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BACKGROUND Demographers have emphasized the importance of age in explaining the spread of COVID-19 and its impact on mortality However, the relationship between COVID-19 mortality and age should be contextualized in relation to other causes of death OBJECTIVE To compare the age pattern of COVID-19 mortality with other causes of death and across countries, and to use these regularities to impute age-specific death counts in countries with limited data METHODS The COVID-19 mortality doubling time in a Gompertz context was compared with 65 major causes of death using US vital statistics COVID-19 fatality doubling time was similarly compared across 27 countries and used for estimating death counts by age in Israel as a case in point RESULTS First, COVID-19 mortality increases exponentially with age at a Gompertz rate near the median of aging-related causes of death, as well as pneumonia and influenza Second, COVID-19 mortality levels are 2 8 to 8 2 times higher than pneumonia and influenza across the adult age range Third, the relationship between both COVID-19 mortality and fatality and age varies considerably across countries CONCLUSIONS The increase in COVID-19 mortality with age resembles the population rate of aging Country differences in the age pattern of COVID-19 mortality and fatality may point to differences in underlying population health, standards of clinical care, or data quality CONTRIBUTION This study underscores the need to contextualize the age pattern of COVID-19 mortality in relation to other causes of death Furthermore, it demonstrates how to estimate agespecific COVID-19 deaths in countries with limited data availability

Never before in history has population aging been a driving factor in epidemics to the same extent as with the current COVID-19 pandemic, with its dramatic shift in mortality towards older age groups. The paper presents the results of an analysis of the COVID-19-related mortality data for Spain, Italy, and Sweden, which show that within the 30- to 90-year age range, the logarithms of mortality rate depend on age linearly, and all regression lines are strictly parallel to the lines corresponding to the dependencies of the general mortality on age in accordance with the Gompertz law. In all cases, irrespective of the countries and epidemic stages, the mortality doubling times within this age range are close to 7.5 years. The probabilities of infection with the SARS-CoV-2 coronavirus, the causative agent of COVID-19, and of the development of the clinical symptoms of infection depend on age to a much lesser extent. Based on these observations, three main points are proposed for discussion: (1) Older people have become the main victims not only of SARS-CoV-2 itself but also of the measures undertaken to prevent its spread; (2) At the same time, older people are not the main force driving the spread of SARS-CoV-2, and (3) Older people can and should participate in the fight against the pandemic and in overcoming its consequences, but not through their selective isolation and other forms of discrimination. People over 65 years of age make up a considerable segment of the population and have at least as much right as other age groups to have their needs and interests be respected and observed, including the right to as high quality of life as is accessible even in extreme situations. The prospects for full control over SARS-CoV-2 are vague. This is why those who are in charge of decisions that concern people over 65 years of age should mind that, unlike the situation in the Middle Ages, the age of 65+ is the individual future of almost everyone.

COVID-19 became pandemic in 2020 and causes higher mortality in males (M) than females (F) and among older people. In some countries, like Belgium, more than half of COVID-19 confirmed or suspected deaths occurring in spring 2020 concerned residents of care homes. The high incidence in this population is certainly linked to its peculiar age structure but could also result from its poorer general health condition and/or from a higher contamination through the staff of care homes, while protection equipment and testing capacity were initially limited. To address these issues, we used data from Wallonia (Belgium) to characterize the distribution of death rates among care home institutions, to compare the dynamics of deaths in and outside care homes, and to analyse how age and sex affected COVID-19 death rates inside and outside care homes. We also used annual death rates as a proxy for the health condition of each population. We found that: (1) COVID-19 death rate per institution varied widely from 0‰ to 340‰ (mean 43‰) and increased both with the size of the institution (number of beds) and with the importance of medical care provided. (2) 65% of COVID-19 deaths in Wallonia concerned residents of care homes where the outbreak started after but at a faster pace than the outbreak seen in the external population. (3) The impact of age on both annual and COVID-19 mortality closely follows exponential laws (i.e. Gompertz law) but mortality was much higher for the population living in care homes where the age effect was lower (mortality rate doubling every 20 years of age increment in care homes, 6 years outside them). (4) Both within and outside care homes, the ratio of M/F death rates was 1.6 for annual mortality but reached 2.0 for COVID-19 mortality, a ratio consistent among both confirmed and suspected COVID-19 deaths. (5) When reported to the annual death rate per sex and age, the COVID-19 relative mortality was little affected by age and reached 24% (M) and 18% (F) of their respective annual rate in nursing homes, while these percentages reduced to 10% (M) and 9% (F) in homes for elderly people (with less medical assistance), and to 5% (M) and 4% (F) outside of care homes. In conclusion, a c. 130x higher COVID-19 mortality rate found in care homes compared to the outside population can be attributed to the near multiplicative combination of: (1) a 11x higher mortality due to the old age of its residents, (2) a 3.8x higher mortality due to the low average health condition of its residents, and (3) probably a 3.5x higher infection rate (1.6x in homes for elderly people) due to the transmission by its staff, a problem more acute in large institutions. Our results highlight that nursing home residents should be treated as a very specific population, both for epidemiological studies and to take preventive measures, due to their extreme vulnerability to COVID-19.

