Since humans aged above ca. 90 years have become numerous enough to matter at all and to make the analysis of mortality kinetics in such populations reasonably reliable, the apparent deceleration of the age-associated increase in death rate among the oldest old has been being attracting relentless attention, e.g. (Vaupel, Carey et al. 1998, Weitz and Fraser 2001, Bebbington, Green et al. 2012, Lai 2012, Barbi, Lagona et al. 2018, Newman 2018, Gavrilov and Gavrilova 2019, Dang, Camarda et al. 2023). The phenomenon, which is often associated with the so-called late-life mortality plateaus, may be explained assuming that either (i) the age-associated increases in the individual frailty decelerate at older ages and (ii) progressive age-associated changes in the composition of a population are such that those who are frailer initially or/and become frailer because of aging more rapidly die out earlier, which results in an increase in the proportion of those who die out at a slower rate (Vaupel, Carey et al. 1998, Avraam, Arnold et al. 2016, Németh and Missov 2018, Böhnstedt and Gampe 2019, Salinari and De Santis 2019). Logically, these two explanations, physiological and compositional (demographic), are not mutually exclusive. However, the latter one predominates currently. One reason for this may be that the age-associated increase in the mortality rate of adults is commonly believed to conform to the Gompertz law (GL) (Gompertz 1825, Gavrilov and Gavrilova 1991, Olshansky and Carnes 1997, Golubev 2009 and 2019, Bronikowski and Flatt 2010, Wrigley-Field 2014, Kirkwood 2015):

$${\mu }\left(t\right)={\mu }_{0}{\text{e}}^{\gamma \times t}$$

which may be written as

$$\text{ln}\left[\mu \left(t\right)\right]={\text{l}\text{n}\mu }_{0}+\gamma \times t$$

where *γ* is interpreted as the rate of aging and the associated increase in frailty, i.e. in the inability to withstand the causes of death. In a ln*µ* -vs-age plot, *γ* is the tangent of the slope of a straight line, which suggests that the age-associated increase in frailty occurs at a constant velocity. In studies where different species or genetic modifications of a species are compared, greater slopes of the linear approximations of ln*µ*(*t*)-vs-*t* datapoint series are interpreted as indicating higher rates of aging (Finch and Pike 1996, de Magalhaes, Cabral et al. 2005).

Because the linearity of the increase in frailty with increasing age is inbuilt in GL, there remain two options available to explain systematic deviations of empirical data on mortality, survivorship, and lifespan distribution from the patterns consistent with GL. One of the options is to introduce additional terms into GL. The simplest embodiment of such option implies that the single additive term does not depend on age. The result is known as the Gompertz-Makeham law (GML) (Makeham 1860, Gavrilov and Gavrilova 1991, Golubev 2004, 2019).

$${\mu }\left(t\right)=C+{\mu }_{0}{\text{e}}^{\gamma \times t}$$

The term *C* in GML may be interpreted as capturing how much the causes of death that produce the age-independent (accidental or extrinsic) mortality contribute to the total death rate.

Importantly, *C* makes the ln*µ*-vs-age plots deviate from linearity so that the apparent rate of aging increases with increasing age, as shown in Fig. 1, which presents the results of numerical modeling, using Mathcad 13, at the values of the GMM parameters g = 0.1 and µ0 = 0.00001 and variable *C* within ranges typical for humans.

Figure 1. Dependencies of mortality rate (upper right panel) and aging rate (lower left panel) on age at different levels of the Makeham parameter *C*, which is assumed to be either constant or decreasing linearly with increasing age. The dashed lines in the upper panel show the linear approximations of the plots constructed at C > 0.

It is easy to see in Fig. 1 that approximating the plots related to C > 0 with straight lines, i.e., assuming that the differences of *C* from zero are negligible, will make the slopes of the resulting lines decrease with increasing *C* making the impression that aging rates are different in the three cases where actually only *C* are different, that is the differences relate actually not to the properties of the organisms in the modeled populations but to the conditions they are in. Moreover, even a very small *C*, which produces seemingly negligible effect on the respective ln*µ*-vs-age plot, can distort the apparent dependence of aging rate on age quite markedly. When there is no *C* at work, that is *C* = 0, the apparent aging rate does not change with increasing age. The effect of *C* makes the apparent rate of aging significantly underestimated at lower ages, its estimates increasing only by middle ages up to the values deduced according to GMM.

