Predicting human lifespan limits

doi:10.4236/ns.2010.29120

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Vol.2, No.9, 984-989 (2010) Natural Science

http://dx.doi.org/10.4236/ns.2010.29120

Predicting human lifespan limits

Byung Mook Weon1,2*, Jung Ho Je2*

1Department of Physics, School of Engineering and Applied Sciences, Harvard University, Cambridge, USA; *Corresponding Author:

[email protected]

2X-ray Imaging Center, Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang,

Korea; *Corresponding Author: [email protected]

Received 17 June 2010; revised 22 July 2010; accepted 28 July 2010.

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 age-dependent

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 sur-

vival dynamics.

Keywords: Human Lifespan; Maximum Survival

Tendency; Lifespan Limit

1. INTRODUCTION

Humans live longer now. Indeed the life expectancy and

the observed maximum age at death have significantly

increased during recent decades [1-5]. Such increase is

mainly attributable to non-biological aspects such as the

intricate interplay of advances in income, nutrition, edu-

cation, sanitation, and medicine [6,7]. Biologists and

gerontologists are hunting for a variety of useful ways to

prolong life in animals, including mice and worms [8].

Their research suggests that human lifespan may be re-

markably pliable [8]. Can the strategies for animals help

humans live longer? So far it is not practical, useful, or

ethical to extend healthy life merely by modifying hu-

man genes [9] or by restricting food intake [10]. The

theoretical maximum lifespan (called ω) in humans is

still a subject of considerable debate [6] and the life ex-

tension is one of the great challenges in the 21st century

[8]. Many scientists believe that human lifespan has an

inherent upper limit, although they disagree on whether

it is 85 or 100 or 150 [8]. The maximum human lifespan

is generally postulated to be around 125 years [7,11,12],

whereas the record of the oldest ages at death is increas-

ing today [4]. Conventional analysis or theoretical model

has not yet come up with a plausible explanation for this

disagreement.

Recently, based on extended Weibull model with

age-dependent stretched exponent [13,14], we suggested

a mathematical model for human survival dynamics, S(x)

= exp(–(x/α)β(x)), which denotes survival probability with

characteristic life α and age-dependent exponent β(x),

and showed maximum survival tendency, dS(x)/dx → 0

[15]. In this study, we further develop the model and

explore the dynamic patterns with year and country in

predicting human lifespan limits (ω) for industrialized

countries during recent three decades: ω = 0.458q +

54.241 where the upper x-intercept q = h + (k/p)1/2 for

the quadratic model β(x) = –p(x – h)2 + k (where p, h,

and k are variable with year and country). We aim to

examine the lifespan puzzle—whether human lifespan is

approaching a limit or not. Our analysis strategy has

practical implications for aging research in biology,

medicine, statistics, economy, public policy, and culture.

2. METHODS

We examine the survival dynamics of Sweden female’s

survival curves during recent three decades, from 1977

to 2007. The reliable demographic data were taken from

the periodic life tables (1 × 1) at the Human Mortality

Database (http://www.mortality.org). We analyze the

survival curves by using a general expression of human

survival probability (S(x)) as a function of age (x) [15]:







(x)

–S(x) expx/ 



Here, the characteristic life (α) corresponds to the spe-

cific age of S(α) = exp(–1) and the age-dependent stret-

B. M. Weon et al. / Natural Science 2 (2010) 984-989

985

ched exponent (or beta function, β(x)) reflects the flexi-

bility of the survival curve [15]. The survival function

allows the cumulative hazard function M(x) (= –log S(x))

on a restricted range. The breakdown of the positivity of

the hazard function m(x) (= ∂M(x)/∂x) enables us to

estimate a maximum limitation of human lifespan. Intui-

tively, our survival model approximates the Gompertz

model [16] with a linear expression for β(x) as well as

the Weibull model [17] with a constant β(x) through an

approximation of ‘log m(x) ~ β(x)’.

Figure 1 illustrates the evolution of the beta function

β(x), which is a pure mathematical conversion of the

survival probability S(x), for Swedish females from

1977 to 2007. The smooth survival data points above

94-97 years were chosen for modeling the beta function

(solid lines). Here the discontinuities of the beta curves

(dashed lines) near the characteristic lives (around 85-90

years) are due to the mathematical feature of the sug-

gested model [15]. Apparently the curvatures of the beta

curves seem to become more “negative” at the highest

ages and the vertex points move upward year by year.

Such trends directly connote the emergence of the

“maximum survival tendency” [15].

