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This paper provides a flexible multi-factor framework to address some ongoing challenges in mortality modeling, with a special focus on the mortality curvature and possible mortality plateau for extremely old ages. We extend the Gompertz law Gompertz (1825) by proposing a multi-factor exponential model. The proposed framework is based on the Laguerre approximating functions, and is able to capture flexible mortality patterns, and allows for a convenient estimation and prediction algorithm. An extensive empirical analysis is conducted using the proposed framework with a merged mortality database containing a large number of countries and regions with credible old-age mortality data. We find that the proposed exponential model leads to superior goodness-of-fit to historical data, and better out-of-sample forecast performance. Moreover, the exponential model predicts more balanced mortality improvements across ages, and thus leads to higher projected remaining life expectancy for the old ages than existing Gompertz-based mortality models. Finally, the modeling capacity of the proposed exponential model is further demonstrated by a multi-population extension, and an illustrative example of estimation and forecast is provided.

Gompertzian tumor growth can be reproduced by mitosis, related to nutrient supply, with local spatial cell correlations. The global energy constraint alone does not reproduce in vivo data by the observed values of the nutrient expenditure for the cell activities. The depletion of the exponential growth, described by the Gompertz law, is obtained by mean field spatial correlations or by a small word network among cells. The well-known interdependence between the two parameters of the Gompertz growth naturally emerges and depends on the cell volume and on the tumor density.

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.

Aging organisms die out in accordance with the “Gompertz law,” i.e., the probability of their death increases with age. Survival curve construction is the main tool for gerontologists to study aging and test antiaging drugs. The analysis of survival curves includes obtaining some indices characterizing aging of the population, for example, the average and maximum lifespan, the mortality rate, and the aging rate. Testing geroprotectors can be correctly performed only by obtaining such curves. The dying out of stationary cell populations—bacteria, yeast, and mammalian cell cultures—also occurs in accordance with the Gompertz equation. In this regard, it is reasonable to use the construction of survival curves and their analysis to study the “aging” of non-subcultured cell cultures and testing anti-aging drugs on them. We used this approach in our experiments, due to which we were able to detect the positive anti-aging effect of the Quinton Marine Plasma on stationary phase aging culture of Chinese hamster cells.

Clean foams tend to age with time through sequential coalescence events. This study evaluates aging dynamics in clean foams by measuring bubble populations from coalescence simulation experiments and adopting biological population dynamics analysis. The population dynamics of bubbles in clean foams during coalescence show that the mortality rates of individual bubbles change exponentially with time, regardless of initial simulation conditions, consistent with the Gompertz mortality law commonly observed in biological aging. This result would be beneficial in understanding the aging dynamics of clean foams.

Biological age acceleration (BAA) models based on blood tests or DNA methylation emerge as a de facto standard for quantitative characterizations of the aging process. We demonstrate that deep neural networks trained to predict morbidity risk from wearable sensor data can provide a high-quality and cheap alternative for BAA determination. The GeroSense BAA model was trained and validated using steps per minute recordings from 103,830 one-week long and 2,599 of up to 2 years-long longitudinal samples and exhibited a superior association with life-expectancy over the average number of steps per day in, e.g., groups stratified by professional occupations. The association between the BAA and effects of lifestyles, the prevalence of future incidence of diseases was comparable to that of BAA from models based on blood test results. Wearable sensors let sampling of BAA fluctuations at time scales corresponding to days and weeks and revealed the divergence of organism state recovery time (resilience) as a function of chronological age. The number of individuals suffering from the lack of resilience increased exponentially with age at a rate compatible with Gompertz mortality law. We speculate that due to the stochastic character of BAA fluctuations, its mean and auto-correlation properties together comprise the minimum set of biomarkers of aging in humans.

