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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

In this paper we provide an interval of existence of negative critical mortality rate parameters A and α in Gompertz survival model, in the absence of age specific mortality data.

In this paper we have proven uniqueness theorems for estimation of the age-dependent parameter @a in the Gompertz survival model with a mortality deceleration rate, in the absence of age-specific mortality data. We obtained estimations dependent on and uniform in the population size N.

In nematodes, environmental or physiological perturbations alter death's scaling of time. In human cancer, genetic perturbations alter death's curvature of time. Those changes in scale and curvature follow the constraining contours of death's invariant geometry. I show that the constraints arise from a fundamental extension to the theories of randomness, invariance and scale. A generalized Gompertz law follows. The constraints imposed by the invariant Gompertz geometry explain the tendency of perturbations to stretch or bend death's scaling of time. Variability in death rate arises from a combination of constraining universal laws and particular biological processes., Comment: F1000Research 5:2076

Health-care slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. This paper solves the problem of optimal dynamic consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz' law. 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. Health spending steadily increases with age, both in absolute terms and relative to total spending. Differential access to healthcare can account for observed longevity gains across cohorts.

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

In this paper we provide an interval of existence of mortality rate parameters A and @a, and their asymptotic expressions in a Gompertz survival model with mortality deceleration rate, in the absence of age specific mortality data.

The exponential increase with age in mortality rate, the Gompertz law, indicates that the decrease in vitality and viability linked to aging depends on phenomena with exponential or logarithmic dynamics. Gompertz slope (alpha) is assumed to be a measure of aging rate, provided the studied cohort is homogeneous and in a supporting environment. The law provides no clue about the cause of aging, but may be formally correlated with various physical or mathematical functions. A possible correlation between the Ogston-Laurent exclusion equation and human aging is examined. An increase with age of an inert cross-linked insoluble protein network is assumed to result in a logarithmic decrease in water volume available to colloidal macromolecules. In this model, alpha is assumed to be a measure of the rate of accumulation of the polypeptide network.

Aging and mortality is usually modeled by the Gompertz-Makeham distribution, where the mortality rate accelerates with age in adult humans. The resulting parameters are interpreted as the frailty and decrease in vitality with age. This fits well to life data from 'westernized' societies, where the data are accurate, of high resolution, and show the effects of high quality post-natal care. We show, however, that when the data are of lower resolution, and contain considerable structure in the infant mortality, the fit can be poor. Moreover, the Gompertz-Makeham distribution is consistent with neither the force of natural selection, nor the recently identified 'late life mortality deceleration'. Although actuarial models such as the Heligman-Pollard distribution can, in theory, achieve an improved fit, the lack of a closed form for the survival function makes fitting extremely arduous, and the biological interpretation can be lacking. We show, that a mixture, assigning mortality to exogenous or endogenous causes, using the reduced additive and flexible Weibull distributions, models well human mortality over the entire life span. The components of the mixture are asymptotically consistent with the reliability and biological theories of aging. The relative simplicity of the mixture distribution makes feasible a technique where the curvature functions of the corresponding survival and hazard rate functions are used to identify the beginning and the end of various life phases, such as infant mortality, the end of the force of natural selection, and late life mortality deceleration. We illustrate our results with a comparative analysis of Canadian and Indonesian mortality data.

A new probabilistic model of aging that can be applied to organisms is suggested and analyzed. Organisms are subject to shocks that follow the generalized Polya process (GPP), which has been recently introduced and characterized in the literature. Distinct from the nonhomogeneous Poisson process that has been widely used in applications, the important feature of this process is the dependence of its future behavior on the number of previous events (shocks). The corresponding survival and the mortality rate functions are derived and analyzed. The general approach is used for justification of the Gompertz law of human mortality.

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.

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.

In this paper we investigate the optimal harvesting problems of a single species with Gompertz law of growth. Based on continuous harvesting models, we propose impulsive harvesting models with constant harvest or proportional harvest. By using the discrete dynamical systems determined by the stroboscopic map, we discuss existence, stability and global attractivity of positive periodic solutions, and obtain the maximum sustainable yield and the corresponding optimal population level. At last, we compare the maximum sustainable yield of impulsive harvest with that of continuous harvest, and point out that proportional harvest is superior to constant harvest.

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.