There is great interest among gerontologists, demographers, and actuaries in the question concerning the limits to human longevity. Attempts at getting answers to this important question have stimulated many studies on late-life mortality trajectories, often with opposing conclusions. One group of researchers believes that mortality stops growing with age at extreme old ages, and that hence there is no fixed limit to the human life span. Other studies found that mortality continues to grow with age up to extreme old ages. Our study suggests a possible solution to this controversy. We found that mortality deceleration is best observed when older, less accurate life span data are analyzed, while in the case of more recent and reliable data there is a persistent mortality growth with age. We compared the performance (goodness of fit) of two competing mortality models – the Gompertz model and the Kannisto (“mortality deceleration”) model – at ages of 80–105 years using data for 1880–1899 single-year birth cohorts of US men and women. The mortality modeling approach suggests a transition from mortality deceleration to the Gompertzian mortality pattern over time for both men and women. These results are consistent with the hypothesis about disappearing mortality deceleration over time due to improvement in the accuracy of age reporting. In the case of more recent data, mortality continues to grow with age even at very old ages. This observation may lead to more conservative estimates of future human longevity records.

Since chronological age is not a complete and accurate indicator of organism aging, the concept of 9biological age9 has emerged as a well-accepted way to quantify the aging process in humans and laboratory animals. In this study, we performed a systematic statistical evaluation of the relationships between locomotor activity and biological age, mortality risk, and frailty using human physical activity records from the 2003-2006 National Health and Nutrition Examination Survey (NHANES) and UK BioBank (UKBB) databases. These records are from subjects ranging from 5 to 85 years old and include 7-day long continuous tracks of activity provided by wearable monitors as well as data for a comprehensive set of clinical parameters, lifestyle information and death records, thus enabling quantitative assessment of frailty and mortality. We proposed a statistical description of the locomotor activity tracks and transformed the provided time series into vectors representing individual physiological states for each participant. Using this data, we performed an unsupervised multivariate analysis and observed development and aging as a continuous trajectory consisting of distinct phases, each corresponding to subsequent human life stages. Therefore, we suggest the distance measured along this trajectory as a definition of the biological age. Consistent with the Gompertz law, mortality, estimated with the help of a proportional hazard model, was found to be an exponential function of biological age as quantified herein. However, we observed that the significant contribution of clinical frailty to mortality risk can be independent of biological age. We used the biological age and mortality models to show that some lifestyle variables, such as smoking, produce a reversible increase in all-cause mortality without a significant effect on biological age. In contrast, medical conditions, such as type 2 diabetes mellitus (T2DM) or hypertension, are associated with significant aging acceleration and a corresponding increase in mortality as well. The results of this work demonstrate that significant information relevant to aging can be extracted from human locomotor activity data and highlight the opportunity provided by explosive deployment of wearable sensors to use such information to encourage lifestyle modifications and clinical development of therapeutic interventions against the aging process.

One of the many interests of the late Dietrich Stauffer was the modelling of mortality. Here we review the features of mortality data for various biological species. Age specific mortality (death rate) leads to a discussion of possible models of the death rate, including that of Gompertz, raising the interesting question: is our lifetime finite or could we contemplate living for ever? The answer judging from many different data sources is that without radical changes in our biology it seems death above age 120 is extremely unlikely. We then show how a toy model, linking mortality to the immune system, can predict the general variation of the death rate with time, spanning both infant and adult phases. The outcome provides underpinning support for the many nutritionists and medical experts who increasingly advocate the benefits to mortality of a healthy lifestyle. Age specific mortality within social networks is also shown to be significantly affected by both psychological and physical shocks. The review concludes with the description of novel experiments using soap films for the study of failure. These allow for a reproduction of high infant mortality, the so-called bath-tub curve of mortality, and the Gompertz law.

