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J.theor.Biol. (2001) 213, 527}545
doi:10.1006/jtbi.2001.2430, available online at http://www.idealibrary.com on
The Reliability Theory of Aging and Longevity
LEONID A. GAVRILOV*AND NATALIA S. GAVRILOVA
Center on Aging,NORC/;niversity of Chicago, 1155 East 60th Street,Chicago,I¸60637, ;.S.A.
(Received on 31 August 2000, Accepted in revised form on 8January 2001)
Reliability theory is a general theory about systems failure. It allows researchers to predict the
age-related failure kinetics for a system of given architecture (reliability structure) and given
reliability of its components. Reliability theory predicts that even those systems that are
entirely composed of non-aging elements (with a constant failure rate) will nevertheless
deteriorate (fail more often) with age, if these systems are redundant in irreplaceable elements.
Aging, therefore, is a direct consequence of systems redundancy. Reliability theory also
predicts the late-life mortality deceleration with subsequent leveling-o!, as well as the late-life
mortality plateaus, as an inevitable consequence of redundancy exhaustion at extreme old ages.
The theory explains why mortality rates increase exponentially with age (the Gompertz law) in
many species, by taking into account the initial -aws (defects) in newly formed systems. It also
explains why organisms &&prefer'' to die according to the Gompertz law, while technical devices
usually fail according to the Weibull (power) law. Theoretical conditions are speci"ed when
organisms die according to the Weibull law: organisms should be relatively free of initial #aws
and defects. The theory makes it possible to "nd a general failure law applicable to all adult
and extreme old ages, where the Gompertz and the Weibull laws are just special cases of this
more general failure law. The theory explains why relative di!erences in mortality rates of
compared populations (within a given species) vanish with age, and mortality convergence is
observed due to the exhaustion of initial di!erences in redundancy levels. Overall, reliability
theory has an amazing predictive and explanatory power with a few, very general and realistic
assumptions. Therefore, reliability theory seems to be a promising approach for developing
a comprehensive theory of aging and longevity integrating mathematical methods with speci"c
biological knowledge.
2001 Academic Press
1. Introduction
Extensive empirical studies on species aging and
longevity have proved successful in establishing
many important facts and details about the aging
process (Finch, 1990; Jazwinski, 1996, 1998) that
have yet to be explained and understood. Empiri-
cal observations on this issue have become so
numerous and abundant that a special encyclo-
pedia, ¹he Macmillan Encyclopedia of Aging,is
now required for even partial coverage of the
accumulated facts (Ekerdt, 2002). To transform
these numerous observations into a comprehens-
ive body of knowledge, a general theory of species
aging and longevity is required.
Attempts to develop a fundamental quantitat-
ive theory of aging, mortality, and lifespan have
deep historical roots. In 1825, the British actuary
Benjamin Gompertz discovered a law of mortal-
ity (Gompertz, 1825), known today as the Gom-
pertz law (Strehler, 1978; Finch, 1990; Gavrilov
0022}5193/01/240527#19 $35.00/0 2001 Academic Press
& Gavrilova, 1991; Olshansky & Carnes, 1997).
Speci"cally, he found that the force of mortality
(known in modern science as mortality rate, haz-
ard rate, or failure rate) increases in geometrical
progression with the age of adult humans. Ac-
cording to the Gompertz law, human mortality
rates double over about every 8 years of adult
age. Gompertz also proposed the "rst mathemat-
ical model to explain the exponential increase in
mortality rate with age (Gompertz, 1825). More-
over, he found that at advanced ages mortality
rates increase less rapidly than an exponential
function, thus forestalling two centuries ago the
recent fuss over &&late-life mortality deceleration''
(Fukui et al., 1993, 1996; Khazaeli et al., 1996;
Vaupel et al., 1998; Partridge & Mangel, 1999),
&&mortality leveling o!'' (Carey & Liedo, 1995;
Clark & Guadalupe, 1995; Vaupel et al., 1998),
and &&late-life mortality plateaus'' (Mueller &
Rose, 1996; Tower, 1996; Pletcher & Curtsinger,
1998; Wachter, 1999). For a more in-depth analy-
sis of the previous extensive studies on mortality
leveling-o!(Makeham, 1867; Brownlee, 1919;
Perks, 1932; Greenwood & Irwin, 1939; Mildvan
& Strehler, 1960; Strehler, 1960; Economos, 1979,
1980, 1983, 1985; Gavrilov & Gavrilova, 1991),
see the recent thoughtful review by Olshansky
(1998).
The Gompertz law of exponential increase in
mortality rates with age is observed in many
biological species (Strehler, 1978; Finch, 1990),
including humans, rats, mice, fruit #ies, #our
beetles, and human lice (Gavrilov & Gavrilova,
1991), and, therefore, some general theoretical
explanation for this phenomenon is required.
Many attempts to provide such theoretical
underpinnings for the Gompertz law have been
made (see reviews in Strehler, 1978; Gavrilov
& Gavrilova, 1991), and the problem now is to
"nd out which of these models is correct.
A comprehensive theory of species aging and
longevity should provide answers to the follow-
ing questions:
(1) Why do most biological species deteriorate
with age (i.e. die more often as they grow older)
while some primitive organisms do not demon-
strate such a clear age dependence for mortality
increase (Haranghy & BalaHzs, 1980; Finch, 1990;
Martinez, 1998)?
(2) Speci"cally, why do mortality rates in-
crease exponentially with age in many adult spe-
cies (Gompertz law)? How should we handle
cases when the Gompertzian mortality law is not
applicable?
(3) Why does the age-related increase in
mortality rates vanish at older ages? Why do
mortality rates eventually decelerate compared
to predictions of the Gompertz law, occasionally
demonstrate leveling-o!(late-life mortality
plateau), or even a paradoxical decrease at
extreme ages?
(4) How do we explain the so-called compen-
sation law of mortality (Gavrilov & Gavrilova,
1991)? This paradoxical phenomenon refers to
the observation that high mortality rates in dis-
advantaged populations (within a given species)
are compensated for by a low apparent &&aging
rate'' (longer mortality doubling period). As a re-
sult of this compensation, the relative di!erences
in mortality rates tend to decrease with age with-
in a given biological species. This is true for
male}female comparisons, for international com-
parisons of di!erent countries within the same
sex, as well as for within-species comparisons of
animal stocks (Gavrilov & Gavrilova, 1991). The
theory of aging and longevity has to explain this
paradox of mortality convergence.
Following a long-standing tradition of biolo-
gical thought, the search for a general biological
theory to explain aging and longevity has been
made mainly in terms of evolutionary biology
(Medawar, 1946, 1952; Williams, 1957, 1966;
Hamilton, 1966; Rose, 1991; Carnes & Ol-
shansky, 1993; Charlesworth, 1994) and genetics
(Finch, 1990; Jazwinski, 1996, 1998; Finch &
Tanzi, 1997; Carnes et al., 1999). However, the
attempts to explain &&late-life mortality plateaus''
using evolutionary theory (Mueller & Rose, 1996)
have failed so far because they required highly
specialized and unrealistic assumptions (see criti-
cal reviews by Charlesworth & Partridge, 1997;
Pletcher & Curtsinger, 1998; Wachter, 1999). It
looks like the evolutionary theory is more appro-
priate to explain early successes of biological
species (e.g. reproductive success), rather than
their later failures (aging and death). There seems
to be a missing piece in the theoretical arsenal of
evolutionary biologists trying to explain aging,
528 L. A. GAVRILOV AND N. S. GAVRILOVA
and this missing piece is about the general theory
of system failures. This theory, known as the
theory of reliability (Lloyd & Lipow, 1962;
Barlow et al., 1965; Barlow & Proschan, 1975;
Kaufmann et al., 1977; Crowder et al., 1991; Aven
& Jensen, 1999; Rigdon & Basu, 2000), allows
researchers to understand many puzzling features
of mortality and lifespan (Gavrilov, 1978, 1987;
Gavrilov et al., 1978; Abernethy, 1979; D[oubal,
1982; Gavrilov & Gavrilova, 1991, 1993; Bains,
2000) not readily explainable otherwise (i.e. the
Gompertz law, mortality plateaus, and the com-
pensation law of mortality).