This paper deals with the combined bioeconomic harvesting of two competing fish species, each of which obeys the Gompertz law of growth. The catch-rate functions are chosen so as to reflect saturation effects with respect to stock abundance as well as harvesting effort. The stability of the dynamical system is discussed and the existence of a bionomic equilibrium is examined. The optimal harvest policy is studied with the help of Pontryagin's maimum principle. The results are illustrated with the help of a numerical example.

A square lattice is introduced into the Penna model for biological aging in order to study the evolution of diploid sexual populations under certain conditions when one single locus in the individual's genome is considered as identifier of species. The simulation results show, after several generations, the flourishing and coexistence of two separate species in the same environment, i.e., one original species splits up into two on the same territory (sympatric speciation). As well, the mortalities obtained are in a good agreement with the Gompertz law of exponential increase of mortality with age., 5 pages including 3 encapsulated postscript (*.eps) figures; To appear in European Physical Journal B

A bioeconomic model is developed for the selective harvesting of a single species, inshore–offshore fishery, assuming that the growth of the species is governed by the Gompertz law. The dynamical system governing the fishery is studied in depth; the local and global stability of its non-trivial steady state are examined. Existence of a bionomic equilibrium is established under different parametric considerations. The optimal harvest policy is discussed by invoking Pontryagin's Maximum Principle. Lastly, the results are illustrated with the help of a numerical example.

The present paper deals with a problem of selective harvesting of two competing fish species in a randomly fluctuating environment. Here species are assumed to have Gompertz law of growth and interaction. The growth parameters of the species are assumed to be perturbed by white noise characterised by a Gaussian distribution with mean zero and unit spectral density. The dynamic behaviour of the stochastic system is studied and the fluctuations in the populations are measured both analytically and numerically by computer simulation. It has been observed that spectral density, denoting the intensity of fluctuation of each population to be an increasing function of each time lag.

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.

The Gompertz law of mortality quantitatively describes the mortality rate of humans and almost all multicellular animals. However, its underlying kinetic mechanism is unclear. The Gompertz law cannot explain the mortality plateau at advanced ages and cannot give an explicit relationship between temperature and mortality. In this study a reaction kinetics model with a time dependent rate coefficient is proposed to describe the survival and senescence processes. A temperature-dependent mortality function was derived. The new mortality function becomes the Gompertz mortality function with the same relationship of parameters prescribed by the Strehler–Mildvan correlation when age is smaller than a characteristic value δ, and reaches the mortality plateau when age is greater than δ. A closed-form analytical expression for describing the relationship of average lifespan with temperature and other equations are derived from the new mortality function. The derived equations can be used to estimate the limit of average lifespan, predict the maximal longevity, calculate the temperature coefficient of lifespan, and explain the tendency of the survival curve. This prediction is consistent with the most recently reported mortality trajectories for single-year birth cohorts. This study suggests that the senescence process results from the imbalance between damaging energy and protecting energy for the critical chemical substance in the body. The rate of senescence of the organism increases while the protecting energy decreases. The mortality plateau is reached when the protecting energy decreases to its minimal levels. The decreasing rate of the protecting energy is temperature-dependent. This study is exploring the connection between the biochemical mechanism and demography., Highlights • A temperature-dependent mortality function is derived. • The function reaches a mortality plateau as age>δ (a life characteristic value). • An expression for the temperature-dependent average lifespan is derived. • The equation for estimating the limit of average lifespan is derived.

A mechanistic mathematical model for radiotherapy is developed to study the effect of tumor repopulation on radiation treatment outcome. The Two-Lesion Kinetic (TLK) model is adopted to describe the interaction between ionizing radiation and malignant cells, and extended in this work to take into account tumor proliferation. The investigated kinetics for tumor cells growth includes exponential, logistic and Gompertz laws, while standard fractionation treatments are considered as case studies. Scaling analysis is used for non-dimensionalizing the system of ordinary differential equations describing the process under investigation. Numerical solution of the proposed model is shown as a function of the dimensionless variables and groups generated by the scaling procedure. The latter allows minimizing the number of parameters needed to describe the radiotherapy process when DNA damage formation and repair, and tumor repopulation phenomena take place simultaneously. The effectiveness in tumor eradication is mapped as a function of two dimensionless groups that take into account the radiation dose rate as well as the DNA lesions repair and malignant cells repopulation rates. The present study illustrates that outcomes of the radiotherapy treatment are strongly affected by the relative rates of phenomena simultaneously occurring. In particular, it appears that in order to achieve an effective treatment by keeping as low as possible the total dose to be administered, the fractionation scheme should be optimized on the base of the ratio between the DNA repair mechanisms and tumor cell growth rates. Model results also show that exponential and logistic kinetics yield similar results in terms of treatments outcomes. By comparison, model simulations with the Gompertz law indicate that repopulation described by this growth kinetics result in a significantly poorer prognosis for tumor eradication than either exponential or logistic models.