Age is the most important single risk factor for the incidence of chronic diseases and ultimately, death. The mortality rate increases exponentially with age and doubles approximately every eight years, as described by the Gompertz law of mortality. The incidence of specific diseases, such as cancer or stroke, also accelerates after the age of about forty, and doubles at a rate that mirrors the mortality rate doubling time. It is therefore entirely plausible to think that there is a single underlying process, the driving force behind the progressive reduction of the organism's fitness leading to the increasing susceptibility to diseases and death, that is aging itself. There is, however, no fundamental law of nature requiring exponential morbidity and mortality risk trajectories. The acceleration of mortality is thus the most important characteristics of aging process. It varies dramatically even among closely related mammalian species and hence appears to be a tunable phenotype. Here we follow how big data from large human medical studies, and analytical approaches borrowed from physics of complex dynamic systems can help to reverse engineer the underlying biology behind Gompertz mortality law. With this approach we hope to generate predictive models of aging and for systematic discovery of biomarkers of aging followed by identification of novel therapeutic targets for future anti-aging interventions.

The growing number of persons living beyond age 80 underscores the need for accurate measurement of mortality at advanced ages and understanding the old-age mortality trajectories. It is believed that exponential growth of mortality with age (Gompertz law) is followed by a period of deceleration, with slower rates of mortality increase at older ages. This pattern of mortality deceleration is traditionally described by the logistic (Kannisto) model, which is considered as an alternative to the Gompertz model. Mortality deceleration was observed for many invertebrate species, but the evidence for mammals is controversial. We compared the performance (goodness-of-fit) of two competing models—the Gompertz model and the logistic (Kannisto) model using data for three mammalian species: 22 birth cohorts of U.S. men and women, eight cohorts of laboratory mice, and 10 cohorts of laboratory rats. For all three mammalian species, the Gompertz model fits mortality data significantly better than the “mortality deceleration” Kannisto model (according to the Akaike’s information criterion as the goodness-of-fit measure). These results suggest that mortality deceleration at advanced ages is not a universal phenomenon, and survival of mammalian species follows the Gompertz law up to very old ages.

Deviations from the Gompertz law of exponential mortality increases in late-middle and early-old age are commonly neglected in overall mortality analyses. In this study, we examined mortality increase patterns between ages 40 and 85 in 16 low-mortality countries and demonstrated sex differences in these patterns, which also changed across period and cohort. These results suggest that the interaction between aging and death is more complicated than what is usually assumed from the Gompertz law and also challenge existing biodemographic hypotheses about the origin and mechanisms of sex differences in mortality. We propose a two-mortality model that explains these patterns as the change in the composition of intrinsic and extrinsic death rates with age. We show that the age pattern of overall mortality and the population heterogeneity therein are possibly generated by multiple dynamics specified by a two-mortality model instead of a uniform process throughout most adult ages.

Age affects mortality from diseases differently than it affects mortality from external causes, such as accidents. Exclusion of the latter leads to the “all-diseases” category. The age trajectories of mortality from all diseases are studied in the five most populated countries of the EU, and the shape of these 156 age trajectories is investigated in detail. The arithmetic mean of ages where mortality reaches a minimal value is 8.47 years with a 95% confidence interval of [8.08, 8.85] years. Two simple deterministic models fit the age trajectories on the two sides of the mortality minimum. The inverse relationship is valid in all cases prior to this mortality minimum and death rates exactly decreased to three thousandths of its original size during the first 3000 days. After the mortality minimum, the standard Gompertz model fits the data in 63 cases, and the Gompertz model extended by a small quadratic element fits the remaining 93 cases. This analysis indicates that the exponential increase begins before the age of 15 years and that it is overshadowed by non-biological causes. Therefore, the existence of a mechanism switching that would explain the exponential increase in mortality after the age of 35 years is unlikely.

Using a new distribution capable of exhibiting all the possible modes of accelerating and decelerating mortality, we conduct a systematic investigation of late-life mortality in humans. We check the insensitivity of the distribution to age cutoffs in the data relative to the logistic mortality model and propose a method to forecast evolution in the characteristic deceleration ages of the distribution. A number of data sets have been explored, with a particular emphasis on those originating from Scandinavia. Although those from Australia, Canada, and the USA are compatible with Gompertzian mortality, those from the other countries examined are not. We find in particular that the onset of mortality deceleration is being progressively delayed in Western societies but that there is evidence of mortality plateauing at earlier ages.

BACKGROUND Half a century ago Strehler and Mildvan (1960) have published the seminal paper that, based on some assumptions (postulates), theoretically 'justified' the Gompertz law of mortality. OBJECTIVE We wish to discuss assumptions and limitations of the original Strehler-Mildvan model (as well as of the Strehler-Mildvan correlation) and consider some modifications and departures from this model. METHODS We use the framework of stochastic point processes for analyzing the original Strehler-Mildvan model. We also suggest the 'lifesaving approach' for describing the departure from rectangularization to shifts in survival curves for human mortality that has been observed in the second half of the previous century. RESULTS We show that the Strehler-Mildvan model can be justified only under the additional assumption that the process of shocks (demands for energy) follows the Poisson pattern. We also suggest a modification that accounts for the oldest-old mortality plateau.