In principle, the age-dependent beta function origi-

nates from the “maximum survival tendency”, which is a

fundamental biological feature of human survival dy-

namics by minimizing its death rate (dS(x)/dx → 0) [15].

Figure 1. The evolution of Sweden female’s survival curves

(dashed lines) from 1977 to 2007. The beta function β(x) is

plotted as a conversion of the survival probability, S(x) = exp

(–(x/α)β(x)), where the characteristic life α corresponds to the

specific age of S(α) = exp(–1) and the age-dependent beta

function β(x) reflects the flexibility of the survival curve. Ap-

parently the β(x) curvature becomes more negative and the

vertex point moves upward year by year from 1977 to 2007.

The inset describes that the maximum lifespan (ω) is deter-

mined at the specific age of β(x) = f(x), which is defined as f(x)

= –xln(x/α) dβ(x)/dx. This feature suggests that the ω value can

be found between the vertex point “ν(h, k)” and the upper

x-intercept “q” point.

The maximum survival tendency is characterized as a

“negative” slope of the beta function as d2β(x)/dx2 < 0

for the phase of x > α.

We find that a quadratic model, β(x) = –β0 + β1x –β2x2

(where β0, β1, β2 > 0), is appropriate to describe the

maximum survival tendency from the modern survival

curves for the highest ages (for x > α), as marked by the

solid lines in Figure 1. This agrees to our previous ob-

servation [15]. In this study, we modify the quadratic

model for β(x) as:

(x)p(h) kx





Here the coefficient p (= β2 = (–1/2) d2β(x)/dx2) de-

notes the curvature of the quadratic curve and the vertex

point ν(h, k) indicates the maximum value of the quad-

ratic curve. The curvature and the vertex point give an

upper x-intercept (Figure 1), as can be defined as the

“q” point:

1/2

qh(k/p)

The quadratic beta function based on the maximum

survival tendency can be entirely described by quantify-

ing the ν(h, k) and the q points.

3. RESULTS

The intrinsic definition of β(x) and S(x) leads to a mathe-

matical limitation of the survival age, beyond which

none can be alive. The theoretical limitation of the maxi-

mum lifespan (ω) is determined at the specific age of β(x)

= f(x) as seen in Figure 1. Here f(x) is the mathematical

constraint of β(x) as defined as f(x) = –xln(x/α) dβ(x)/dx

[15]. This feature suggests that the ω value can be found

between the ν(h, k) and the q points.

We observe the evolution of the quadratic beta func-

tions from the survival curves of Sweden females, as

seen in Figure 2. The p and the k parameters linearly

increase by period (P): p = 6.8897 × 10–5 P – 0.1346 and

k = 6.389 × 10–2 P – 116.581. By contrary, the h pa-

rameter does not significantly change (average ~ 95.482

years), obviously since 1985, as marked by the gray area

in Figure 2. The linear increases of the p and the k pa-

rameters indicate that the curvature of the quadratic

function becomes more negative and the vertex point

moves upward from 1977 to 2007. Our model suggests

that the upper x-intercept (q-value) may significantly

decrease by period, following the scaling of (k/p)1/2. The

obtained q values (squares) well follow the trend line

(solid line) which is estimated from the p and the k pa-

rameters. The parameter estimation for Sweden females

is summarized at Table 1. The correlation coefficients

(r2) between data and model are higher than 0.994, sug-

gesting the feasibility of the model. As a result, we see

in Figure 2 that the q parameters gradually decrease by

B. M. Weon et al. / Natural Science 2 (2010) 984-989

986

Figure 2. The evolution of the quadratic beta function parameters estimated from Figure 1. The p and the k parame-

ters increase linearly by period, while the h parameter does not significantly change, obviously since 1985 (gray area).

These evolutions lead to a gradual decrease of the q parameter by period, following the scaling of (k/p)1/2.

Table 1. Estimations for Sweden female survival datasets.

