Demographic Indicators of Probability Models

Describing mortality dynamics using average indicators without considering variability can yield average results, impeding the analysis of survival-curve patterns during periods of significant mortality spikes, especially at the oldest or youngest ages. Therefore, instead of the generally accepted Gompertz method, other methods are increasingly used, which rely on comparisons of various demographic indicators. In humans, chronic phenoptosis, in contrast to age-independent acute phenoptosis, manifests as a rectangularization of the survival curve with a simultaneous increase in the life expectancy at birth due to the advancement of social, scientific, and technological progress. Rectangularization is difficult to notice solely by examining the optimal coefficients in the Gompertz—Makeham equation, primarily because of the inevitable calculation errors. This can be avoided by calculating demographic indicators based on the spread of the life expectancy: Keyfitz entropy, Gini coefficient, and coefficient of variation of lifespan. We examine several sub-Gompertzian models of mortality growth with age, which describe the aging of nematodes and insects. Within the sub-Gompertzian model of aging, the increase in mortality with age in invertebrates is quantified as a rectangularization of the survival function estimated by these demographic indicators. On the other hand, the increasing rectangularization of the survival function with the development of scientific and technological progress, demonstrated by a decrease in the Keyfitz entropy, along with a simultaneous increase in the life expectancy in humans, also aligns well with the hypothesis of an age-dependent increase in mortality in mammals overall. Calculations on aging models demonstrate the effectiveness of using Keyfitz entropy and the Gini coefficient as important demographic indicators. The use of these indicators seems preferable, especially for nematodes, where the sub-Gompertzian model of aging is applicable, and for vertebrates, primarily mammals, with certain restrictions, the Gompertz–Makeham law is applicable. Approaches that consider dynamic age-related shifts in improved survival, such as studying imbalances in lifespan, enhance our understanding of the mechanisms of aging. This, in turn, will contribute to the development of more accurate methods for assessing the effects of biologically active substances used in gerontology, such as anti-aging drugs and geroprotectors.