The death toll of the current COVID-19 pandemic is strongly biased toward the elderly. COVID-19 case fatality rate (CFR) increases with age exponentially, its doubling time being about 7 years, irrespective of countries and epidemic stages. The same age-dependent mortality pattern known as the Gompertz law is featured by the total mortality and its main constituents attributed to cardiovascular, metabolic, neurological, and oncological diseases. Among patients dying of COVID-19, most have at least one of these conditions, whereas none is found in most of those who pass it successfully. Thus, gerontology is indispensable in dealing with the pandemic, which becomes a benchmark for validating the gerontological concepts and advances. The two basic alternative gerontological concepts imply that either aging results from the accumulation of stochastic damage, or is programmed. Based on these different grounds, several putative anti-aging drugs have been proposed as adjuvant means for COVID-19 prevention and/or treatment. These proposals are reviewed in the context of attributing the molecular targets of these drugs to the signaling pathways between the sensors of resource availability and the molecular mechanisms that allocate resources to storage, growth and reproduction or to self-maintenance and repair. Each of the drugs appears to reproduce only a part of the physiological responses to reduced resource availability caused by either dietary calories restriction or physical activity promotion, which are the most robust means of mitigating the adverse manifestations of aging. In the pathophysiological terms, the conditions of the endothelium, which worsen as age increases and may be significantly improved by the physical activity, is a common limiting factor for the abilities to withstand both physical stresses and challenges imposed by COVID-19. However, the current anti-epidemic measures promote sedentary indoor lifestyles, at odds with the most efficient behavioral interventions known to decrease the vulnerability to both the severe forms of COVID-19 and the prevalent aging-associated diseases. To achieve a proper balance in public health approaches to COVID-19, gerontologists should be involved in crosstalk between virologists, therapists, epidemiologists, and policy makers. The present publication suggests a conceptual background for that.

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.

Epidemiologists commonly use an adjusted hazard ratio or incidence density ratio, or a standardized mortality ratio, to measure a difference in all-cause mortality rates. They seldom translate it into an age-, time-, or probability-based measure that would be easier to communicate and to relate to. Several articles have shown how to translate from a standardized mortality ratio or hazard ratio to a longevity difference, a difference in actuarial ages, or a probability of being outlived. In this paper, we describe the settings where these translations are and are not appropriate and provide some of the heuristics behind the formulae. The tools that yield differences in “effective age” and in longevity are applicable when both 1) the mortality rate ratio (hazard ratio) is constant over age and 2) the rates themselves are log-linear in age. The “probability/odds of being outlived” metric is applicable whenever the first condition holds, and thus it provides no direct information on the magnitude of the effective age/longevity difference.

This paper studies optimal consumption, investment, and healthcare spending under Epstein-Zin preferences. Given consumption and healthcare spending plans, Epstein-Zin utilities are defined over an agent's random lifetime, partially controllable by the agent as healthcare reduces mortality growth. To the best of our knowledge, this is the first time Epstein-Zin utilities are formulated on a controllable random horizon, via an infinite-horizon backward stochastic differential equation with superlinear growth. A new comparison result is established for the uniqueness of associated utility value processes. In a Black-Scholes market, the stochastic control problem is solved through the related Hamilton-Jacobi-Bellman (HJB) equation. The verification argument features a delicate containment of the growth of the controlled morality process, which is unique to our framework, relying on a combination of probabilistic arguments and analysis of the HJB equation. In contrast to prior work under time-separable utilities, Epstein-Zin preferences facilitate calibration. The model-generated mortality closely approximates actual mortality data in the US and UK; moreover, the efficacy of healthcare can be calibrated and compared between the two countries.

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.

On large time scales the progress of human technology follows an exponential growth trend that is termed accelerating change. The exponential growth trend is commonly considered to be the amalgamated effect of consecutive technology revolutions – where the progress carried in by each technology revolution follows an S-curve, and where the aging of each technology revolution drives humanity to push for the next technology revolution. Thus, as a collective, mankind is the ‘intelligent designer’ of accelerating change. In this paper we establish that the exponential growth trend – and only this trend – emerges universally, on large time scales, from systems that combine together two elements: randomness and amalgamation. Hence, the universal generation of accelerating change can be attained by systems with no ‘intelligent designer’.