The purpose of this article is to introduce the
ideas and methods of reliability theory to a wider
audience interested in understanding the mecha-
nisms of aging, mortality, survival, and longevity.
It is also important to review and summarize the
recent scienti"c literature on reliability approach
to the problem of biological aging (Miller, 1989;
Gavrilov & Gavrilova, 1991; Abernethy, 1998;
Bains, 2000). The main emphasis here is made on
the accomplishments of the reliability approach
rather than previous occasional mistakes, be-
cause some questionable models (Murphy, 1978;
Skurnick & Kemeny, 1978; Koltover, 1983, 1997;
Witten, 1985) were already reviewed elsewhere
(Gavrilov, 1984, 1987; Gavrilov & Gavrilova,
1991). This theoretical review article also elabor-
ates further some results and ideas published
in the book on a related topic (Gavrilov &
Gavrilova, 1991).
2. Reliability Theory: General Overview
Reliability theory is a body of ideas, math-
ematical models, and methods directed to
predict, estimate, understand, and optimize the
lifespan distribution of systems and their
components (adapted from Barlow et al., 1965).
Reliability of the system (or component) refers to
its ability to operate properly according to a spe-
ci"ed standard (Crowder et al., 1991). Reliability
is described by the reliability function S(x), which
is the probability that a system (or component)
will carry out its mission through time x(Rigdon
& Basu, 2000). The reliability function (also
called the survival function) evaluated at time xis
just the probability P, that the failure time X,
is beyond time x. Thus, the reliability function is
de"ned in the following way:
S(x)"P(X'x)"1!P(X)x)"1!F(x),
(1)
where F(x) is a standard cumulative distribution
function in the probability theory (Feller, 1968).
The best illustration for the reliability function
S(x) is a survival curve describing the proportion
of those still alive by time x(the lVcolumn in life
tables). Failure rate j(x), also called the hazard
rate h(x), is de"ned as the relative rate for reliabil-
ity function decline:
j(x)"! dS(x)
S(x)dx"!d[logCS(x)]
dx. (2)
Failure rate is equivalent to mortality force,
k(x), in demography. In those cases when the
failure rate is constant (does not increase with
age), we have a non-aging system (component)
that does not deteriorate (does not fail more
often) with age. The reliability function of non-
aging systems (components) is described by the
exponential distribution:
j(x)"j"const, (3a)
S(x)"Sexp(!jx). (3b)
This failure law describes &&lifespan'' distribu-
tion of atoms of radioactive elements and it is
also observed in many wild populations with
high extrinsic mortality (Finch, 1990; Gavrilov
& Gavrilova, 1991).
If failure rate increases with age, we have an
aging system (component) that deteriorates (fails
more often) with age. There are many failure laws
for aging systems and the Gompertz law with
exponential increase of the failure rates with age
is just one of them (see Gavrilov & Gavrilova,
1991). In reality, system failure rates may contain
both non-aging and aging terms as, for example,
in the case of the Gompertz}Makeham law of
mortality (Makeham, 1860; Strehler, 1978; Gav-
rilov & Gavrilova, 1991):
k(x)"A#Rexp(ax),
where parameters A,R,a'0. (4)
RELIABILITY THEORY 529
In this formula, the "rst, age-independent term
(Makeham parameter, A) designates the con-
stant, &&non-aging'' component of the failure rate
(presumably due to extrinsic causes of death,
such as accidents and acute infections), while the
second, age-dependent term (the Gompertz func-
tion, Re?V) designates the &&aging'' component,
presumably due to deaths from age-related
degenerative diseases like cancer and heart
disease.
The compensation law of mortality in its strong
form refers to mortality convergence, when higher
values for the parameter a(in the Gompertz
function) are compensated by lower values of the
parameter Rin di!erent populations of a given
species:
ln(R)"ln(M)!Ba, (5)
where Band Mare universal species-speci"c in-
variants. Sometimes, this relationship is also
called the Strehler}Mildvan correlation (Strehler
& Mildvan, 1960; Strehler, 1978), although that
particular correlation was largely an artifact of
the opposite biases in parameters estimation
caused by not taking into account the age-inde-
pendent mortality component, the Makeham
term A(see Gavrilov & Gavrilova, 1991). Para-
meter B is called the species-speci"c lifespan (95
years for humans), and parameter Mis called the
species-speci"c mortality rate (0.5 yr\ for hu-
mans). These parameters are the coordinates for
the convergence of all the mortality trajectories
into one single point (within a given biological
species), when extrapolated by the Gompertz
function (Gavrilov & Gavrilova, 1991). In those
cases when the compensation law of mortality is
not observed in its strong form, it may still be
valid in its weak form*i.e. the relative di!erences
in mortality rates of compared populations tend
to decrease with age in many species. An explana-
tion of the compensation law of mortality is
a great challenge for many theories of aging and
longevity (Strehler, 1978; Gavrilov & Gavrilova,
1991).
There are some exceptions both from the
Gompertz law of mortality and the compensa-
tion law of mortality that have to be understood
and explained. In some cases, the organisms die
according to the =eibull (power)law (Hirsch
& Peretz, 1984; Janse et al., 1988; Hirsch et al.,
1994; Eakin et al., 1995; Van#eteren et al., 1998):
k(x)"jx?for x*0, where j,a'0. (6)
The Weibull law is more commonly applicable
for technical devices (Barlow & Proschan, 1975;
Rigdon & Basu, 2000) while the Gompertz law is
more common for biological systems (Strehler,
1978; Finch, 1990; Gavrilov & Gavrilova, 1991).
Both the Gompertz and the Weibull failure laws
have fundamental explanation rooted in reliabil-
ity theory (Barlow & Proschan, 1975) and are the
only two theoretically possible limiting extreme
value distributions for systems whose lifespans are
determined by the "rst failed component (Gum-
bel, 1958; Galambos, 1978). In other words, as the
system becomes more and more complex (con-
tains more vital components, each being critical
for survival), its lifespan distribution may asymp-
totically approach one of the only two theoret-
ically possible limiting distributions*either
Gompertz or Weibull (depending on the early
kinetics of failure of system components). The
two limit theorems in the statistics of extremes
(Gumbel, 1958; Galambos, 1978) make the Gom-
pertz and the Weibull failure laws as fundamental
as are some other famous limiting distributions
known in regular statistics, e.g. the normal distri-
bution and the Poisson distribution. It is puzzl-
ing, however, why organisms prefer to die
according to the Gompertz law, while technical
devices typically fail according to the Weibull
law. One possible explanation of this mystery is
suggested in the next section of the article.
The phenomena of mortality increase with age
and the subsequent mortality leveling-o!are the-
oretically predicted to be an inevitable feature
of all reliability models that consider aging as
a progressive accumulation of random damage
(Gavrilov & Gavrilova, 1991). The detailed
mathematical proof of this prediction for some
particular models is provided in the next two
sections of this article. In short, if the destruction
of an organism occurs not in one but in two or
more sequential random stages, this is su$cient
for the phenomenon of aging (mortality increase)
to appear and then to vanish at older ages. Each
stage of destruction corresponds to one of the
organism's vitally important structures being
530 L. A. GAVRILOV AND N. S. GAVRILOVA
damaged. In the simplest organisms with unique,
critical structures, this damage usually leads to
their deaths. Therefore, defects in such organisms
do not accumulate, and the organisms themselves
do not age*they just die when damaged. In
more complex organisms with many vital struc-
tures and signi"cant redundancy, every occur-
rence of damage does not lead to death because
of this redundancy. Defects do accumulate, there-
fore, giving rise to the phenomenon of aging
(mortality increase). Thus, aging is a direct conse-
quence (trade-o!) of systems redundancy that
ensures increased reliability and lifespan of or-
ganisms. As defects accumulate, the redundancy
in the number of elements "nally disappears. As
a result of this redundancy exhaustion, the
organism degenerates into a system with no
redundancy, that is, a system with elements
connected in series, with the result being that any
new defect leads to death. In such a state, no
further accumulation of damage can be achieved,
and the mortality rate levels o!. The next two
sections provide mathematical proof for these
ideas.
