Abstract:
Recent discoveries show steady improvements in life expectancy during modern decades. Does this support that humans continue to live longer in future? We recently put forward the maximum survival tendency, as found in survival curves of industrialized countries, which is described by extended Weibull model with agedependent stretched exponent. The maximum survival ten
dency suggests that human survival dynamics may possess its intrinsic limit, beyond which survival is inevitably forbidden. Based on such tendency, we develop the model and explore the patterns in the maximum lifespan limits from industrialized countries during recent three decades. This analysis strategy is simple and useful to interpret the complicated human survival dynamics.

Abstract:
Recent discoveries show steady improvements in life expectancy during modern decades. Does this support that humans continue to live longer in future? We recently put forward the maximum survival tendency, as found in survival curves of industrialized countries, which is described by extended Weibull model with agedependent stretched exponent. The maximum survival ten dency suggests that human survival dynamics may possess its intrinsic limit, beyond which survival is inevitably forbidden. Based on such tendency, we develop the model and explore the patterns in the maximum lifespan limits from industrialized countries during recent three decades. This analysis strategy is simple and useful to interpret the complicated human survival dynamics.

Abstract:
General functions for human survival and mortality may support a possibility of general mechanisms in human ageing. We discovered that the survival and mortality curves could be described very simply and accurately by the Weibull survival function with age-dependent shape parameter. The age-dependence of shape parameter determines the shape of the survival and mortality curves and tells the nature of the ageing rate. Especially, the progression of shape parameter with age may be explained by the increase of interaction among vital processes or the evolution of susceptibility to faults with age. Age-related diseases may be attributed to the evolution of susceptibility to faults with age.

Abstract:
Trends in human longevity are puzzling, especially when considering the limits of human longevity. Partially, the conflicting assertions are based upon demographic evidence and the interpretation of survival and mortality curves using the Gompertz model and the Weibull model; these models are sometimes considered to be incomplete in describing the entire curves. In this paper a new model is proposed to take the place of the traditional models. We directly analysed the rectangularity (the parts of the curves being shaped like a rectangle) of survival curves for 17 countries and for 1876-2001 in Switzerland (it being one of the longest-lived countries) with a new model. This model is derived from the Weibull survival function and is simply described by two parameters, in which the shape parameter indicates 'rectangularity' and characteristic life indicates the duration for survival to be 'exp(-1)'. The shape parameter is essentially a function of age and it distinguishes humans from technical devices. We find that although characteristic life has increased up to the present time, the slope of the shape parameter for middle age has been saturated in recent decades and that the rectangularity above characteristic life has been suppressed, suggesting there are ultimate limits to human longevity. The new model and subsequent findings will contribute greatly to the interpretation and comprehension of our knowledge on the human ageing processes.

Abstract:
We wish to verify that the mortality deceleration (or decrease) is a consequence of the bending of the shape parameter at old ages. This investigation is based upon the Weon model (the Weibull model with an age-dependent shape parameter) for human survival and mortality curves. According to the Weon model, we are well able to describe the mortality decrease after the mortality plateau, including the mortality deceleration. Furthermore, we are able to simply define the mathematical limit of longevity by the mortality decrease. From the demographic analysis of the historical trends in Switzerland (1876-2001) and Sweden (1861-2001), and the most recent trends in the other eleven developed countries (1996-2001), we confirm that the bending of the shape parameter after characteristic life is correlated with the mortality deceleration (or decrease). As a consequence, this bending of the shape parameters and the mortality deceleration is associated with the mathematical limit on longevity. These results suggest that the mathematical limit of longevity can be induced by the mortality deceleration (or decrease) in nature. These findings will give us a breakthrough for studying the mortality dynamics at the highest ages.

Abstract:
In recent we introduced, developed and established a new concept, model, methodology and principle for studying human longevity in terms of demographic basis. We call the new model the "Weon model", which is a general model modified from the Weibull model with an age-dependent shape parameter to describe human survival and mortality curves. We demonstrate the application of the Weon model to the mortality dynamics and the mathematical limit of longevity (the mortality rate to be mathematically zero, implying a maximum longevity) in the Section I. The mathematical limit of longevity can be induced by the mortality dynamics in nature. As a result, we put forward the complementarity principle, which explains the recent paradoxical trends that the mathematical limit decreases as the longevity increases, in the Section II. Our findings suggest that the human longevity can be limited by the complementarity principle.

Abstract:
A fundamental question in aging research concerns the demographic trajectories at the highest ages, especially for supercentenarians (persons aged 110 or more). We wish to demonstrate that the Weon model enables scientists to describe the demographic trajectories for supercentenarians. We evaluate the average survival data from the modern eight countries and the valid and complete data for supercentenarians from the International Database on Longevity (Robine and Vaupel, (2002) North American Actuarial Journal 6, 54-63). The results suggest that the Weon model predicts the maximum longevity to exist around ages 120-130, which indicates that there is an intrinsic limit to human longevity, and that the Weon model allows the best possible description of the demographic trajectories for supercentenarians.

Abstract:
Accurate demographic functions help scientists define and understand longevity. We summarize a new demographic model, the Weon model, and show the application to the demographic data for Switzerland (1876-2002). Particularly, the Weon model simply defines the maximum longevity, which is induced in nature by the mortality dynamics. In this study, we reconsider the definition of the maximum longevity and the effectiveness for longevity by the combined effect of the survival and mortality functions. The results suggest that the mortality function should be zero at the maximum longevity, since the density function is zero but the survival function is not zero. Furthermore, the effectiveness for longevity can be maximized at the characteristic life by the complementarity between the survival and mortality functions, which suggests that there may be two parts of rectangularization for longevity. The historical trends for Switzerland (1876-2002) implies that there may be a fundamental limiting force to restrict the increase of the effectiveness. As a result, it seems that the density function is essential to define and understand the mortality dynamics, the maximum longevity, the effectiveness for longevity, the paradigm of rectangularization and the historical trends of the effectiveness by the complementarity between the survival and mortality functions.

Abstract:
Are there limits to human longevity? We suggest a new demographic model to describe human demographic trajectories. Specifically, the model mathematically defines the limits of longevity. Through the demographic analysis of trends for Sweden (between 1751 and 2002), Switzerland (between 1876 and 2002) and Japan (between 1950 and 1999), which are the longest-lived countries, we would like to demonstrate whether or not there is the ultimate limit to longevity. We analyse the trends of new demographic indicators, the characteristic life and the shape parameter, and calculate the mathematical limits of longevity. We find out the surprising phenomenon that the mathematical limits of longevity decrease as the longevity tendency increases in recent decades. These paradoxical trends will be explained by the complementarity of longevity, which is attributable to the nature of biological systems for longevity. According to the regression analysis, the ultimate limit for humans is estimated to be approximately 124 years, which may be considered to be the biological limit for humans.

Abstract:
The Gompertz model since 1825 has significantly contributed to interpretation of ageing in biological and social sciences. However, in modern research findings, it is clear that the Gompertz model is not successful to describe the whole demographic trajectories. In this letter, a new demographic model is introduced especially to describe human demographic trajectories, for example, for Sweden (2002). The new model is derived from the Weibull model with an age-dependent shape parameter, which seems to indicate the dynamical aspects of biological systems for longevity. We will discuss the origin of the age-dependent shape parameter. Finally, the new model presented here has significant potential to change our understanding of definition of maximum longevity.