Ignoring *C* in analyzing mortality or survival data is a source of much confusion (Golubev 2004) because it may lead to the erratic conclusion that the mean (median) lifespan has changed because of a change in the rate of aging whereas actually it has changed mainly because of changes in the age-independent (extrinsic, accidental) mortality, which is captured by *C*. Moreover, changes in *C* are associated with negative correlations between the estimates of *g* and *µ*0 made assuming that *C* is zero. Thus, *C* is an important factor of the so-called Strehler-Mildvan correlation (Gavrilov and Gavrilova 1991). However, such correlations will result from whatever deviations of data from conformance to a model, including purely accidental or stochastic deviations, when data are treated using not only GM or GML (Tarkhov, Menshikov et al. 2017) but any nonlinear two- or three-parametric model to which the data do not conform exactly (Golubev, Panchenko et al. 2018).

These caveats taken into account, it may be shown that *C* is what mainly decreased over the last century in developed countries, whereas aging rate *g* and the initial mortality *µ*0 remained relatively constant, and their negatively correlated deviations from constancy are attributable to data-treatment artifacts (Golubev 2012, Golubev 2019). For example, the difference between the plots showing ln*µ* vs age dependencies in French women in periods 1920 and 2020 (Fig. 1) is caused by a higher contribution of *C* to mortality in 1920. The contribution is more significant at younger ages because the age-dependent mortality becomes increasingly dominant at later ages. Importantly, in the plot related to the 1920 cohort, the downward slope at ages from 20 to 40 years may be attributable to the historical decrease in *C*.

The other option to deal with the deviations of empirical data from GL and GML, upon the assumption that *g* is constant, is to assume that the values of the parameter *µ*0 or *γ*, or both are not the same for all subjects in a population and feature(s) some form of distribution. Clearly, any real population or cohort is heterogeneous, and its heterogeneity likely involves the properties that influence frailty and its age-associated changes. It is less clear how the resulting heterogeneity must be reflected in the parameters of mortality and survival as they depend on age. The most popular assumption is that *µ*0 is gamma-distributed (Vaupel, Carey et al. 1998, Missov and Németh 2016, Németh and Missov 2018, Böhnstedt and Gampe 2019, Salinari and De Santis 2019). The resulting relationships are known as gamma-Gompertz and gamma-Gompertz-Makeham models. Both produce age-associated decreases in the slopes of ln*µ*-vs-age plots. The ages beyond which the decreases become noticeable depend on the numerical values of the parameters of a gamma-distribution.

### Preliminaries: a generalized Gompertz-Makeham law

Strictly linear changes, including physiological decline during aging as reflected in the dependence of ln*µ* on age, are the least biologically plausible and thus are always only more or less acceptable, in practical terms, approximations to any real state of things. Based on this and more specific considerations derived from the possible constrains imposed by physics and chemistry on the primordial living objects upon their emergence from the physicochemical world, a generalized Gompertz-Makeham law (GGML) has been suggested (Golubev 2009, 2019):

$$\mu \left(t\right)=C\left(t\right)+{{\mu }}_{\text{o}}{\text{e}}^{\text{f}\left(t\right)}$$

From this standpoint, the canonical GL and GML are not laws but models, which may be derived from GGML upon the assumptions that f(*t*) = *γt* and *C* = 0 (GM) or f(*t*) = *γt* and *C* = const > 0 (GMM).

One implication of the GGML is that *C* is not just a constant added to GM to improve its fit to data, but is an essential term that captures indispensable aspects of the reality and thus has a substantial meaning and can never be actually equal to zero. Yet, a smaller *C* compared with the Gompertzian term of GMM will make the relationship between ln*µ* and f(*t*) closer to linearity (see Fig. 1) thus allowing more accurate inferences about the real age-dependent trajectory of functional decline, provided the heterogeneity of a population under examination may be ignored. Importantly, the relative contribution of *C* to the total mortality and, thus, any possible *C*-caused bias decreases with increasing age because of the increasing age-associated mortality. However, analyzing later ages is associated with greater noise because of smaller sample sizes. It is only in relatively recent times that the late-age human populations became sufficiently large and least affected by the age-independent mortality captured by *C*.

In the present attempt to derive, based on the GGML, the late-age patterns of the averaged individual physiological decline from the patterns of the late-age demographic (actuarial) mortality, Human Mortality Database (HMD) data related to France, Japan, and Sweden are used. This choice is based on that these populations are large and relatively homogenous, the estimates of *C* there are among the smallest in the world and, at the same time, the populations there markedly differ one from another in a number of other aspects. The period data relate to 2020, the last year before the COVID-19 pandemic, and to 1920, the starting year of the cohorts that reached the age of 100 years in 2020. The cohort data are used assuming that the counts of emigrants and immigrants and of the natives who are still alive are too small to care of licensing.