Mortality deceleration is the observed but yet to be understood phenomenon that the increase in the late-life death rate slows down after a certain species-related advanced age. Various definitions of onsets of mortality deceleration are examined. A new distribution based on the Strehler-Mildvan theory of aging takes on the required shapes. The application is done on mortality data from the 1892 cohort of Swedish women and on Mediterranean fruit flies.

In a heterogeneous cohort, the change with age in the force of mortality or some other kind of hazard or intensity of attrition depends on how the hazard changes with age for the individuals in the cohort and on how the composition of the cohort changes due to the loss of those most vulnerable to attrition. Here we prove that the change with age for the cohort equals the average of the change in the hazard for the individuals in the cohort minus the variance in the hazard across individuals. The variance captures the compositional change. This very general and remarkably elegant relationship can be applied to understand and to analyze changes with age in many kinds of demographic hazards, including, e.g., the lifetable aging rate or the intensity of first births.

We study the effects of demographic shocks and changes in the pension system on the macroeconomic performance of an advanced small open economy facing a given world interest rate. We construct an overlapping-generations model which includes a realistic description of the mortality process. Individual agents choose their optimal retirement age, taking into account the time- and age profiles of wages, taxes, and the public pension system. The early retirement provision in most pension systems acts as a trap, inducing most workers to retire well before the normal retirement age. Simulations show that pension reform must be drastic for it to have any effects on the retirement behaviour of workers.Keywords: Retirement; Pensions; Ageing; Demography; Gompertz-Makeham Law of mortality; Overlapping generations; Small open economyJEL classification codes: E10; J26; H55; D91; F41; J11

Gompertz's law tells us that for humans above the age of 35 the death rate increases exponentially with a doubling time of about 10 years. Here, we show that the same law continues to hold even for ages over 100. Beyond 106 there is so far no statistical evidence available because the number of survivors is too small even in the largest nations. However assuming that Gompertz's law continues to hold beyond 106, we conclude that the mortality rate becomes equal to 1 at age 120 (meaning that there are 1,000 deaths in a population of one thousand). In other words, the upper bound of human life is near 120. The existence of this fixed-point has interesting implications. It allows us to predict the form of the relationship between death rates at age 35 and the doubling time of Gompertz's law. In order to test this prediction, we first carry out a transversal analysis for a sample of countries comprising both industrialized and developing nations. As further confirmation, we also develop a longitudinal analysis using historical data over a time period of almost two centuries. Another prediction arising from this fixed-point model, is that, above a given population threshold, the lifespan of the oldest person is independent of the size of her national community. This prediction is supported by available empirical evidence., Comment: 14 pages, 7 figures

Background: The Gompertz force of mortality (hazard function) is usually expressed in terms of a, the initial level of mortality, and b, the rate at which mortality increases with age. Objective: We express the Gompertz force of mortality in terms of b and the old-age modal age at death M, and present similar relationships for other widely-used mortality models. Our objective is to explain the advantages of using the parameterization in terms of M. Methods: Using relationships among life table functions at the modal age at death, we express the Gompertz force of mortality as a function of the old-age mode. We estimate the correlation between the estimators of old (a and b) and new (M and b) parameters from simulated data. Results: When the Gompertz parameters are statistically estimated from simulated data, the correlation between estimated values of b and M is much less than the correlation between estimated values of a and b. For the populations in the Human Mortality Database, there is a negative association between a and b and a positive association between M and b. Conclusions: Using M, the old-age mode, instead of a, the level of mortality at the starting age, has two major advantages. First, statistical estimation is facilitated by the lower correlation between the estimators of model parameters. Second, estimated values of M are more easily comprehended and interpreted than estimated values of a. Background: The Gompertz force of mortality (hazard function) is usually expressed in terms of a, the initial level of mortality, and b, the rate at which mortality increases with age.Objective: We express the Gompertz force of mortality in terms of b and the old-age modal age at death M, and present similar relationships for other widely-used mortality models. Our objective is to explain the advantages of using the parameterization in terms of M.Methods: Using relationships among life table functions at the modal age at death, we express the Gompertz force of mortality as a function of the old-age mode. We estimate the correlation between the estimators of old (a and b) and new (M and b) parameters from simulated data.Results: When the Gompertz parameters are statistically estimated from simulated data, the correlation between estimated values of b and M is much less than the correlation between estimated values of a and b. For the populations in the Human Mortality Database, there is a negative association between a and b and a positive association between M and b.Conclusions: Using M, the old-age mode, instead of a, the level of mortality at the starting age, has two major advantages. First, statistical estimation is facilitated by the lower correlation between the estimators of model parameters. Second, estimated values of M are more easily comprehended and interpreted than estimated values of a.