This paper deals with the selective harvesting of two species from a food chain model of three species, in which prey and predator obey the Gompertz law of growth. Initially the dynamical behaviour of the system was studied under deterministic environment. In deterministic case, the local stability of the system was also studied; we investigated the condition of global stability and the existence of the bionomic equilibrium was examined. The optimal harvesting policy is studied with the help of Pontryagin’s maximum principle. In the second part of the problem, we investigated the stability condition of the system under stochastic environment. Then a comparison is made between deterministic and stochastic cases.

This paper develops a mathematical model for the nonselective harvesting of a prey-predator system in which both the prey and the predator obey the Gompertz law of growth and some prey avoid predation by hiding. The steady states of the system are determined, and the dynamical behaviour of both species is examined. The possibility of existence of bionomic equilibria is discussed. The optimal harvest policy is formulated and solved as a control problem with the help of Pontryagin's maximal principle. Finally, the results are illustrated with the help of a numerical example.

A specific mortality rate process governed by the non-homogeneous Poisson process of point events is considered and its properties are studied. This process can describe the damage accumulation in organisms experiencing external shocks and define its survival characteristics. It is shown that, although the sample paths of the unconditional mortality rate process are monotonically increasing, the population mortality rate can decrease with age and, under certain assumptions, even tend to zero. The corresponding analysis is the main objective of this paper and it is performed using the derived conditional distributions of relevant random parameters. Several meaningful examples are presented and discussed.

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.

Gompertz law of mortality quantitatively describes the mortality rate of humans and almost all multicellular animals. However, this law was found empirically nearly 200 years ago and its underlying kinetic mechanism is still unclear. Research has shown that the mortality rate does not follow the Gompertz law at very advanced ages and the Gompertz law cannot explain why temperature can affect lifespan. The author proposed a reaction kinetic model with a time dependent rate coefficient to describe the survival and senescence processes. A new mortality rate equation is established. From this new mortality rate equation, we can obtain the Gompertz law of mortality and show that the Gompertz law is only true when age is less than a characteristic value of life, can derive some important equations reported in Nature (Shaw RF, et al. Temperature and life-span in poikilothermous animals. Nature 196:454-457, 1962) and Science (Strehler BL, et al. General theory of mortality and aging. Science 132(3418):14-21, 1960...

Some modifications of the simple asexual Penna model, enriched by epigenetic contributions, are presented. The standard bit-string Penna model of biological aging and population evolution is based on an inherited DNA structure which defines the future life of a newly born individuals, when genes are activated by the biological clock, and the predefined genetic death is fully controlled by the number of defected genes. Epigenomes allow to introduce additional mechanism of gene activation or silencing without affecting the DNA genome itself. It may be either inherited or may reflect external, environmental factors. In the presented model, information read from the introduced epigenome may alter gene expression that may be stopped or re-activated. We concentrate on the influence of epigenetics on the age \(a\) distribution of genetic mortality \(m(a)\). Changes in \(m(a)\) are strong for the case of inherited epigenetic contribution with nearly perfect inheritance and ‘positive’ epigenome that partly ignores the ‘bad’ mutations. We conclude that the epigenetic contribution may influence population structure \(m(a)\) and could be, at least partly, responsible for deviation of \(m(a)\) distribution from the Gompertz law. In short, we claim that proposed epigenetic contribution may be seen as a candidate for possible explanation of observed deviation from the Gompertz law, also among senior members of society. A very simple model was used in this paper and many crucial mechanisms of biological aging were omitted. Therefore, further work based on a more realistic models is necessary.