3. Reliability Theory of Aging for Highly
Redundant Systems Replete with Defects
In this section, we will show that the exponen-
tial growth in mortality rate, as well as other
aging phenomena (late-life mortality deceleration
and compensation law of mortality), follows nat-
urally from a simple reliability model and two
general features of biosystems.
The "rst fundamental feature of biosystems is
that, in contrast to technical (arti"cial) devices
which are constructed out of previously manufac-
tured and tested components, organisms form
themselves in ontogenesis through a process of
self-assembly out of de novo forming and ex-
ternally untested elements (cells). The second
property of organisms is the extraordinary degree
of miniaturization of their components (the
microscopic dimensions of cells, as well as the
molecular dimensions of information carriers like
DNA and RNA), permitting the creation of
a huge redundancy in the number of elements.
Thus, we expect that for living organisms, in
distinction to many technical (manufactured) de-
vices, the reliability of the system is achieved not
by the high initial quality of all the elements but
by their huge numbers (redundancy). As will be
shown later, this feature of organisms provides an
explanation why the failure rate grows as an
exponential rather than a power function of age,
and it also enables researchers to understand the
other mortality phenomena (e.g. compensation
law of mortality).
Figure 1 presents a scheme explaining the
causes of cardinal di!erences in reliability struc-
ture between technical devices and biological
systems.
The fundamental di!erence in the manner in
which the system is formed (external assembly
in the case of technical devices and self-assembly
in the case of biosystems) has two important
consequences. First, it leads to the macroscopic-
ity of technical devices in comparison with bio-
systems, since technical devices are assembled
&&top-down'' with the participation of a macro-
scopic system (man) and must be suitable for this
macroscopic system to use (i.e. commensurate
with man). Organisms, on the other hand, are
assembled &&bottom-up'' from molecules and cells,
resulting in an exceptionally high degree of
miniaturization of the component parts. Second,
since technical devices are assembled under
the control of man, the opportunities to pretest
components (external quality control) are incom-
parably greater than in the self-assembly of bio-
systems. The latter inevitably leads to organisms
being &&littered'' with a great number of defective
elements. As a result, the reliability of technical
devices is assured by the high quality of elements,
with a strict limit on their numbers because of
size and cost limitations [Fig. 2(a)], while the
reliability of biosystems is assured by an excep-
tionally high degree of redundancy to overcome
the poor quality of some elements [Fig. 2(b)].
It is interesting to note that the uniqueness of
individuals, which delights biologists so much,
may be caused by &&littering'' the organisms with
defects and thus forming a unique pattern of
individual damage. Our past experience working
with dilapidated computer equipment in Russia
gave rise to the same thought: the behavior of this
equipment could only be described by resorting
to such &&human'' concepts as character, freaks,
personality, and change of mood. As will be
shown later, ideas of this kind proved to be useful
RELIABILITY THEORY 531
FIG. 1. Schema explaining why biological systems and technical devices may have di!erent solutions to the problem of
achieving consistent reliability. Owing to the di!erent ways in which systems are created (self-assembly of organisms vs.
external assembly of machines) two opposite strategies are used to achieve high reliability*either huge redundancy in
numbers of loose components (biosystems) or high standards for each unique component (technical devices).
in constructing a mathematical model of aging
and longevity for biological systems.
The idea that living organisms are starting
their lives with a large number of defects has deep
historical roots. Biological justi"cation for this
idea was discussed by Dobzhansky (1962).
He noted that, from the biological perspective,
Hamlet's&&thousand natural shocks that -esh is
heir to'' was an underestimate and that in reality,
&&the shocks are innumerable'' (Dobzhansky,
1962, p. 126). Recent studies found that the
troubles in human life start from the very begin-
ning: the cell-cycle checkpoints (which ensure
that a cell will not divide until DNA damage
is repaired and chromosomal segregation is
complete) do not operate properly at the early,
cleavage stage of human embryo (Handyside
& Delhanty, 1997). This produces mosaicism in
532 L. A. GAVRILOV AND N. S. GAVRILOVA
FIG. 2. Diagrams illustrating the di!erences in reliability structure between (a) technical devices and (b) biological systems.
Each block diagram represents a system with mserially connected blocks (each being critical for system survival, "ve blocks in
these particular illustrative examples) built of nelements connected in parallel (each being su$cient for the block being
operational). Initially defective non-functional elements are indicated by crossing (x). The reliability structure of technical
devices (a) is characterized by relatively low redundancy in elements (because of cost and space limitations), each being
initially operational because of strict quality control. Biological species, on the other hand, have a reliability structure (b) with
huge redundancy in small, often non-functional elements (cells). The theoretical consequences of these di!erences are
discussed in the text.
the pre-implantation embryo, where some
embryonic cells are genetically abnormal
(McLaren, 1998), and this may have devastating
consequences in later life. Another potential
source of extensive initial damage is the birth
process itself. During birth, the child is "rst de-
prived of oxygen by compression of the umbilical
cord (Mo!ett et al., 1993) and su!ers severe hy-
poxia (often with ischemia and asphyxia). Then,
just after birth, the newborn child is exposed to
oxidative stress because of acute reoxygenation
when breathing begins. It is known that acute
reoxygenation after hypoxia may produce an ex-
tensive oxidative damage through the same
mechanisms that also produce ischemia-reperfu-
sion injury (IRI) and asphyxia-reventilation in-
jury (Martin et al., 2000). Thus, returning to
Hamlet's metaphor, we may add that humans
&&su+er the slings and arrows of outrageous for-
tune'' and have &&a sea of troubles'' from the very
beginning of their lives.
Also, the system may behave as if it has a large
number of initial defects when some of its compo-
nents are apparently non-functional for whatever
reason (because of impaired regulation, disrupted
communication between components, or &&self-
ish'' behavior of DNA, cells, and tissues, etc.). An
apparent lack of any function is typical for many
structures of living organisms, starting from the
molecular level (e.g. non-functional, sel"sh DNA
and inactive pseudogenes, see Gri$ths et al.,
1996), up to the level of the human brain (see
Finger et al., 1988).
We begin to consider our model by "rst
analysing the simplest case when all the elements
of the system are initially functional (which is
typical for technical devices) and have a constant
failure rate k. If these non-aging elements are
organized into blocks of nmutually substitutable
elements so that the failure of a block occurs only
when all the elements of the block fail (parallel
construction in the reliability theory context), the
failure rate of a block k@(n,k,x) can be written as
follows (see Appendix A):
k@(n,k,x)"! dS@(n,k,x)
S@(n,k,x)dx"nke\IV(1!e\IV)L\
1!(1!e\IV)L
(7)
+nkLxL\ when x;1/k(early-life period
approximation, when 1!e\IV+kx)
RELIABILITY THEORY 533
+kwhen x<1/k(late-life period approxi-
mation, when 1!e\IV+1).
Thus, the failure rate of a block initially grows
as a power function of age (the Weibull law).
Then the tempo at which the failure rate grows
declines, and the failure rate asymptotically ap-
proaches an upper limit equal to k. Here, we
should pay attention to three signi"cant points.