Datasets α (yrs) β0 β1 β2 r2 h (yrs)k q (yrs) ω (yrs)

2007 88.57223 26.32347 0.78810 0.00409 0.99915 96.3411.64 149.70 122.86

2005 88.52680 17.85141 0.61727 0.00325 0.99949 94.9611.46 154.34 125.19

2002 87.83450 19.58927 0.65550 0.00346 0.99872 94.7311.46 152.27 123.75

2000 87.81336 17.81521 0.60247 0.00314 0.99864 95.9311.08 155.35 125.45

1997 87.66873 17.06434 0.58731 0.00307 0.99541 95.6511.02 155.58 125.59

1995 87.34920 10.82163 0.46078 0.00245 0.99955 94.0410.84 160.56 127.73

1992 86.91140 12.47482 0.49175 0.00260 0.99857 94.5710.78 158.95 126.81

1990 86.53163 13.17982 0.50236 0.00265 0.99926 94.7810.63 158.11 126.21

1987 86.30050 9.860300 0.42559 0.00225 0.99820 94.5810.26 162.12 128.30

1985 85.86844 11.91555 0.46436 0.00243 0.99576 95.5510.27 160.55 127.37

1982 85.58855 2.065850 0.27060 0.00151 0.99858 89.6010.06 171.21 132.05

1980 85.06656 2.594650 0.27414 0.00150 0.99487 91.389.93 172.75 132.66

1977 84.83856 2.584840 0.26922 0.00147 0.99463 91.579.74 172.98 132.95

B. M. Weon et al. / Natural Science 2 (2010) 984-989

987

period in Sweden female’s life tables during recent three

decades.

The most interesting observation is that the maximum

lifespan limits (ω) have a linear relationship with the

upper x-intercept (q) parameter, as clearly seen in Fig-

ure 3. The high linearity between the ω and the q values

is found for both cases of Sweden females (between

1977 and 2007) and modern industrialized countries

(Austria, Belgium, Bulgaria, Canada, Czech, Denmark,

England, Estonia, Finland, France, West Germany,

Hungary, Iceland, Ireland, Italy, Japan, Latvia, Lithuania,

Netherlands, Norway, Poland, Russia, Scotland, Slove-

nia, Spain, Sweden, Switzerland, and USA; for females

between 2005 and 2007). The parameter estimation for

modern industrialized countries is summarized at Table

2. The correlation coefficients (r2) between data and

model are higher than 0.954, suggesting again the feasi-

bility of the model. Interestingly, we find in Figure 3

that the three-decade variation of Sweden female

(squares) is similar to the national variation of the other

countries (circles). This similarity suggests that the

Figure 3. The linear relationship of the upper x-intercept q and

the theoretical lifespan ω values. The three-decade variation

(open and closed squares) for Sweden females is similar to the

national variation (circles) for modern industrialized countries.

Interestingly, the maximum lifespan ω values linearly decrease

with the upper x-intercept q values at a rate of ω = 0.458q +

54.241. The ω values approach ~125 years for Sweden females

during the latest decade from 1997 to 2007 (closed squares).

dataset of Sweden female can be indeed “representative”

for human survival tendency as suggested [4]. In Figure

3, we see that the ω values for all the datasets linearly

decrease as the q values decrease (r2 = 0.9445):

0.458q 54.241





It is interesting that the ω values shift toward ~125 years

(close squares) for Sweden females during the latest

decade from 1997 to 2007.

4. DISCUSSION

The overall evolutions of the q values (Figure 2) and ω

values (Figure 3) suggest that the human lifespan would

be reaching an upper limit. Our study implies that the

observed maximum lifespan limit is able to continue to

climb until it encounters a theoretical forbidden barrier

of human lifespan, as suggested [18]. The life-extension

strategies such as aggressive anti-aging therapies may

allow more people to reach the limit of the natural hu-

man lifespan and thus the period of disease or senes-

cence will be compressed against the natural barrier at

the end of life, as expected [19,20].

The lifespan limit estimation may support current ag-

ing theories that presume the existence of the biological

limit to human lifespan [21-23]. Based on our estimation,

it is predictable that many countries will face increasing

issues of aging populations, age-related diseases, and

healthcare costs [8]. The rise of human longevity will

accelerate the population growth rate [24] and probably

the steady rise in the achieved maximum lifespan [4] or

the life expectancy [25] will reduce in the coming half

century. The forthcoming trends may cause an ethical

issue on fair distribution of healthcare resources [26].

Aging research requires new approaches to figure out the

complex biology of aging [27].