We propose a stochastic model of aging to explain deviations from exponential growth in mortality rates commonly observed in empirical studies. Mortality rate plateaus are explained as a generic consequence of considering death in terms of first passage times for processes undergoing a random walk with drift. Simulations of populations with age-dependent distributions of viabilities agree with a wide array of experimental results. The influence of cohort size is well accounted for by the stochastic nature of the model.

There is a widely observed non-linear kinetics in the ageing of biological systems, which is characterised by three successive stages, (1) the ageing rate is firstly high, but decreases quickly to a minimum, from which (2) it remains nearly constant during the major part of the process until (3) it starts increasing again up to the final collapse of the system. Such kinetics are also encountered in the ageing of inert systems. It is shown that a model useful for the follow-up of operating inert systems allows to find back typical curves and laws related to the ageing of biological systems (mortality rate curves, survival curves, growth curves, Gompertz law, …). In this model, ageing is seen as a multifactorial process. The classical concepts of lifespan, longevity and life expectancy are given new light using the model, which also gives clues to explain both the discrepancy in the age of death of individuals in a given population and the wide range of lifespans of species encountered in nature. Finally, the model shows in which directions accelerated senescence testing protocols should be orientated for a better understanding of the underlying phenomena and for life prediction purposes.

The Gompertz transform of the distribution function for the age at death expresses mortality in a form R = R0ealpha t where R0 is themortality at time zero and α is the rate of increase ofmortality, frequently taken as the rate of ageing. The slope ofthe line α is frequently used as a measure of the rate of ageing.It is argued that it is incorrect to use α in this way. Tosupport this contention, a paradox is produced whereby selectionfor longevity increases α, which could lead to the absurdconclusion that selection for longevity increases the rate ofageing.

A new formulation of Gompertz' law of mortality is proposed. Explicit formulas for moment generating function and moments of this formulation are derived. The new formulation is applied to the theory of heterogeneous populations and formulas for stop-loss transformations are derived.

A single species fishery model has been developed using the Gompertz law of population growth and the CPUE (Catch-per-unit-effort) hypothesis. The dynamical and the bionomic steady states were determined and their natures were examined from the biological as well as economic view points. The optimal harvest policy is discussed by taking the fishing effort as a dynamic control variable. The results are compared with those of the Schaefer model [10].

Despite considerable progress in understanding tumour development, the law of growth for human tumours is still a matter of some dispute. In this study, we used large-scale mammography screening trial data to deduce the growth law of primary breast cancer. We compared the empirical tumour population size distributions of primary breast cancer inferred from these data to the distributions that correspond to various possible theoretical growth functions. From this, we showed that the data are inconsistent with the exponential, logistic and Gompertz laws, but support power law growth (exponent approximately 0.5). This law indicates unbounded growth but with slowing mass-specific growth rate and doubling time. In the clinical size ranges, it implies a greater decline in the mass-specific growth rate than would be predicted by the Gompertz law using the accepted parameters. This suggests that large tumours would be less sensitive to cycle-specific therapies, and be better treated first by non-cell cycle-specific agents. We discussed the use of our study to estimate the sensitivity of mammography for the detection of small tumours. For example, we estimated that mammography is about 30% less sensitive in the detection of tumours in the 1 to 1.5-cm range than it is in detecting larger tumours.

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 Guidelines recommend incorporating life expectancy (LE) into clinical decision-making for preventive interventions such as cancer screening. Previous research focused on mortality risk (e.g. 28% at 4 years) which is more difficult to interpret than LE (e.g. 7.3 years) for both patients and clinicians. Our objective was to utilize the Gompertz Law of Human Mortality which states that mortality risk doubles in a fixed time interval to transform the Lee mortality index into a LE calculator. Methods We examined community-dwelling older adults age 50 and over enrolled in the nationally representative 1998 wave of the Health and Retirement Study or HRS (response rate 81%), dividing study respondents into development (n = 11701) and validation (n = 8009) cohorts. In the development cohort, we fit proportional hazards Gompertz survival functions for each of the risk groups defined by the Lee mortality index. We validated our LE estimates by comparing our predicted LE with observed survival in the HRS validation cohort and an external validation cohort from the 2004 wave of the English Longitudinal Study on Ageing or ELSA (n = 7042). Results The ELSA cohort had a lower 8-year mortality risk (14%) compared to our HRS development (23%) and validation cohorts (25%). Our model had good discrimination in the validation cohorts (Harrell’s c 0.78 in HRS and 0.80 in the ELSA). Our predicted LE’s were similar to observed survival in the HRS validation cohort without evidence of miscalibration (Hosmer-Lemeshow, p = 0.2 at 8 years). However, our predicted LE’s were longer than observed survival in the ELSA cohort with evidence of miscalibration (Hosmer-Lemeshow, p