First, a block constructed out of non-aging ele-
ments is now behaving like an aging object: i.e.
aging is a direct consequence of the redundancy
of the system (redundancy in the number of ele-
ments). Second, at very high ages, the phenom-
enon of aging apparently disappears (failure rate
levels-o!), as redundancy in the number of ele-
ments vanishes. The failure rate approaches an
upper limit which is totally independent of the
initial number of elements, but coincides with the
rate of their loss (parameter k). Third, the blocks
with di!erent initial levels of redundancy (para-
meter n) will have very di!erent failure rates in
early life, but these di!erences will eventually
vanish as failure rates approach the upper limit
determined by the rate of elements'loss (para-
meter k). Thus, the compensation law of mortal-
ity (in its weak form) is an expected outcome of
this model.
If in its turn, a system is constructed out of
mirreplaceable blocks in such a way that the
failure of any of the blocks leads to the failure of
the whole system (series construction in the relia-
bility theory context), the failure rate of the sys-
tem is equal to the sum of the failure rates of all
the blocks:
kQ(x)"K
H
k@(j)"mk@"mkne\IV(1!e\IV)L\
1!(1!e\IV)L
(8)
+mnkLxL\ when x;1/k(early-life period
approximation, when 1!e\IV+kx)
+mk when x<1/k(late-life period approxi-
mation, when 1!e\IV+1).
In this model, the failure rate grows as a power
function rather than as an exponential function
of age. Therefore, this kind of a model seemed
incapable of describing the exponential growth of
the failure rate in biological systems for a long
time, and attention was drawn to the search
for more complex failure scenarios such as the
avalanche-like failure models (Le Bras, 1976;
Gavrilov & Gavrilova, 1991).
In this section, we demonstrate that the relia-
bility model presented above has been unde-
servedly rejected merely because it started
by analysing initially ideal structures in which all
the elements are functional from the outset.
This standard assumption may be justi"ed for
technical devices manufactured from pre-tested
components, but it is not justi"ed for living
organisms, presumably replete with defects, for
the reasons described earlier [see previous dis-
cussion of relevant publications (Dobzhansky,
1962; Finger et al., 1988; Mo!ett et al., 1993;
Gri$ths et al., 1996; Handyside & Delhanty,
1997; McLaren, 1998; Martin et al., 2000) and
Figs 1 and 2]. It is therefore worthwhile to exam-
ine another particular case of the model in which
initially functional elements occur very rarely
with a low probability q. (This interpretation of
the assumption could be relaxed. See the end of
this section.)
In this case, the distribution of the blocks in the
organism according to the number iof initially
functional elements they contain is described by
the Poisson law with parameter j"nq, corre-
sponding to the mean number of initially func-
tional elements in a block. As shown in Appendix
B, the failure rate in this case is given by
kQ"kjmce\He\IV L
G
jG\(1!e\IV)G\
(i!1)!(1!(1!e\IV)G). (9)
Thus, in the early life period (when x;1/k
and, therefore, 1!e\IV+kx), the mortality ki-
netics follows the exponential Gompertzian law:
kQ(x)+kjmce\HL
G
(jkx)G\
(i!1)!
"R(e?V!e(x))+Rexp(ax), (10)
where R"cmjke\H,a"jk,e(x)"
GL>
(jkx)G\/(i!1)! . e(x) is close to zero for large
nand small x(in early life period).
534 L. A. GAVRILOV AND N. S. GAVRILOVA
In the late-life period (when x<1/kand, there-
fore, 1!e\IV+1), the failure rate levels-o!and
the mortality plateau is observed
kQ(x)+mk. (11)
Thus, the failure rate of an organism initially
(for x;1/k) grows exponentially with age fol-
lowing the Gompertz law. If the background,
age-independent component of mortality (A) also
exists in addition to Gompertz function, we ob-
tain the well-known Gompertz}Makeham law
described earlier. At advanced ages, the rate of
mortality decelerates and asymptotically ap-
proaches an upper limit equal to mk.
The model explains not only the exponential
increase in mortality rate with age and the sub-
sequent leveling o!, but also the compensation
law of mortality.
Indeed, according to the previous notations:
ln(R)"ln(kjcm)!j, (12a)
a"jk, (12b)
i.e. the quantities ln(R) and aare parametrically
linked via the common parameter j, which
allows us to present ln(R) as a function of ain
explicit form:
ln(R)"ln(cma)!a
k"ln(M)!Ba, (13)
where M"cma,B"1/k.
Thus, the compensation law of mortality is
observed: lower mortality rate due to decreased
parameter Ris compensated by higher mortality
acceleration due to increased parameter a.Ac-
cording to this model, the compensation law is
inevitable whenever di!erences in mortality arise
from di!erences in the parameter j(the mean
number of initially functional elements in the
block), while the &&true aging rate'' (rate of ele-
ments'loss, k) is similar in di!erent populations
of a given species (presumably because of homeo-
stasis). In this case, the species-speci"c lifespan
estimated from the compensation law as an ex-
pected age at mortality convergence (95 years for
humans, see Gavrilov & Gavrilova, 1991) charac-
terizes the mean lifetime of the elements (1/k).
The model also predicts certain deviations
from the exact mortality convergence in a speci"c
direction because the parameter Mproved to be
a function of the parameter aaccording to this
model (see earlier). This prediction could be tes-
ted in future studies.
It also follows from this model that even small
progress in optimizing the processes of ontogen-
esis and increasing the numbers of initially
functional elements (j) can potentially result in
a remarkable fall in mortality and a signi"cant
improvement in lifespan. This optimistic predic-
tion is supported by experimental evidence of
increased o!spring lifespan in response to protec-
tion of parental germ cells against oxidative
damage just by feeding the future parents with
antioxidants (Harman & Eddy, 1979). Increased
lifespan is also observed among the progeny of
parents with a low respiration rate (proxy for the
rate of oxidative damage to DNA of germ cells,
see Gavrilov & Gavrilova, 1991). The model also
predicts that early life events may a!ect survival
in later adult life through the level of initial dam-
age. This prediction proved to be correct for such
early life indicators as parental age at a person's
conception (Gavrilov & Gavrilova, 1997a, b;
2000) and the month of person's birth (Doblham-
mer, 1999; Gavrilov & Gavrilova, 1999). The idea
of fetal origins of adult degenerative diseases and
early life programming of late-life health and
survival is being actively discussed in the scient-
i"c literature (Lucas, 1991; Barker, 1992, 1998;
Kuh & Ben-Shlomo, 1997; Leon et al., 1998;
Lucas et al., 1999; Blackwell et al., 2001).
The model assumes that most of the elements
in the system are initially non-functional. ¹his
interpretation of the assumption can be relaxed,
however,because most non-functional elements
(e.g.cells)may have already died and eliminated by
the time the adult organism is formed. In fact, the
model is based on the hypothesis that the number
of functional elements in the blocks is described
by the Poisson distribution, and the fate of defec-
tive elements and their death in no way a!ects the
conclusions of the model. Therefore, those
readers who resist the idea that they are built-up
of unreliable trash (or feel uncomfortable with the
biological justi"cation for this idea provided
earlier), can choose a more comfortable inter-
pretation for the same model and formulae,
RELIABILITY THEORY 535
namely that stochastic events in early develop-
ment determine later-life aging and survival
through variation in initial redundancy of organs
and tissues (see, for example, Finch & Kirkwood,
2000). We believe that, with these reservations
mentioned above, the earlier criticism of the sug-
gested model as based on &&biologically indefens-
ible assumptions'' that are &&highly unlikely'' (see
Pletcher & Neuhauser, 2000, p. 530) could be also
relaxed.
The conclusions of the model are valid for any
value of the parameter j(mean number of ini-
tially functional elements), no matter how large.