Gompertz empirical law of mortality is often used in practical research to parametrize survival fraction as a function of age with the help of just two quantities: the Initial Mortality Rate (IMR) and the Gompertz exponent, inversely proportional to the Mortality Rate Doubling Time (MRDT). The IMR is often found to be inversely related to the Gompertz exponent, which is the dependence commonly referred to as Strehler-Mildvan (SM) correlation. In this paper, we address fundamental uncertainties of the Gompertz parameters inference from experimental Kaplan-Meier plots and show, that a 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. Therefore, an analysis of consequent repeats of the same experiments in the same biological conditions yields the whole degenerate manifold of possible Gompertz parameters. We find that whenever the average lifespan of species greatly exceeds MRDT, small random variations in the survival records produce large deviations in the identified Gompertz parameters along the line, corresponding to the set of all possible IMR and MRDT values, roughly compatible with the properly determined value of average lifespan in experiment. The best fit parameters in this case turn out to be related by a form of SM correlation. Therefore, we have to conclude that the combined property, such as the average lifespan in the group, rather than IMR and MRDT values separately, may often only be reliably determined via experiments, even in a perfectly homogeneous animal cohort due to its finite size and/or low age-sampling frequency, typical for modern high-throughput settings. We support our findings with careful analysis of experimental survival records obtained in cohorts ofC. elegansof different sizes, in control groups and under the influence of experimental therapies or environmental conditions. We argue that since, SM correlation may show up as a consequence of the fitting degeneracy, its appearance is not limited to homogeneous cohorts. In fact, the problem persists even beyond the simple Gompertz mortality law. We show that the same degeneracy occurs exactly in the same way, if a more advanced Gompertz-Makeham aging model is employed to improve the modeling. We explain how SM type of relation between the demographic parameters may still be observed even in extremely large cohorts with immense statistical power, such as in human census datasets, provided that systematic historical changes are weak in nature and lead to a gradual change in the mean lifespan.

We provide a prediction of the corona virus flu propagation in selected European countries. This prediction is based on estimating sub-exponential growth rates with linear regressions (using Gompertz' law).

This chapter develops a methodology for valuing simple cash-flow streams that last a lifetime, which are part of most Defined Benefit (DB) pensions. The focus is on the longevity-contingent building blocks of: (1) immediate, (2) temporary, and (3) deferred income annuities. The chapter begins with a discussion of the value of a longevity-contingent claim and how it differs from the market price versus the manufacturing cost of the product. The algorithms and user-defined R functions are mostly based on the Gompertz law of mortality, although a number of alternative continuous and discrete mortality models are discussed as well. The chapter concludes with a mathematical derivation and implementation of a closed-form expression for the Gompertz Annuity Valuation Model.

Continuing from where the prior Chap. 7 left off, this chapter explains how to construct and work with (a.k.a. cook) the remaining lifetime random variable: Tx. The approach to lifetime randomness is based on the underlying mortality hazard rate λx, which is the continuous-time (and probabilistic) analog of the 1-year death rate qx. This chapter models and constructs Tx variables for a variety of given mortality hazard rates λx. This then sets the stage for the main intellectual objective, which is to introduce (and justify) the Benjamin Gompertz law of mortality. That important and famous law is experienced via a number of simulation exercises and experiments in R. The Gompertz model is the computational backbone for many of the subsequent computations (and recipes) in the book.

Background Clinical studies often track dose-response curves of subjects over time. One can easily model the dose-response curve at each time point with Hill equation, but such a model fails to capture the temporal evolution of the curves. On the other hand, one can use Gompertz equation to model the temporal behaviors at each dose without capturing the evolution of time curves across dosage. Results In this article, we propose a parametric model for dose-time responses that follows Gompertz law in time and Hill equation across dose approximately. We derive a recursion relation for dose-response curves over time capturing the temporal evolution and then specify a regression model connecting the parameters controlling the dose-time responses with individual level proteomic data. The resultant joint model allows us to predict the dose-response curves over time for new individuals. Conclusion We have compared the efficacy of our proposed Recursive Hybrid model with individual dose-response predictive models at desired time points. We note that our proposed model exhibits a superior performance compared to the individual ones for both synthetic data and actual pharmacological data. For the desired dose-time varying genetic characterization and drug response values, we have used the HMS-LINCS database and demonstrated the effectiveness of our model for all available anticancer compounds. Electronic supplementary material The online version of this article (10.1186/s12859-019-2831-4) contains supplementary material, which is available to authorized users.