The feasibility of the model is further obtained

through a mathematical verification [28]: our data exist

between 0.4 < (ω/α)ln(ω/α) < 0.8, which are consistent

with their mathematical expectation between 0.410986

and 0.829297. Another verification is obtained from

mortality patterns, which are defined as μ(x) = –dln

S(x)/dx = d[(x/α)β(x)]/dx or ∫μ(x) = (x/α)β(x). For simpli-

fication one defines δ(x) = μ(x)/∫μ(x) = β(x)/x +

ln(x/α)dβ(x)/dx. The point where the mortality curve

starts to decline is obtained from δ(x)2 + dδ(x)/dx = 0 by

solving dμ(x)/dx = 0. This condition can be tested by

graphical analysis or numerical simulation. For instance,

taking the parameters: α = 88.57223 years, β0 =

26.32347, β1 = 0.78810, and β2 = 0.00409 from 2007

Swedish female’s data (Table 1), we obtain the point as

~111 years. Above that point, the mortality curve de-

creases and eventually reaches zero at the maximum

lifespan ~122.86 years. With the quadratic pattern of

B. M. Weon et al. / Natural Science 2 (2010) 984-989

988

Table 2. Estimations for international female survival datasets.

Datasets α (yrs) r2 h (yrs) k q (yrs) ω (yrs)

Austria (2005) 88.07717 0.99973 92.06 11.51 157.23 126.16

Belgium (2006) 88.17957 0.99960 95.29 11.23 154.80 125.23

Bulgaria (2005) 82.71487 0.95438 94.92 9.55 150.53 121.06

Canada (2005) 88.93178 0.99963 100.00 10.56 157.10 127.34

Czech (2006) 85.58519 0.99742 94.70 10.51 153.61 123.58

Denmark (2006) 86.67277 0.98848 100.58 9.96 158.11 127.16

England (2005) 87.57264 0.99952 96.30 10.45 162.70 129.24

Estonia (2007) 85.47782 0.99604 95.36 9.79 165.86 129.85

Finland (2007) 88.76823 0.99834 93.56 11.93 153.32 124.54

France (2006) 90.37934 0.99963 95.20 11.72 159.00 128.33

Germany (2006) 88.11166 0.99869 89.77 11.92 156.22 125.34

Hungary (2005) 83.79004 0.99779 96.27 9.07 169.96 131.53

Iceland (2006) 88.78873 0.99560 99.22 11.61 142.67 119.77

Ireland (2006) 87.53789 0.99960 90.75 10.44 174.44 134.53

Italy (2005) 89.22443 0.99977 92.76 11.62 159.09 127.66

Japan (2006) 91.59045 0.99942 98.17 11.47 161.51 130.31

Latvia (2007) 83.82117 0.99683 95.46 9.43 165.53 129.20

Lithuania (2007) 84.63518 0.99738 97.72 9.82 153.30 123.59

Netherlands (2006) 87.85314 0.99636 95.65 11.17 155.48 125.55

Norway (2006) 88.49842 0.99951 94.48 11.57 154.52 125.19

Poland (2007) 87.63163 0.99919 91.29 11.48 156.14 125.31

Russia (2006) 81.29562 0.98645 98.59 8.61 155.26 123.63

Scotland (2006) 86.15168 0.99884 100.50 9.49 159.68 127.78

Slovenia (2006) 87.42881 0.99890 94.93 10.81 157.52 126.32

Spain (2006) 89.47929 0.99946 89.37 11.89 163.90 129.74

Sweden (2007) 88.57223 0.99915 96.34 11.64 149.70 122.86

Switzerland (2006) 89.69486 0.99806 97.52 11.75 149.62 123.38

USA (2005) 87.43102 0.99878 103.37 9.27 159.62 128.74

β(x), the mortality pattern tends to decrease after a pla-

teau and ultimately approach zero, well matching typical

human mortality patterns. These results show the feasi-

bility of the model.

5. CONCLUSIONS

To conclude, we develop a human survival dynamics

model as S(x) = exp(–(x/α)β(x)) with β(x) = –p(x – h)2 + k

(where p, h, and k are variable with year and country),

and explore the pattern of the parameters, q = h + (k/p)1/2

and ω = 0.458q + 54.241, which are useful in predicting

human lifespan limits (ω). We show generality and fea-

sibility of the model for modern industrialized countries

during recent three decades. Based on statistical ap-

proach, we suggest that human lifespan is approaching a

true limit around 125 years. This estimate may shed light

on the central puzzle in aging research: whether biologi-

cal lifespan limits exist or not. Our model and prediction

method would be useful to assess the complicated hu-

man survival dynamics [29], which would be essential to

study on biology, medicine, statistics, economy, public

B. M. Weon et al. / Natural Science 2 (2010) 984-989

989

policy, and culture.

6. ACKNOWLEDGEMENTS

We are grateful to the Human Mortality Database (http://www.mortality.

org) for allowing anyone to access the demographic data for research.

This work was supported by the Creative Research Initiatives (Func-

tional X-ray Imaging) of MEST/NRF.

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