This is important, because it is known that as
parameter jincreases, the Poisson distribution
approximates to the normal distribution (Feller,
1968). Thus, the proposed model in fact en-
compasses a wide spectrum of distributions of
blocks according to their degree of redundancy,
starting with a marked positive (right-sided)
skewness (for small values of j) and ending with
distributions which are close to the symmetrical
normal distribution (for large values of j). In other
words, the proposed model might also be called
the model of series-connected blocks with varying
degrees of redundancy. This rather general model
is generalized even further in the next section.
4. Reliability Theory of Aging for Partially
Damaged Redundant Systems
In the preceding section, we examined a
reliability model for a system consisting of
mseries-connected blocks, each of which
contains nparallel-connected elements. It was
shown that the behavior of such a system
depends decisively on the initial conditions. If the
system is initially ideal, i.e. if the probability
qthat the elements are initially functional is
unity, the model leads to a power function for
failure rate increase with age (the Weibull law).
On the other hand, if the system is from the very
beginning replete with defects and the probability
for a given element being initially functional is
close to zero, the model leads to an exponential
function for failure rate increase with age (the
Gompertz law). In both cases, however, there
exists an upper limit to the failure rate which is
determined by the product of the number of
blocks (m) and the failure rate of the elements (k).
In this section, we shall examine the more general
case in which the probability of an element being
initially functional can assume any possible
value: 0(q)1.
In the general case, the distribution of blocks in
the organism according to the number of initially
functional elements is described by the binomial
distribution rather than the Poisson law. In this
case (see Appendix C), the failure rate is given by
kQ+cmknq L
G n!1
i!1(qkx)G\ (1!q)L\\G\.
(14)
The sum represented in this expression is the
binomial formula for the expression [(1!q)
#qkx]L\. It is therefore possible to write
kQ+cmn(qk)L1!q
qk #xL\
"cmn(qk)L(x#x)L\, (15)
where x"(1!q)/qk and xis a parameter
named the initial virtual age of the system,I<AS
(Gavrilov & Gavrilova, 1991). Indeed, this para-
meter has the dimension of time, and corresponds
to the age by which an initially ideal system
would have accumulated as many defects as
a real system already has at the initial moment in
time (at x"0). In particular, when q"1, i.e.
when all the elements are functional at the begin-
ning, the initial virtual age of the system is zero
and the failure rate grows as a power function of
age (the Weibull law), as described in the previous
section. However, when the system is not initially
ideal (q(1), we obtain the so-called binomial law
of mortality (Gavrilov & Gavrilova, 1991):
k(x)"A#(b#cx)L. (16)
In the case when x'0, there is always an initial
period of time, such that x;xand the following
approximation to the binomial law is valid:
kQ+cmn(qk)LxL\
1#x
xL\
+cmn(qk)LxL\
exp n!1
xx. (17)
536 L. A. GAVRILOV AND N. S. GAVRILOVA
FIG. 3. The dependence of the logarithm of mortality
force (failure rate) on age (computer simulation test). Note
that at early ages (20}60 yr), the data simulated with the
binomial law of mortality (k"b(x#x)L) are very close to
the linear relationship corresponding to the Gompertz law:
k"Re?V. At older ages, however, mortality deceleration is
observed, i.e. the mortality rates are increasing with age at
a slower pace compared to the Gompertz law (straight line
on a semi-log scale). The parameters of the binomial law of
mortality in this illustrative example are: x"100 yr;
n"10; b"10\ yr\. Although the choice of parameters
is arbitrary in this computer simulation, the obtained mor-
tality trajectory proved to be very close to the actual traject-
ory observed for human populations (see Gavrilov &
Gavrilova, 1991, p. 150).
Hence, for any value of q(1, there is always
a period of time xwhen the number of newly
formed defects is much less than the original
number, and the failure rate grows exponentially
with age
kQ"Rexp(ax), (18a)
where
R"cmn(qk)LxL\
"cmnqk(1!q)L\ (18b)
a"n!1
x
"kq(n!1)
1!q. (18c)
So, if the system is not initially ideal (q(1),
the failure rate in the initial period of time
grows exponentially with age according to the
Gompertz law. A numerical example provided in
Fig. 3 shows that the binomial law can reproduce
the important features of observed mortality
curves: exponential increase in mortality rates at
younger ages (20}60 yr), and &&mortality deceler-
ation'' in later life, when mortality rates increase
with age at a slower pace compared to the Gom-
pertz law. The mathematical proof of this state-
ment (asymptotic analyses) is provided later
[Formulae (21)}(23)].
The model discussed here not only provides an
explanation for the exponential increase in the
failure rate with age, but it also explains the
compensation law of mortality. Indeed, accord-
ing to the above formulae:
ln(R)"ln(cmnqk)!(n!1) ln1
1!q, (19a)
a"kq
1!q(n!1), (19b)
i.e. the quantities ln(R) and aare parametrically
linked via the quantity (n!1), allowing ln(R)to
be represented as a function of a:
ln(R)"ln(cma(1!q)#cmkq]!a(1!q)
kq ln1
1!q
"ln(M)!Ba, (20a)
where
M"cma(1!q)#cmkq (20b)
B"(1!q)
kq ln1
1!q. (20c)
Thus, the compensation law of mortality is
observed whenever di!erences in mortality are
caused by di!erences in initial redundancy (the
number of elements in a block, n), while the other
parameters, including the &&true aging rate'' (rate
of elements'loss k) are similar in populations
of a given species (presumably because of
homeostasis*stable body temperature, glucose
concentration, etc.). For lower organisms with
poor homeostasis there may be deviations from
this law. Our analysis of data published by
Pletcher et al. (2000) revealed that in Drosophila,
RELIABILITY THEORY 537
this law holds true for male-female comparisons
(maintaining the same temperature), but not for
experiments conducted at di!erent temperatures,
presumably because temperature may in#uence
the rate of element loss.
The length of the period when the failure rate
grows exponentially depends on the value of q.In
general, the q-dependent behavior of the system
in the age interval 0(x;1/kcan be reduced to
the following three scenarios:
(1) 0(q)1/2. This case corresponds to the
situation when less than half the total number of
elements is initially functional. In this case,
1!q
q*1 and, therefore, x*1/k. (21)
Therefore, over the entire interval when x;1/k,
the condition x;xis also valid. In this case, the
failure rate grows exponentially with age
throughout the interval under consideration.
(2) 1/2(q(1. This case corresponds to the
situation when more than half of all elements are
initially functional. In this case,
1!q
q(1 and, therefore, x(1/k. (22)
In these circumstances, the age-dependence of
mortality in the age interval under consideration
(0(x;1/k) consists of two stages:
(a) the "rst stage of the initial period, when
x;xand consequently the binomial law of
mortality reduces to the Gompertz law.
(b) the second stage of the initial period, when
x+xand only the binomial law of mortal-
ity in its full form is valid, without any ap-
proximations.
(3) 1/2;q(1. This case corresponds to the
situation when only a small proportion of ele-
ments is initially defective. In this case,
1!q
q;1 and, therefore, x;1/k. (23)
The age-dependence of mortality then consists
of three stages:
(a) the "rst stage of the initial period, when
x;xand the binomial law of mortality
reduces to the Gompertz law.
(b) the second stage of the initial period, when
x+xand only the binomial law of mortal-
ity is applicable.
(c) the third stage of the initial period, when
x;x;1/kand the binomial law of mortal-
ity reduces to the power law for failure rate
increase with age (the Weibull law).
As qtends to unity, the length of the "rst stage
of the initial period, with exponential growth in
the failure rate, is sharply reduced, and the length
of the third stage is sharply increased. In the
extreme case of an initially ideal system (q"1),
we come to the Weibull law, valid over the entire
age interval 0(x;1/k. This case has been de-
scribed in detail in the previous section.