Bacteria follow the human aging law of exponential mortality, with an aging rate tuned by feast-or-famine life history., Natural selection is thought to shape the evolution of aging patterns, although how life-history trajectories orchestrate the inherently stochastic processes associated with aging is unclear. Tracking clonal growth-arrested Escherichia coli cohorts in an homogeneous environment at single-cell resolution, we demonstrate that the Gompertz law of exponential mortality characterizes bacterial lifespan distributions. By disentangling the rate of aging from age-independent components of longevity, we find that increasing cellular maintenance through the general stress pathway reduces the aging rate and rescales the lifespan distribution at the expense of growth. This trade-off between aging and growth underpins the evolutionary tuning of the general stress response pathway in adaptation to the organism’s feast-or-famine lifestyle. It is thus necessary to involve both natural selection and stochastic physiology to explain aging patterns.

Ageing is considered as a snowballing phenotype of the accumulation of damaged dysfunctional or toxic proteins and silent mutations (polymorphisms) that sensitize relevant proteins to oxidative damage as inborn predispositions to age-related diseases. Ageing is not a disease, but it causes (or shares common cause with) age-related diseases as suggested by similar slopes of age-related increase in the incidence of diseases and death. Studies of robust and more standard species revealed that dysfunctional oxidatively damaged proteins are the root cause of radiation-induced morbidity and mortality. Oxidized proteins accumulate with age and cause reversible ageing-like phenotypes with some irreversible consequences (e.g. mutations). Here, we observe in yeast that aggregation rate of damaged proteins follows the Gompertz law of mortality and review arguments for a causal relationship between oxidative protein damage, ageing and disease. Aerobes evolved proteomes remarkably resistant to oxidative damage, but imperfectly folded proteins become sensitive to oxidation. We show that α-synuclein mutations that predispose to early-onset Parkinson's disease bestow an increased intrinsic sensitivity of α-synuclein to in vitro oxidation. Considering how initially silent protein polymorphism becomes phenotypic while causing age-related diseases and how protein damage leads to genome alterations inspires a vision of predictive diagnostic, prognostic, prevention and treatment of degenerative diseases.

Tumor regrowth and heterogeneity are important clinical parameters during radiotherapy, and the probability of treatment benefit critically depends on the tumor progression pattern in the interval between the fractional irradiation treatments. We propose an analytic, easy-to-use method to take into account clonal subpopulations with different specific growth rates and radiation resistances. The different strain regrowth effects, as described by Gompertz law, require a dose-boost to reproduce the survival probability of the corresponding homogeneous system and for uniform irradiation. However, the estimate of the survival fraction for a tumor with a hypoxic subpopulation is more reliable when there is a slow specific regrowth rate and when the dependence on the oxygen enhancement ratio of radiotherapy is consistently taken into account. The approach is discussed for non-linear two-population dynamics for breast cancer and can be easily generalized to a larger number of components and different tumor phenotypes.

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.

Purpose The purpose of this paper is to present a clear path to securitize the longevity risk with two distinct swaps in order to inspire a new Chinese life market. Design/methodology/approach Studies on longevity risk securitization consist of three aspects, respectively, instrument design, pricing methodology and mortality projection. The swaps designed are referenced, respectively, to vanilla and complex survivor swaps (Dowd et al., 2006; Lin and Cox, 2005). Methods applied are RHH model and Gompertz law for mortality projection, as well as two-factor Wang transformation for pricing. Findings This paper figures out the market price of risk in Chinese annuity market, checks for the sensitivity of the price to parameters and tests the hedging effects by Monte Carlo simulation. Originality/value Based on the theoretical and numerical results, this paper suggests an effective way to possibly witness the birth of New Life Market in China.

Analysis of the age-specific mortality dynamics in the male imago of the Drosophila melanogaster Canton-S strain demonstrated pronounced deviations from the Gompertz law. Four successive observations carried out in large drosophila cohorts made it possible to determine that the survival curves of the objects within the cohort lifetime consist of five discrete regions (phases) divided by sharp inflections. All of the indicated regions are linear. A rigid genetic determination for the disclosed character of the investigated curves was hypothesized.

This paper solves the problem of optimal dynamic investment, consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz law and investment opportunities are constant. Healthcare slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Healthcare spending steadily increases with age, both in absolute terms and relative to total spending. The optimal stochastic control problem reduces to a nonlinear ordinary differential equation with a unique solution, which has an explicit expression in the old-age limit. The main results are obtained through a novel version of Perron’s method.