In 1825, the actuary Benjamin Gompertz read a paper, ‘On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies’, to the Royal Society in which he showed that over much of the adult human lifespan, age-specific mortality rates increased in an exponential manner. Gompertz's work played an important role in shaping the emerging statistical science that underpins the pricing of life insurance and annuities. Latterly, as the subject of ageing itself became the focus of scientific study, the Gompertz model provided a powerful stimulus to examine the patterns of death across the life course not only in humans but also in a wide range of other organisms. The idea that the Gompertz model might constitute a fundamental ‘law of mortality’ has given way to the recognition that other patterns exist, not only across the species range but also in advanced old age. Nevertheless, Gompertz's way of representing the function expressive of the pattern of much of adult mortality retains considerable relevance for studying the factors that influence the intrinsic biology of ageing. This commentary was written to celebrate the 350th anniversary of the journalPhilosophical Transactions of the Royal Society.

The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws. The mathematical model is a set of six coupled differential equations governing the time evolution of both the tumour cell population (cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunal enterocyte population, to be shielded from unwanted side effects during a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent free platinum Pt (the active drug) infusion law being given, the mathematical model allows to compute the time evolution of both tumour and villi populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicity rhythm. The model has been derived using, as far as possible, experimental data. We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L ∞ criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.

We have simulated the effect of the diversity of the late expressed genes in the genetic pool of population on the phenotypes of individuals in the late ages. Using Penna model based on the Monte Carlo method we have obtained for the oldest fractions of populations lower mortality rates than predicted by the exponential Gompertz function. Such deviations from the expected exponential increase of mortality are the characteristic for populations which are not in equilibrium with the environment, or if a relatively high probability of reversions was assumed, or if the population is heterogeneous. In such populations, the genes expressed in the late ages, are under the very weak selection pressure and thus, highly-polymorphic. As an effect, the probability of the genetically-determined death of the oldest organisms does not grow as fast as predicted by the Gompertz exponential curve describing mortality during earlier periods of life.

Biodemography is a promising scientific approach based on using demographic data and methods for getting insights into biological mechanisms of observed processes. Recently, new important developments have happened in biodemographic studies of aging and longevity that call into question conventional aging theories and open up novel research directions. Recent studies found that the exponential increase of the mortality risk with age (the famous Gompertz law) continues even at extreme old ages in humans, rats, and mice, thus challenging traditional views about old-age mortality deceleration, mortality leveling-off, and late-life mortality plateaus. This new finding represents a challenge to many aging theories, including the evolutionary theory that explains senescence by a declining force of natural selection with age. Innovative ideas are needed to explain why exactly the same exponential pattern of mortality growth is observed not only at reproductive ages, but also at very-old postreproductive ages (up to 106 years), long after the force of natural selection becomes negligible (when there is no room for its further decline). Another important recent development is the discovery of long-term ‘memory' for early-life experiences in longevity determination. Siblings born to young mothers have significantly higher chances to live up to 100 years, and this new finding, confirmed by two independent research groups, calls for its explanation. As recent studies found, even the place and season of birth matter for human longevity. Beneficial longevity effects of young maternal age are observed only when children of the same parents are compared, while the maternal age effect often could not be detected in across-families studies, presumably being masked by between-family variation. It was also found that male gender of centenarian has a significant positive effect on the survival of adult male biological relatives (brothers and fathers) but not of female relatives. Finally, large gender differences are found in longevity determinants for males and females, suggesting a higher importance of occupation history for male centenarians as well as a higher importance of home environment history for female centenarians.

The widely-known Gompertz law of mortality states the exponential increase of mortality with age in human populations. Such an exponential increase is observed at the adulthood span, roughly after the reproductive period, while mortality data at young and extremely old ages deviate from it. The heterogeneity of human populations, i.e. the existence of subpopulations with different mortality dynamics, is a useful consideration that can explain age-dependent mortality patterns across the whole life-course. A simple mathematical model combining the heterogeneity of populations with an assumption that the mortality in each subpopulation grows exponentially with age has been proven to be capable of reproducing the entire mortality pattern in a human population including the observed peculiarities at early- and late-life intervals. In this work we fit this model to actual (Swedish) mortality data for consecutive periods and consequently describe the evolution of mortality dynamics in terms of the evolution of the model parameters over time. We have found that the evolution of the model parameters validates the applicability of the compensation law of mortality to each subpopulation separately. Furthermore, our study has indicated that the population structure changes so that the population tends to become more homogeneous over time. Finally, our analysis of the decrease of the overall mortality in a population over time has shown that this decrease is mainly due to a change in the population structure and to a lesser extent to a reduction of mortality in each of the subpopulations, the latter being represented by an alteration of the parameters that outline the exponential dynamics.

