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V.N.
Krutko, M.B. Slavin, T.M.
Smirnova. MATHEMATICAL FOUNDATIONS OF GERONTOLOGY
URSS. M.: 2002. 384 p. (Russian) The monograph presents in a
systematic form the most significant fundamental achievements in the field of
mathematical modeling, theory and methods of describing and analyzing the
processes of aging and mortality at the level of an individual organism and
population. Reliability theory is used
as the basis for systematization. A variant of the "general theory of
health" is proposed. Methods of statistical
analysis of indicators of aging, mortality and life expectancy are presented
in an accessible form adapted for practical use. The book can be recommended
both to scientists and specialists in the field of mathematical biology,
gerontology, sociology, demography, preventive medicine, hygiene, sanitation,
human ecology, as well as to teachers and students studying biomedical and
social disciplines. The monograph was prepared with the support
of the Ministry of Industry and Science of the Table of contents introduction: THE SUBJECT,
HISTORY AND METHOD OF MATHEMATICAL GERONTOLOGY CHAPTER 1. INDICATORS
CHARACTERIZING AGING AND STATISTICAL METHODS OF THEIR ANALYSIS 1.1. General and special
mortality rates 1.1.1. Determination of mortality rates 1.1.2. Methods of data standardization 1.2. Survival tables 1.2.1. Types of survival
tables and methods of their calculation 1.2.2. Use of survival
tables for the analysis of experimental data on life expectancy 1.3. The distribution of
life expectancy and its statistical estimates 1.3.1. Functions
characterizing the distribution of life expectancy 1.3.2. Statistical estimates
of distributions and their use as characteristics of life expectancy 1.3.2.1. Statistical estimates of
samples and their relation to the characteristics of random variables 1.3.2.2. Types of distributions most
often used in the analysis of life expectancy 1.3.3. Statistical
evaluation of the survival function and related functions 1.3.3.1. The case of small samples 1.3.3.2. The case of survival tables 1.3.4. Point estimates of
the distribution of life expectancy 1.4. Methods for comparing
life expectancy 1.4.1. General approaches to
statistical analysis of differences between samples 1.4.2. Comparison of life
expectancy for the proportional risks model 1.4.3. Comparison of life
expectancy distributions using the Kolmogorov-Smirnov
criterion 1.4.4. Comparison of life expectancy using
generalizations of the Wilcoxon criterion CHAPTER 2. MATHEMATICAL
MODELS IN GERONTOLOGY 2.1. Mathematical methods
for the analysis of aging at the individual level 2.1.1. Mathematical models
of biological age 2.1.2. Dynamic models of
aging of the body 2.1.2.1. Modeling of aging based on the
model of homeostasis of the body 2.1.2.2. The “shagreen
skin” aging model
2.1.3. General theory of health 2.1.3.1. The conceptual basis of the
general theory of health 2.1.3.2. Formal description of the
environment-health system 2.1.3.3. Generalized model of the
environment-organism system 2.1.3.4. Criteria of quality and
optimality in health management 2.2. Analytical models of
population aging 2.2.1. Methodological foundations for the
construction of analytical models of aging 2.2.2. Gompertz
and Gompertz-Makem models 2.2.3. Analysis of the
patterns of aging within the Gompertz model 2.2.3.1. The nature of differences in
life expectancy 2.2.3.2. Historical dynamics of
mortality 2.2.3.3. Regional and gender
differences in aging 2.2.3.4. Using the Gompertz
model in experimental studies 2.2.3.5. Modifications and
generalizations of the Gompertz model 2.2.3.6. Streler-Mildwan
correlation and Streler evaluation criteria 2.2.4. Gompertz-Makem
model 2.2.4.1. Theoretical justification of
the model 2.2.4.2. The concept of historical
stability of the age component of mortality and the possibility of increasing
human life expectancy 2.2.4.3. Compensatory effect of
mortality and species invariants of life expectancy 2.2.5. Modeling of damage
and repair processes 2.2.5.1. Deterministic approach 2.2.5.2. Stochastic approach 2.2.6. Models of death risk
distribution among peers and relatives 2.2.6.1. One-dimensional case 2.2.6.2. The two-dimensional case CHAPTER 3. RELIABILITY
THEORY AS A VARIANT OF THE CONCEPTUAL MATHEMATICAL BASE OF GERONTOLOGY 3.1. Reliability in living
and inanimate nature 3.2. Concepts of aging,
wear, durability and life expectancy 3.3. Basic terms of
reliability theory 3.4. Quantitative indicators
used in reliability theory and in gerontology 3.5. Statistics of the
survival of human populations in comparison with the statistics of the
“survival” of technical products 3.6. Time distributions of
failure occurrence used in the theory of reliability of non-recoverable
products 3.7. Model of gradual
approach to failure 3.8. Reliability models
taking into account operating conditions and the environment 3.9. Models of reliability
of devices with redundancy
3.10. Technical level,
quality and operational age of technical products in comparison with gerontological analogues 3.11. Comparison of modeling
tasks in reliability theory and in gerontology 3.12. Theory of restoration
and prospects of its adaptation to gerontology
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