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.

Clinical studies often track dose-response curves of subjects over time. One can easily model dose-response curve at each time point with Hill equation, but such a model fails to capture the temporal evolution of curves. On the other hand, one can use Gompertz equation to model the dose-time curves at each time point without capturing the evolution of time curves across dosage. In this article, we propose a parametric model for dose-time responses that follows Gompertz law in time and approximately follows Hill equation across dose. We derive a recursion relation for dose-response curves over time capturing the temporal evolution. We then specify a regression model connecting the parameters controlling the dose-time responses with individual level proteomic data. The resultant joint model allows us to predict the dose-response curves over time for new individuals. We illustrate the superior performance of our proposed model as compared to the individual models using data from the HMS-LINCS database.

Natural selection has long been hypothesised to shape ageing patterns, but whether and how ageing contributes to life-history evolution remains elusive. The complexity of various ageing-associated molecular mechanisms and their inherent stochasticity hinder reductionist approaches to the understanding of functional senescence,i.e. reduced fecundity and increased mortality. Recent bio-demographic work demonstrated that high-precision statistics of life-history traits such as mortality rates could be used phenomenologically to understand the ageing process. We adopted this approach to study cellular senescence in growth-arrestedE. colicells, where damages to functional macromolecules are no longer diluted by fastde novobiosynthesis. We acquired high-quality longitudinal physiological and life history data of large environmentally controlled clonalE. colipopulations at single-cell resolution, using custom-designed microfluidic devices coupled to time-lapse microscopy. We show thatE. colilifespan distributions follow the Gompertz law of mortality, a century-old actuarial observation of human populations, despite developmental, cellular and genetic differences between bacteria and metazoan organisms. Measuring the shape of the hazard functions allowed us to disentangle quantitatively the demographic effects of ageing, which accumulate with time, from age-independent genetic longevity-modulating interventions. A pathway controlling cellular maintenance, the general stress response, not only promotes longevity but also temporally scales the whole distribution by reducing ageing rate. We further show thatE. coli, constrained by the amount of total biosynthesis, adapt to their natural feast-or-famine lifestyle by modulating the amount of maintenance investment, rendering ageing rate a highly evolvable life-history trait.

In a number of earlier chapters I made reference to the Gompertz-Makeham (or just plain Gompertz) Law of Mortality. In particular, I used this law of mortality in Chapter 7 when I introduced and described Jared's tontine payout rate and in Chapter 2 when I discussed the probability density function of the “present value” of the tontine versus annuity cash-flow payout. Well, in this appendix I offer a brief explanation of this well-known law, as well as the analytic representation I used plus a bit of information about the person after whom it is named. For those interested the technical details, see Milevsky (2012), which is a popular book covering the most important equations in the field of retirement income planning. For starters, age-dependent mortality rates for adults – for example, those displayed in Table 5.3 but continued to older ages – seem rather arbitrary at first, but there is actually an underlying pattern to them. In particular, for people between the ages of twenty and ninety – mortality rates not only increase consistently every year with age, they actually increase by approximately 9% every year. In mathematical symbols, if the starting mortality or death rate per year was q percent at age y, (for example, q = 2% at age fifty) then it is q (1 + z ) percent in year ( y + 1) and then q (1 + z ) 2 percent in year ( y + 2), and then q(1 + z ) 3 percent in year ( y + 3), and so on, where z is approximately 9%. Human adult mortality rates – regardless of what particular group of humans or population you select and whether it is in the seventeenth or nineteenth or twenty-first century – are an exponentially increasing function of age with a (growth rate) parameter of 9%. What this also means (mathematically) is that if you take the logarithms of these annual mortality rates denoted by q , they can be approximated quite nicely by a straight line and determined by a slope and an intercept.

In this article, we propose a three-factor model for mortality modeling in which the dynamic of the entire term structure of mortality rates can be expressed in closed form as a function of three variables x, t, and y. Due to this feature, we are able to project mortality rates across age (x), across time (t), and for y years (y ⩾ 1) after t. Our proposal differs from most existing models where only the one-year mortality rate is considered (y = 1). The model is characterized by three parameters that are calibrated yearly. We describe the stochastic dynamic of the three factors with correlated autoregressive processes. We generate stochastic scenarios accounting for the historical mortality trend in a consistent manner with the Gompertz law. Using population mortality data for Italy, the U.S., and the U.K., the model’s forecasting capability is assessed, and a comparative analysis with other models is provided.