Background: Life table entropy is a quantity frequently used in demography; e.g., as a measure ofheterogeneity in age at death, or as the elasticity of life expectancy with regards to proportional changes in age-specific mortality. It is therefore instructive to calculate its valuefor the widely used Gompertz-Makeham mortality model. Objective: I present and prove a simple expression of life table entropy for the Gompertz-Makeham model, which ties together the parameters of the model with demographically relevant quantities. Comments: The relationship shows that entropy is easily calculated from the parameters of the given model, life expectancy and the average age in the stationary population. The latter enters the equation only if the Makeham term c is different from zero.

We propose a simple model describing the dynamics of a system of two populations — more numerous natives and less numerous immigrants. The immigrants' birth rate is higher than that of the natives. Several modifications of this model taking into account changes of the birth rates due to external factors and/or possibility of contacts between the populations, are also introduced. The model is studied within two approaches — by solving a set of differential equations and through a Monte Carlo simulations. We show that the question of which population will eventually dominate depends on such factors as the probability of producing offsprings of mixed origin, assimilation of the immigrants, the ratio of the birth rates, initial numbers of the populations and the average age of an individual. In all, but two extreme cases, both populations will survive.

There are many theories of mortality, but no consensus even on the basic problem: is it genetically determined? In a general case, the problem is mathematically unsolvable. Yet, in the case of mortality, a physical approach yields its universal law. The law predicts, e.g., that mortality and senescence may decrease with age. Experiments verify it. I suggest experiments, which are supposed to produce genetic Methuselas, who live, e.g., over 20 η ( η is the mean life span), but whose biological age is less than η /2. If the universal mortality law is convincingly proven, it may lead to a quantitative model and theory of mortality and aging.

Background: Survival models accounting for unobserved heterogeneity (frailty models) play an importantrole in mortality research, yet there is no article that concisely summarizes useful relationships. Objective: We present a list of important mathematical relationships that govern populations in which individuals differ from each other in unobserved ways. For some relationships we present proofs that, albeit formal, tend to be simple and intuitive. Methods: We organize the article in a progression, starting with general relationships and then turning to models with stronger and stronger assumptions. Results: We start with the general case, in which we do not assume any structure of the underlyingbaseline hazard, the frailty distribution, or their link to one another. Then we sequentially assume, first, a relative-risk model; second, a gamma distribution for frailty; and, finally, a Gompertz and Gompertz-Makeham specification for baseline mortality. Comments: The article might serve as a handy overall reference to frailty models, especially for mortalityresearch.

The accounts of medical trials provide very detailed information about the patients' health conditions. In contrast, only minimal data are usually given about demographic factors. Yet, some of these factors can have a notable impact on the overall death rate, thereby changing the outcome and conclusions of the trial. This paper focuses on two of these variables. The first is marital status; this effect, which will be referred to as the Bertillon effect, may change death rates by over 100%. The second is the age of the oldest patients; because of the exponential nature of Gompertz's law, changes in the distribution of ages in the oldest age group can have dramatic consequences on the overall number of deaths. It will be seen that randomization alone can hardly take care of these problems. Appropriate remedies are easy to formulate however. First, the marital status of patients as well as the age distribution of those over 65 should be documented for both study groups. Then, thanks to these data and based on the Bertillon and Gompertz laws, it will become possible to perform appropriate corrections. Such corrections will notably improve the reliability and accuracy of the trial's conclusions, especially for trials which include a large proportion of elderly subjects., 9 pages, 4 figures

International audience; This paper examines and demonstrates the importance of the adult modal age at death (M) in longevity research. Unlike life expectancy at birth (e0) and median age at death, M is determined solely by old-age mortality as far as mortality follows a bathtub curve. It represents the location of old-age death heap in the age distribution of deaths, and captures mortality shifts more accurately than conditional life expectancies such as e65. Although M may not be directly determined from erratic mortality data, a recently developed method for deriving M from the P-spline-smoothed mortality curve based on penalised Poisson likelihood is highly effective in estimating M. Patterns of trends and differentials in M can be noticeably different from those in other lifespan measures, as indicated in some examples. In addition, major mathematical models of adult mortality such as the Gompertz, logistic and Weibull models can be reformulated using M, which plays a critical role as the mortality level parameter in those models.

The classical Gompertz assumption that the number of individuals surviving in a population decreases with age at an exponentially increasing rate often fits actual data poorly. In particular, observed mortality rates of the older individuals may be lower than predicted. However, such data can be very well fit by the alternative assumption that the number dying increases at an exponentially decreasing rate. Thus, when the appropriate form of the Gompertz kinetics is used, the older individuals die according to the same rule as the younger.