The failure rate of the blocks asymptotically
approaches an upper limit which is independent
of the number of initially functional elements and
is equal to k. Therefore, the failure rate of a sys-
tem consisting of mblocks in series tends asymp-
totically with increased age to an upper limit mk,
independent of the values of nand q.
Thus, the reliability model described here pro-
vides an explanation for a general pattern of
aging and mortality in biological species: the ex-
ponential growth of failure rate in the initial
period, with the subsequent mortality deceler-
ation and leveling-o!, as well as the compensa-
tion law of mortality. In addition, the model
clari"es the conditions under which we observe
not the exponential law, but the power law for
failure rate increase with age (the Weibull law).
Finally, the model allows researchers to treat two
at "rst sight mutually exclusive laws, the Gom-
pertz law and the Weibull law, as special cases of
a more general binomial law of mortality which
follows from this model.
According to the proposed model, the fate of
non-functional elements and their death has no
e!ect whatsoever on the model's conclusions.
Therefore, the model will be valid even when all
the non-functional elements have already died by
the time the adult organism has been formed, and
538 L. A. GAVRILOV AND N. S. GAVRILOVA
the adult organism consists only of functional
elements (cells). What is important is that a trace
nevertheless remains in the form of the binomial
distribution of blocks according to the number of
functional elements within the organism. In fact,
this is the essence of the model, and consider-
ations of initially defective elements are only one
of the possible explanations for the existence of
variability in the initial degree of redundancy.
For this reason, the proposed model might also
be called the model of series-connected blocks
with varying degrees of redundancy.
Taking into account these notes, the basic con-
clusion of the model might be reformulated as
follows: if vital components of a system di+er in
their degree of redundancy,the mortality rate ini-
tially grows exponentially with age (according to
the Gompertz law)with subsequent leveling-o+in
later life. This statement is valid regardless of the
shape of the binomial distribution of blocks in
the organism according to their degree of re-
dundancy: whether there is negative (left-sided)
skewness, zero skewness (a normal distribution),
or positive (right-sided) skewness. The only e!ect
of the shape of the distribution is that in the case
of a negative (left-sided) skewness of the distribu-
tion, the exponential growth in the failure rate
may last only a short time, while in the case of
a zero or positive skewness the period of ex-
ponential growth in the failure rate is signi"-
cantly longer.
The model may also help in resolving an ap-
parent contradiction between exponential in-
crease in total mortality with age, as opposed to
non-exponential (e.g. power) increase in mortal-
ity from particular causes of death. Indeed, the
classi"cation of diseases and causes of death is
largely based on the anatomical principle and
registration of the damage to particular organs
and tissues of the organism. One of the interest-
ing features of the model is that each component
(block) may fail according to the Weibull (power)
law, but this in turn may lead to exponential
increase of failure rate for the whole organism.
Indeed, it turned out that the Gompertz
function is a better descriptor for &&all-causes'' of
death and combined disease categories, while the
Weibull function is a better descriptor of purer,
single causes-of-death (Juckett & Rosenberg,
1993).
5. Concluding Remarks
Reliability theory allows researchers to predict
the age-related failure kinetics for a system of
given architecture (reliability structure) and given
reliability of its components. As shown in this
article, the theory provides explanations of the
fundamental problems regarding species aging
and longevity that were posed in the introduction
of this paper:
(1) Reliability theory explains why most bio-
logical species deteriorate with age (i.e. die
more often as they grow older) while some primi-
tive organisms do not demonstrate such a clear
age dependence for mortality increase. The
theory predicts that even those systems that are
entirely composed of non-aging elements (with
a constant failure rate) will nevertheless deterior-
ate (fail more often) with age, if these systems
are redundant in irreplaceable elements. Aging,
therefore, is a direct consequence of systems
redundancy. The &&apparent aging rate'' (the rela-
tive rate of age-related mortality acceleration
corresponding to parameter ain the Gompertz
law) increases, according to reliability theory,
with higher redundancy levels. Therefore, a
negligible &&apparent aging rate'' in primitive or-
ganisms (Haranghy & BalaHzs, 1980; Finch, 1990;
Martinez, 1998) with little redundancy is a
predicted observation for reliability theory.
At this point, however, evolutionary biologists
have good reasons to argue with our suggested
explanation for negligible senescence. This is be-
cause evolutionary theory also predicts negligible
senescence among some primitive organisms, but
for a completely di!erent reason (lack of parent}
o!spring asymmetry, see Charlesworth, 1994,
pp. 246}247). Evolutionary biologists believe that
e!ective repair mechanisms are responsible for
the negligible aging rate in some species and give
the following arguments: &&2unicellular organ-
isms, such as bacteria, which propagate simply by
binary "ssion, and the germ lines of multicellular
organisms, have been able to propagate themsel-
ves without senescence over billions of years,
showing that biological systems are capable of
ongoing repair and maintenance and so can
avoid senescence at the cellular level. Senescence
cannot, therefore, just be an unavoidable
RELIABILITY THEORY 539
cumulative result of damage.'' (Charlesworth,
2000, p. 927). It is important, however, not to
overlook two key related questions posed by
reliability theory:
What are the actual death rates observed in
populations of species with negligible senescence,
as well as among germ cells?
Are these death rates really negligible (indicat-
ing perfect repair) or, on the contrary, quite high
(indicating low redundancy of these cells and
organisms)?
Debates on these issues may be expected in the
future, but there are promising opportunities for
merging the reliability and the evolutionary the-
ories (Miller, 1989).
(2) The reliability theory explains why mortal-
ity rates increase exponentially with age in many
adult species (Gompertz law) by taking into ac-
count the initial -aws (defects) in newly formed
systems. It also explains why organisms &&prefer''
to die according to the Gompertz law, while
technical devices usually fail according to the
Weibull (power) law. Moreover, the theory pro-
vides a sound strategy for handling those cases
when the Gompertzian mortality law is not ap-
plicable. In this case, the second best choice
would be the Weibull law, which is also funda-
mentally grounded in reliability theory. Theoret-
ical conditions are speci"ed when organisms die
according to the Weibull law: organisms should
be relatively free of initial #aws and defects.
In those cases when none of these two
mortality laws is appropriate, reliability theory
o!ers more general failure law applicable to adult
and extreme old ages. The Gompertz and the
Weibull laws are just special cases of this
unifying more general law (see earlier Sections 3
and 4).
(3) Reliability theory also explains why the age-
related increase in mortality rates vanishes at
older ages. It predicts the late-life mortality decel-
eration with subsequent leveling-o!, as well as
the late-life mortality plateaus, as an inevitable
consequence of redundancy exhaustion at extreme
old ages. This is a very general prediction of
reliability theory: it holds true for systems built of
elements connected in parallel, for hierarchical
systems of serial blocks with parallel elements
(see earlier Sections 3 and 4), for highly intercon-
nected networks of elements (Bains, 2000), and
for systems with avalanche-like random failures
(Gavrilov & Gavrilova, 1991).
The reliability theory also predicts that the
late-life mortality plateaus will be observed at
any level of initial damage: for initially ideal sys-
tems, for highly redundant systems replete with
defects, and for partially damaged redundant sys-
tems with an arbitrary number of initial defects
(see earlier).
Furthermore, reliability theory predicts para-
doxical mortality decline in late life (before event-
ual leveling-o!to mortality plateau) if the system
is redundant for non-identical components with
di!erent failure rates (Barlow et al., 1965; Barlow
& Proschan, 1975). Thus, in those cases when
&&apparent rejuvenation'' is observed (mortality
decline among the oldest-old) there is no need to
blame data quality or to postulate initial popula-
tion heterogeneity and &&second breath'' in cen-
tenarians. The late-life mortality decline is an
inevitable consequence of age-induced population
heterogeneity expected even among initially
identical individuals, redundant in non-identical
system components (Gavrilov & Gavrilova,
unpublished). Recently, this general explanation
was also supported using computer simulations
(Bains, 2000). Late-life mortality decline was
observed in many studies (Barrett, 1985; Carey
et al., 1992; Khazaeli et al., 1995; Klemera &
Doubal, 1997) and stimulated interesting debates
(Klemera & D[oubal, 1997; Olshansky, 1998)
because of the lack of reasonable explanation.
