. I., and Fedichev, P. O.

(2017) Strehler-Mildvan correlation is a degenerate man-

ifold of Gompertz fit, J.Theor. Biol., 416, 180-189, doi:

10.1016/j.jtbi.2017.01.017.

12. Human Mortality Database. University of Califor-

nia, Berkeley (USA), and Max Planck Institute for De-

mographic Research (Germany), URL: https://www.

mortality.org or http://www.humanmortality.de (accessed

on 6.5.2022).

13. Golubev, A. (2019) A 2D analysis of correlations be-

tween the parameters of the Gompertz-Makeham model

(or law?) of relationships between aging, mortality, and

longevity, Biogerontology, 20, 799-821, doi: 10.1007/

s10522-019-09828-z.

14. Strehler, B. L., and Mildvan, A. S. (1960) General theory

of mortality and aging, Science, 132, 14-21, doi:10.1126/

science.132.3418.14.

15. Gavrilov, L. A., Gavrilova, N. S., and Yaguzhinsky, L.S.

(1978) The main patterns of aging and death of animals

from the point of view of the theory of reliability [in Rus-

sian], J.Gen. Biol., 39, 734-742.

16. Gavrilov, L. A., and Gavrilova, N. S. (2023) On the

Commentary by Anatoly I. Mikhalsky Published in

Biochemistry (Moscow), Vol. 88, No. 1, pp. 162-163

2023), Biochemistry (Moscow), 88, 289, doi: 10.1134/

S0006297923020116.

17. Ho, J. Y., and Hendi, A. S. (2018) Recent trends in life

expectancy across high income countries: retrospective

observational study, Br.Med.J., 362, k2562, doi:10.1136/

bmj.k2562.

18. Gavrilov, L. A., and Gavrilova, N. S. (2001) The reli-

ability theory of aging and longevity, J.Theor. Biol., 213,

527-545, doi:10.1006/jtbi.2001.2430.

19. Olovnikov, A. M. (2022) Planetary metronome as a regu-

lator of lifespan and aging rate: the metronomic hypoth-

esis, Biochemistry (Moscow), 87, 1640-1650, doi:10.1134/

S0006297922120197.

20. Gavrilova, N. S., Gavrilov, L. A., Severin, F. F., and

Skulachev, V. P. (2012) Testing predictions of the pro-

grammed and stochastic theories of aging: comparison of

variation in age at death, menopause, and sexual matu-

ration, Biochemistry (Moscow), 77, 754-760, doi:10.1134/

S0006297912070085.