The study investigated the factors responsible for compression of mortality usually observed in human survival curves. The concept of rectangularisation was adopted to study the changes that occur in human survival curves at the older ages under the assumption that the basic mortality table is occurring in time zero. Simulations were carried out to examine the behavior of survival and cumulative death curves under varying conditions. The result revealed that the survival curve shifts to the right of the base mortality survival curve l୶ as mortality situation improves. Furthermore, the study indicated that expansion of mortality spectrum will occur if mortality curve is allowed to flow smoothly throughout the human life.

Heterogeneity in a population with respect to mortality, or variation in “frailty”; among members of that population, which has been discussed extensively in the literature over the last decade and a half is essential to any realistic model of dependence among causes of death. The main problem then is the development of a mortality model incorporating heterogeneity and cause of death which is both realistic and of manageable proportions. In a recent paper (J. H. Pollard, 1991), it has been shown that many life table results are remarkably insensitive to the strict shape of the mortality curve, at least for more developed populations, and that accurate approximations can in many cases be obtained knowing only the mortality rates at two representative ages (e.g. 50 and 70). These results and the Gompertz “law”; of mortality can be used to develop manageable approximate formulae for the expectation of life under heterogeneity and correlation among the causes of death. The formulae are confirmed by simulation...

It is well known that the rate of aging is constant for populations described by the Gompertz law of mortality. However, this is true only when a population is homogeneous. In this note, we consider the multiplicative frailty model with the baseline distribution that follows the Gompertz law and study the impact of heterogeneity on the rate of aging in this population. We show that the rate of aging in this case is a function of age and that it increases in (calendar) time when the baseline mortality rate decreases.

The Makeham distribution describes the age dynamics of human mortality rather accurately in the age window of about 30-80 years. At more advanced ages the death rates do not increase as fast as predicted by this mortality law – a phenomenon known as the late-life mortality deceleration. Historical decline in human mortality before 1950s was mostly due to the decrease in the age-independent mortality component (Makeham parameter), while the agedependent mortality component was surprisingly stable in history before 1950s. After that a new mortality trend has started, leading to unexpected decline in mortality rates at advanced ages and “de-rectangularization” of the survival curve.

The mortality rates are steadily declining with time. The remaining lifetime for e.g. 65 years old person even 20-30 years ago was substantially smaller than nowadays. Therefore, the age correspondence problem for populations in different mortality regimes is of interest. A simple solution, based on the equality of accumulated mortality rates (or, equivalently, on the equality of probabilities of survival) is considered. Furthermore, the mortality regime with a change point is defined and the procedure of age re-calculation after the change point is suggested. Two age re-calculation models (and their combination) are discussed: the first one accounts for wear accumulation in the process of aging and the other is characterized by a kind of memoryless property.

Many equations can be used to study the relationship between mortality rates and age: Gompertz, Weibull, logistic, polynomial and age-period-cohort equations a.o. All these equations result in highly significant correlations between ln mortality rates and age in the age range 35-84 years. This applies to all developed countries and is independent of the differences in causes of death between populations. The best fit is obtained by a second-degree polynomial equation (R2 > 0.99), closely followed by the Gompertz equation. This equation is preferred in view of its extreme simplicity. A highly significant correlation exists between the intercept and the slope of the Gompertz equations, pointing to a crossing-over age. Beyond that age, around 85 years, populations with high mortality rates have a lower mortality, due to selective survival of the strongest individuals. The polynomial age 2 term may be positive or negative, an expression of the acceleration or de-acceleration of mortality at higher ages and is significantly more often positive in women. The equations used are very useful for the study of the aging process and for examining the relationship between possible causal factors and mortality rates in populations.

Since its proposition in 1995, the Heumann-Hotzel model has remained as an obscure model of biological aging. The main arguments used against it were its apparent inability to describe populations with many age intervals and its failure to prevent a population extinction when only deleterious mutations are present. We find that with a simple and minor change in the model these difficulties can be surmounted. Our numerical simulations show a plethora of interesting features: the catastrophic senescence, the Gompertz law and that postponing the reproduction increases the survival probability, as has already been experimentally confirmed for the Drosophila fly., 11 pages, 5 figures, to be published in Phys. Rev. E

Old-age survival has increased substantially since 1950. Death rates decelerate with age for insects, worms, and yeast, as well as humans. This evidence of extended postreproductive survival is puzzling. Three biodemographic insights—concerning the correlation of death rates across age, individual differences in survival chances, and induced alterations in age patterns of fertility and mortality—offer clues and suggest research on the failure of complicated systems, on new demographic equations for evolutionary theory, and on fertility-longevity interactions. Nongenetic changes account for increases in human life-spans to date. Explication of these causes and the genetic license for extended survival, as well as discovery of genes and other survival attributes affecting longevity, will lead to even longer lives.