Reliability theory predicts that the late-life
mortality decline is an expected scenario of
systems failure.
(4) The theory explains the compensation law
of mortality, when the relative di!erences in mor-
tality rates of compared populations (within
a given species) decrease with age, and mortality
convergence is observed due to the exhaustion of
initial di!erences in redundancy levels. Reliability
theory also predicts that those experimental in-
terventions that change &&true aging rate'' (rate of
elements'loss) will also suppress mortality con-
vergence, providing a useful approach on how to
search for factors a!ecting aging rate.
Overall, reliability theory has an amazing
predictive and explanatory power and requires
only a few general and realistic assumptions. It
o!ers a promising approach for developing
540 L. A. GAVRILOV AND N. S. GAVRILOVA
a comprehensive theory of aging and longevity
that integrates mathematical methods with biolo-
gical knowledge including cell biology (Aber-
nethy, 1998), evolutionary theory (Miller, 1989;
Charlesworth, 1994) and systems repair prin-
ciples (Rigdon & Basu, 2000).
We are most grateful to Mr Brian Whiteley for
useful editorial suggestions and to two anonymous
reviewers for constructive criticism of this work. We
would also like to acknowledge partial support from
the National Institute on Aging grants.
REFERENCES
ABERNETHY, J. D. (1979). The exponential increase in mor-
tality rate with age attributed to wearing-out of biological
components. J.theor.Biol.80, 333}354.
ABERNETHY, J. (1998). Gompertzian mortality originates in
the winding-down of the mitotic clock. J.theor.Biol.192,
419}435, jt980657.
AVEN,T.&JENSEN, U. (1999). Stochastic Models in Reliabil-
ity. New York: Springer-Verlag.
BAINS, W. (2000). Statistical mechanic prediction of non-
Gompertzian ageing in extremely aged populations. Mech.
Ageing Dev.112, 89}97.
BARKER, D. J. P. (1992). Fetal and Infant Origins of Adult
Disease. London: BMJ Publishing Group.
BARKER, D. J. P. (1998). Mothers,Babies,and Disease in
¸ater ¸ife, 2nd Edn. London: Churchill Livingstone.
BARLOW,R.E.&PROSCHAN, F. (1975). Statistical ¹heory of
Reliability and ¸ife ¹esting.Probability Models. New York:
Holt, Rinehart and Winston.
BARLOW, R. E., PROSCHAN,F.&HUNTER, L. C. (1965).
Mathematical ¹heory of Reliability. New York: John Wiley
& Sons, Inc.
BARRETT, J. C. (1985). The mortality of centenarians in
England and Wales. Arch.Gerontol.Geriatr.4, 211}218.
BLACKWELL, D. L., HAYWARD,M.D.&CRIMMINS,E.M.
(2001). Does childhood health a!ect chronic morbidity in
later life? Social Science & Medicine 52, 1269}1284.
BROWNLEE, J. (1919). Notes on the biology of a life-table.
J.Roy.Stat.Soc.82, 34}77.
CAREY,J.R.&LIEDO, P. (1995). Sex-speci"c life table aging
rates in large med#y cohorts. Exp.Gerontol.30, 315}325.
CAREY, J. R., LIEDO, P., OROZCO,D.&VAUPEL,J.W.
(1992). Slowing of mortality rates at older ages in large
Med#y cohorts. Science 258, 457}461.
CARNES,B.A.&OLSHANSKY, S. J. (1993). Evolutionary
perspectives on human senescence. Popul.Dev.Rev.19,
793}806.
CARNES, B. A., OLSHANSKY, S. J., GAVRILOV, L. A., GAV-
RILOVA,N.S.&GRAHN, D. (1999). Human longevity:
nature vs. nurture*fact or "ction. Persp.Biol.Med.42,
422}441.
CHARLESWORTH, B. (1994). Evolution in Age-structured
Populations, 2nd Edn. Cambridge: Cambridge University
Press.
CHARLESWORTH, B. (2000). Fisher, Medawar, Hamilton and
the evolution of aging. Genetics 156, 927}931.
CHARLESWORTH,B.&PARTRIDGE, L. (1997). Ageing: levell-
ing of the grim reaper. Curr.Biol.7, R440}R442.
CLARK,A.G.&GUADALUPE, R. N. (1995). Probing the
evolution of senescence in Drosophila melanogaster with
P-element tagging. Genetica 96, 225}234.
CROWDER, M. J., KIMBER, A. C., SMITH,R.L.&SWEETING,
T. J. (1991). Statistical Analysis of Reliability Data. London:
Chapman and Hall.
DOBLHAMMER, G. (1999). Longevity and month of birth:
evidence from Austria and Denmark. Demogr.Res. [On-
line] 1, 1}22. Available: http://www.demographic-
research.org/Volumes/Vol1/3/default.htm.
DOBZHANSKY, T. (1962). Mankind Evolving.¹he Evolution of
Human Species. New Haven and London: Yale University
Press.
D[OUBAL, S. (1982). Theory of reliability, biological systems
and aging. Mech.Ageing Dev.18, 339}353.
EAKIN, T., SHOUMAN, R., QI, Y. L., LIU,G.X.&WITTEN,
M. (1995). Estimating parametric survival model para-
meters in gerontological aging studies. Methodological
problems and insights. J.Gerontol.Ser.A50, B166}B176.
ECONOMOS, A. C. (1979). A non-gompertzian paradigm for
mortality kinetics of metazoan animals and failure kinetics
of manufactured products. AGE 2, 74}76.
ECONOMOS, A. C. (1980). Kinetics of metazoan mortality.
J.Soc.Biol.Struct.3, 317}329.
ECONOMOS, A. C. (1983). Rate of aging, rate of dying and the
mechanism of mortality. Arch.Gerontol.Geriatr.1, 3}27.
ECONOMOS, A. (1985). Rate of aging, rate of dying and
non-Gompertzian mortality*encore2Gerontology 31,
106}111.
EKERDT, D. J. (ed.) (2002). ¹he Macmillan Encyclopedia of
Aging. New York: Macmillan Reference USA (in press).
FELLER, W. (1968). An Introduction to Probability ¹heory
and its Applications, Vol. 1. New York: Wiley and Sons.
FINCH, C. E. (1990). ¸ongevity,Senescence and the Genome.
Chicago: University of Chicago Press.
FINCH,C.E.&KIRKWOOD, T. B. L. (2000). Chance,Devel-
opment,and Aging. New York, Oxford: Oxford University
Press.
FINCH,C.E.&TANZI, R. E. (1997). Genetics of aging.
Science 278, 407}411.
FINGER, S., LEVERE, T. E., ALMLI,C.R.&STEIN,D.G.
(eds) (1988). Brain Injury and Recovery:¹heoretical and
Controversial Issues. New York: Plenum Press.
FUKUI, H. H., XIU,L.&CURTSINGER, J. W. (1993). Slowing
of age-speci"c mortality rates in Drosophila melanogaster.
Exp.Gerontol.28, 585}599.
FUKUI, H. H., ACKERT,L.&CURTSINGER, J. W. (1996).
Deceleration of age-speci"c mortality rates in chromo-
somal homozygotes and heterozygotes of Drosophila
melanogaster.Exp.Gerontol.31, 517}531.
GALAMBOS, J. (1978). ¹he Asymptotic ¹heory of Extreme
Order Statistics. New York: Wiley.