The aim of this paper is to provide a fractional generalization of the Gompertz law via a Caputo-like definition of fractional derivative of a function with respect to another function. In particular, we observe that the model presented appears to be substantially different from the other attempt of fractional modifications of this model, since the fractional nature is carried along by the general solution even in its asymptotic behavior for long times. We then validate the presented model by employing it as a reference frame to model three biological systems of peculiar interest for biophysics and environmental engineering, namely: dark fermentation, photofermentation and microalgae biomass growth., Comment: 14 pages, 5 figures, matches published version in CNSNS

In US states with small subpopulations, the observed mortality rates are often zero, particularly among young ages. Because in life tables, death rates are reported mostly on a log scale, zero mortality rates are problematic. To overcome the observed zero death rates problem, appropriate probability models are used. Using these models, observed zero mortality rates are replaced by the corresponding expected values. This enables logarithmic transformations and, in some cases, the fitting of the eight-parameter Heligman–Pollard model to produce mortality estimates for ages 0–130 years, a procedure illustrated in terms of mortality data from several states. Copyright © 2014 John Wiley & Sons, Ltd.

Recent findings using records from the U.S. Social Security Administration’s Death Master File discredit the late life mortality deceleration theory and confirm that life expectancies follow the Gompertz law not only until the age of 80, but for many years after. The authors show how these findings have major implications on the valuation of senior life settlements and securities backed by life settlements. They illustrate the sensitivity of valuation to the incorporation of the results of recent research regarding longevity risk.

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.

We believe that cytogerontological models, such as the Hayflick model, though very useful for experimental gerontology, are based only on certain correlations and do not directly apply to the gist of the aging process. Thus, the Hayflick limit concept cannot explain why we age, whereas our “stationary phase aging” model appears to be a “gist model,” since it is based on the hypothesis that the main cause of both various “age-related” changes in stationary cell cultures and similar changes in the cells of aging multicellular organism is the restriction of cell proliferation. The model is applicable to experiments on a wide variety of cultured cells, including normal and transformed animal and human cells, plant cells, bacteria, yeasts, mycoplasmas, etc. The results of relevant studies show that cells in this model die out in accordance with the Gompertz law, which describes exponential increase of the death probability with time. Therefore, the “stationary phase aging” model may prove effective in testing of various geroprotectors (anti-aging factors) and geropromoters (pro-aging factors) in cytogerontological experiments. It should be emphasized, however, that even the results of such experiments do not always agree with the data obtained in vivo and therefore cannot be regarded as final but should be verified in studies on laboratory animals and in clinical trials (provided this complies with ethical principles of human subject research).

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.

Milevsky and Huang (2011) investigated the optimal retirement spending policy for a utility-maximizing retiree facing a stochastic lifetime but assuming deterministic investment returns. They solved the problem using techniques from the calculus of variations and derived analytic expressions for the optimal spending rate and wealth depletion time under the Gompertz law of mortality. Of course, in the real world financial returns are stochastic as well as lifetimes, raising the question of whether their qualitative insights and approximations are generalizable or practical. We solve the retirement income problem when investment returns are indeed stochastic using numerical PDE methods, assuming the principles of stochastic control theory and dynamic programming. But then -- and this is key -- we compare the proper optimal spending rates to the analytic approach presented in Milevsky and Huang (2011) by updating the portfolio wealth inputs to current market values. Our main practical conclusion is that this simplistic approximation when calibrated properly and frequently can indeed be used as an accurate guide for rational retirement spending policy. As a by-product of our PDE-based methodology, our results indicate that even though the wealth depletion time is no longer a certainty under stochastic returns, the expected age at which liquid wealth is exhausted (i.) takes place well before the maximum lifetime and (ii.) is also well approximated by our analytical solution.

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.