The interval 35-85 years is mostly considered as the range of Gomperz law's validity. Looking at the spectrum of causes of death for the interval 10-30 years we find that its main share is caused by various accidents independent of age. These accidents result in the increase and following plateau of the total mortality curve over the age of 15 years. When the number of deaths from age independent causes (accidents, violence, suicide, accidental falls, etc.) is subtracted from the total number of deaths the linear area begins before the age of 20 years. The linear regression of the logarithm of mortality without accidents was performed for the age interval 20-84 years for six populations (men and women in the US in 1989, in Japan in 1990 and in former Czechoslovakia in 1990). Furthermore, the mortality curves of some causes important at the interval 0-15 years were inspected. Unlike other causes, births do not affect the disposition of children to die of malignant neoplasm of the brain and malignant neoplasm of lymphatic and haemopoietic tissue.

SummaryThe Gompertz equation describes an exponential relationship between mortality and ageing. However, mortality in animal and human populations does not increase exponentially at older ages. Age-specific malignant melanoma mortality rate distributions in the white population of the United States from 1979 to 1989 displayed two distinct Gompertzian slopes; one between age 15 and 35 years, and a second between age 45 and 85 years. Rather than indicating a ‘biological’ limitation of the Gompertz law, multiphasic and upper age boundaries to Gompertzian mortality rate dynamics are a mathematical requirement in genetically heterogeneous populations.

It is generally assumed for most species that mortality rates increase monotonically at advanced ages. Mortality rates were found to level off and decrease at older ages in a population of 1.2 million medflies maintained in cages of 7,200 and in a group of approximately 48,000 adults maintained in solitary confinement. Thus, life expectancy in older individuals increased rather than decreased with age. These results cast doubt on several central concepts in gerontology and the biology of aging: (i) that senescence can be characterized by an increase in age-specific mortality, (ii) that the basic pattern of mortality in nearly all species follows the same unitary pattern at older ages, and (iii) that species have absolute life-span limits.

Sir—Thomas Perls and colleagues (May 23, p 1560) show that human longevity is an inherited trait by comparison of the survival rates for the brothers and sisters of centenarians with those of siblings of people who died at age 73. At old age the survival probability for sisters and brothers of centenarians was found to be four times higher than usual. We report this effect in the widespread Penna model for computer simulations of ageing. This model is based on the mutation accumulation hypothesis. We die because of the action of inherited harmful mutations. Although new mutations can become active with equal probability at any age, those that get fixed in the population are mainly active at old age, whereas those that act at a younger age are likely to be eliminated by selection pressure. The genome is represented by a string of many computer bits that correspond to as many age intervals; a bit set to one instead of zero signals a genetic disease (mutation) active from that age on. Children may differ from their parents by one randomly selected bit, which is flipped at birth. Three active mutations kill the individual. Monte Carlo simulations (with random numbers) show reasonable agreement with the steep (exponential) increase of mortality with increasing age, as given by the Gompertz law or the catastrophic senescence of Pacific Salmon. In the computer simulation (asexual version) we measure age not in years but by the intervals corresponding to our computer bits; for human beings such age intervals might correspond to 6 or 7 years. Parents who died in the 16th age interval had 25% of their children surviving at least to age 16, whereas parental death at age 15 meant such survival for only 9% of the children, and with parental death at age 14 only 5% of the children reached age 16. This inherited longevity leads to the correlations of longevity between children of the same parent. The probability of surviving to at least age interval 15 was 44% for siblings of individuals who died at age 16, but only 7% for siblings of individuals who died at age 14. Our figure shows how the average age at death for all siblings of one family is strongly correlated with the age of death of the oldest of them. Thus, the correlation reported by Perls and co-workers is qualitatively confirmed by our computer simulation of hereditary diseases in population genetics. More details, including results from sexual reproduction, will be published elsewhere.

Life table methods are developed for populations whose members differ in their endowment for longevity. Unlike standard methods, which ignore such heterogeneity, these methods use different calculations to construct cohort, period, and individual life tables. The results imply that standard methods overestimate current life expectancy and potential gains in life expectancy from health and safety interventions, while underestimating rates of individual aging, past progress in reducing mortality, and mortality differentials between pairs of populations. Calculations based on Swedish mortality data suggest that these errors may be important, especially in old age.

Increased mortality rates of a constant proportion or corresponding to a constant increase in age produce increasing excess mortality by comparison with rates allowing for a constant addition to the rates and under quite general conditions (monotony or convexity of qx). The comparison of the effect of an age increase with that of a constant proportionate increase in the rates leads, however, to different results if one starts from different mortality formulae. The age increase produces proportionally constant (by Gompertz) or increasing (by de Moivre, Makeham, Lazarus) or decreasing (by Dormoy II) excess mortality; by Dormoy I it represents no excess mortality at all.