Constraints can affect dramatically the behavior of diffusion processes. Recently, we analyzed a natural and a technological system and reported that they perform diffusion-like discrete steps displaying a peculiar constraint, whereby the increments of the diffusing variable are subject to configuration-dependent bounds. This work explores theoretically some of the revealing landmarks of such phenomenology, termed "soft bound". At long times, the system reaches a steady state irreversibly (i.e., violating detailed balance), characterized by a skewed "shoulder" in the density distribution, and by a net local probability flux, which has entropic origin. The largest point in the support of the distribution follows a saturating dynamics, expressed by the Gompertz law, in line with empirical observations. Finally, we propose a generic allometric scaling for the origin of soft bounds. These findings shed light on the impact on a system of such "scaling" constraint and on its possible generating mechanisms., Comment: 9 pages, 6 color figures

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 Gompertz law of dependence of human mortality rate on age is derived from a simple model of death as a result of an exponentially rare escape of abnormal cells from immunological response.

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 "disposable soma" theory for the evolution of senescence suggests that senescence arises from an optimal balancing of resources between reproduction and somatic repair. Dynamic programming models are constructed and analyzed to determine the optimal relationship between reproduction, diversion of resources from repair, and added senescent mortality. Of particular interest is the relationship between the repair-reproduction trade-off and the form of the mortality-rate-versus-age curve predicted. The models analyzed in the greatest detail assume that the relationship between reproduction and added senescent mortality does not change with age. These suggest that mortality should increase at an increasing rate with age, but may approach a linear rate as mortality becomes very high. General results are derived for the shape of the mortality curves early and late in the senescing part of the life span, and mortality curves for specific trade-off functions are illustrated. An exponential increase in death rate with age (Gompertz' Law) corresponds to only one of many possible relationships between reproduction and aging. The "Law" is unlikely to hold generally if the disposable soma theory accounts for a large fraction of the observed senescent increase in mortality with age. However, support for the generality of Gompertz' Law is weak, and other theories have not produced an evolutionary explanation for the law. The disposable soma theory is consistent with some of the exceptions to Gompertz' Law that have been observed.

The analysis of 26 longevity curves of different populations of inbred (Fischer 344) and outbred (Sprague-Dawley) rats highlighted a remarkable between-populations variability in survival parameters. This variability is independent of the breeding characteristics of the strain. The two strains differed in the slope of the survival curves, with Fischer 344 rats showing a higher survival over the second year of life as well as a lower interindividual variability. A model-free approach based on principal component analysis allowed us to quantify these differences and to highlight some limitations of the classical Gompertzian approach.

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.

Dynamic characteristics of mortality experience are intensely studied in demography and in experimental gerontology. The classic Gompertz-Makeham function is also a significant starting point of numerous investigations in this topic. By comparative analysis of human vital statistics, one can observe characteristic secular and topological variations of the Gompertz parameters. Some of these findings are reviewed briefly in the article. Some other related questions are also touched upon. For example, the concept of the maximum life span is criticised. A thorough overview of literature data makes it clear that the general male mortality excess is by far not so unambiguous as is widely supposed. Surprisingly simple stochastic models, published by one of us in the recent past, can serve as theoretical background for the Gompertz law. Further studies would be necessary to check the biological relevance of these models.

Differential female longevity is so far unexplained in evolutionary terms. The theory of evolutionarily necessary aging which goes back to Wallace appears to be up to the task. In this theory, aging minimizes competition between forebear and offspring. The aging equation which is implicit contains the well-known empirical Gompertz law as a special case. Moreover, its parameters are automatically sex-specific. It is shown that the slower aging of the female members of two species of mammals, humans and sperm whales, can be ‘predicted’ on the basis of this equation. A prediction of effective human monogamy under archaic conditions is obtained as a corollary. The analogous if opposite prediction for sperm whales (strong promiscuity) is empirically testable.

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.