In this paper, the uncontrolled environmental factors are perturbed into the intrinsic growth rate factor of deterministic equations of the growth process. The growth process under two different laws which are Verhults and Gompertz’s law are considered, thus leading to stochastic delay differential equations (SDDEs) of logistic and Gompertzian, respectively. Gompertzian deterministic model has been proved to fit well the clinical data of cancerous growth, however the performance of stochastic model towards clinical data is yet to be confirmed. The prediction quality of logistic and Gompertzian SDDEs are evaluating by comparing the simulated results with the clinical data of cervical cancer growth. The parameter estimation of stochastic models is computed by using simulated maximum likelihood method. We adopt 4-stage stochastic Runge-Kutta to simulate the solution of stochastic models.

A new paper claims that there may be no such thing as a maximum life span for naked mole rats. As these burrowing, sausage-shaped rodents grow older, their risk of dying does not seem to increase, defying a mathematical equation called the Gompertz law that describes aging in all known mammalian species. The new finding comes from the lab of Rochelle Buffenstein at Google biotech spinoff Calico in San Francisco, California, who has studied the animals since 1980. Other researchers say it9s too early to say that naked mole rats don9t age, but they agree something very unusual happens in the animals and welcome the treasure trove of life history data revealed in the paper.

Competition is intense among rival technologies, and success depends on predicting their future trajectory of performance. To resolve this challenge, managers often follow popular heuristics, generalizations, or “laws” such as Moore's law. We propose a model, Step And Wait (SAW), for predicting the path of technological innovation, and we compare its performance against eight models for 25 technologies and 804 technologies-years across six markets. The estimates of the model provide four important results. First, Moore's law and Kryder's law do not generalize across markets; neither holds for all technologies even in a single market. Second, SAW produces superior predictions over traditional methods, such as the Bass model or Gompertz law, and can form predictions for a completely new technology by incorporating information from other categories on time-varying covariates. Third, analysis of the model parameters suggests that (i) recent technologies improve at a faster rate than old technologies; (ii) as the number of competitors increases, performance improves in smaller steps and longer waits; (iii) later entrants and technologies that have a number of prior steps tend to have smaller steps and shorter waits; but (iv) technologies with a long average wait time continue to have large steps. Fourth, technologies cluster in their performance by market.

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.

This work presents an optimal harvesting problem associated with a single-species population governed by Gompertz law in a seasonally fluctuating environment. The influence of environmental fluctuation is accommodated by choosing the coefficients in the differential equation to be periodic functions with the same period and restriction on the harvesting effort is accommodated by considering binding constraints on the control variable. Hence, a linear optimal control problem has been considered where the state dynamics is governed by Gompertz equation and the control variable is subject to the binding constraints. With the help of maximum principle and the concept of blocked intervals, an optimal periodic solution has been obtained which is followed by the construction of optimal solution using the theory of most rapid approach. Important results of the study are demonstrated through numerical simulations.

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.

Based upon the deterministic Gompertz law of cell growth, we have proposed a stochastic model of tumour cell growth, in which the size of the tumour cells is bounded. The model takes account of both cell fission (which is an ‘action at a distance’ effect) and mortality too. Accordingly, the density function of the size of the tumour cells obeys a functional Fokker–Planck Equation (FPE) associated with the bounded stochastic process. We apply the Lie-algebraic method to derive the exact analytical solution via an iterative approach. It is found that the density function exhibits an interesting kink-like structure generated by cell fission as time evolves.

Summary We consider two related aspects of the study of old-age mortality. One is the estimation of a parameterized hazard function from grouped data, and the other is its possible deceleration at extreme old age owing to heterogeneity described by a mixture of distinct sub-populations. The first is treated by half of a logistic transform, which is known to be free of discretization bias at older ages, and also preserves the increasing slope of the log hazard in the Gompertz case. It is assumed that data are available in the form published by official statistical agencies, that is, as aggregated frequencies in discrete time. Local polynomial modelling and weighted least squares are applied to cause-of-death mortality counts. The second, related, problem is to discover what conditions are necessary for population mortality to exhibit deceleration for a mixture of Gompertz sub-populations. The general problem remains open but, in the case of three groups, we demonstrate that heterogeneity may be such that it is possible for a population to show decelerating mortality and then return to a Gompertz-like increase at a later age. This implies that there are situations, depending on the extent of heterogeneity, in which there is at least one age interval in which the hazard function decreases before increasing again.