GAVRILOV, L. A. (1978). Mathematical model of aging in
animals. Proc.Natl Acad.Sci.;SSR (Doklady Akademii
Nauk SSSR,Moscow)238, 490}492 (in Russian).
GAVRILOV, L. A. (1984). Does a limit of the life span really
exist? Bio,zika 29, 908}911.
GAVRILOV, L. A. (1987). Critical analysis of mathematical
models of aging, mortality, and life span. In: Population
Gerontology [Populyatsionnaya Gerontologia] (Burlakova,
E. B. & Gavrilov, L. A., eds), Vol. 6, pp. 155}189. Moscow:
VINITI (in Russian).
GAVRILOV,L.A.&GAVRILOVA, N. S. (1991). ¹he Biology of
¸ife Span:A Quantitative Approach. New York: Harwood
Academic Publisher.
RELIABILITY THEORY 541
GAVRILOV,L.A.&GAVRILOVA, N. S. (1993) Fruit #y aging
and Mortalilty. Science 260, 1565.
GAVRILOV,L.A.&GAVRILOVA, N. S. (1997a). Parental age
at conception and o!spring longevity. Rev.Clin.Gerontol.
7, 5}12.
GAVRILOV,L.A.&GAVRILOVA, N. S. (1997b). When father-
hood should stop? Science 277, 17}18.
GAVRILOV,L.A.&GAVRILOVA, N. S. (1999). Season of
birth and human longevity. J.Anti-Aging Med.2,
365}366.
GAVRILOV,L.A.&GAVRILOVA, N. S. (2000). Human lon-
gevity and parental age at conception. In: Sex and ¸ong-
evity:Sexuality,Gender,Reproduction,Parenthood
(T.B.L. Kirkwood & M. Allard, J.-M. Robine eds), pp.
7}31. Berlin, Heidelberg: Springer-Verlag.
GAVRILOV, L. A., GAVRILOVA,N.S.&IAGUZHINSKII,L.S.
(1978). Basic patterns of aging and death in animals from
the standpoint of reliability theory. J.Gen.Biol.(Zh.Ob-
shchej Biol.Moscow)39, 734}742 (in Russian).
GOMPERTZ, B. (1825). On the nature of the function expres-
sive of the law of human mortality and on a new mode of
determining life contingencies. Philos.¹rans.Roy.Soc.
¸ond.A115, 513}585.
GREENWOOD,M.&IRWIN, J. O. (1939). The biostatistics of
senility. Hum.Biol.11, 1}23.
GRIFFITHS, A. J. F., MILLER, J. H., SUZUKI, D. T., LEWON-
TIN,R.C.&GELBART, W. M. (1996). An Introduction to
Genetic Analysis, 6th Edn. New York: W. H. Freeman and
Company.
GUMBEL, E. J. (1958). Statistics of Extremes. New York:
Columbia University Press.
HAMILTON, W. D. (1966). The molding of senescence by
natural selection. J.theor.Biol.12, 12}45.
HANDYSIDE,A.H.&DELHANTY, J. D. A. (1997). Preimplan-
tation genetic diagnosis: strategies and surprises. ¹rends
Genet.13, 270}275.
HARANGHY,L.&BALADZS, A. (1980). Regeneration and reju-
venation of invertebrates. In: Perspectives in Experimental
Gerontology (Shock, N. W., ed.), pp. 224}233. New York:
Arno Press.
HARMAN,D.&EDDY, D. E. (1979). Free radical theory
of aging: bene"cial e!ects of adding antioxidants to the
maternal mouse diet on life span of o!spring: possible
explanation of the sex di!erence in longevity. AGE 2,
109}122.
HIRSCH,H.R.&PERETZ, B. (1984). Survival and aging of
a small laboratory population of a marine mollusc, Aplysia
californica.Mech.Ageing Dev.27, 43}62.
HIRSCH, A. G., WILLIAMS,R.J.&MEHL, P. (1994). Kinetics
of med#y mortality. Exp.Gerontol.29, 197}204.
JANSE, C., SLOB, W., POPELIER,C.M.&VOGELAAR,J.W.
(1988). Survival characteristics of the mollusc ¸ymnaea
stagnalis under constant culture conditions: e!ects of aging
and disease. Mech.Ageing Dev.42, 263}174.
JAZWINSKI, S. M. (1996). Longevity, genes, and aging.
Science 273, 54}59.
JAZWINSKI, S. M. (1998). Genetics of longevity. Exp.Geron-
tol.33, 773}783.
JUCKETT,D.A.&ROSENBERG, B. (1993). Comparison of the
Gompertz and Weibull functions as descriptors for human
mortality distributions and their intersections. Mech.Age-
ing Dev.69, 1}31.
KAUFMANN, A., GROUCHKO,D.&CRUON, R. (1977). Math-
ematical Models for the Study of the Reliability of Systems.
New York: Academic Press.
KHAZAELI, A. A., XIU,L.&CURTSINGER, J. W. (1995). Stress
experiments as a means of investigating age-speci"cmortal-
ity in Drosophila melanogaster.Exp.Gerontol.30, 177}184.
KHAZAELI, A. A., XIU,L.&CURTSINGER, J. W. (1996). E!ect
of density on age-speci"c mortality in Drosophila: a den-
sity supplementation experiment. Genetica 98, 21}31.
KLEMERA,P.&DOUBAL, S. (1997). Human mortality at
very advanced age might be constant. Mech.Ageing Dev.
98, 167}176.
KOLTOVER, V. K. (1983). Theory of reliability, superoxide
radicals, and aging. Adv.Mod.Biol.(;spekhi Sovremennoj
Biol.Moscow)96, 85}100 (in Russian).
KOLTOVER, V. K. (1997). Reliability concept as a trend in
biophysics of aging. J.theor.Biol.184, 157}163.
KUH,D.&BEN-SHLOMO, B. (1997). A¸ife Course Approach
to Chronic Disease Epidemiology. Oxford: Oxford Univer-
sity Press.
LEBRAS, H. (1976). Lois de mortaliteHet age limiteH.Popula-
tion 31, 655}692.
LEON, D. A., LITHELL, H. O., VA
sGEROG, D., KOUPILOVAD, I.,
MOHSEN, R., BERGLUND, L., LITHELL, U.-B.
&MCKEIGUE, P. M. (1998). Reduced fetal growth rate and
increased risk of death from ischaemic heart disease: co-
hort study of 15000 Swedish men and women born
1915}29. Br.Med.J.317, 241}245.
LLOYD,D.K.&LIPOW, M. (1962). Reliability:Management,
Methods,and Mathematics. Englewood Cli!s, NJ: Pren-
tice-Hall, Inc.
LUCAS, A. (1991). Programming by early nutrition in man.
In: ¹he Childhood Environment and Adult Disease (Bock, G.
R. & Whelan, J., eds), pp. 38}55. Chichester: Wiley.
LUCAS, A., FEWTRELL,M.S.&COLE, T. J. (1999). Fetal
origins of adult disease*the hypothesis revisited. Br.Med.
J.319, 245}249.
MAKEHAM, W. M. (1860). On the law of mortality and the
construction of annuity tables. J.Inst.Actuaries 8, 301}310.
MAKEHAM, W. M. (1867). On the law of mortality. J.Inst.
Actuaries 13, 325}358.
MARTIN, L. J., BRAMBRINK, A. M., PRICE, A. C., KAISER, A.,
AGNEW, D. M., ICHORD,R.N.&TRAYSTMAN, R. J. (2000).
Neuronal death in newborn striatum after hypoxia-
ischemia is necrosis and evolves with oxidative stress.
Neurobiol.Dis.7, 169}191.
MARTINEZ, D. E. (1998). Mortality patterns suggest lack of
senescence in hydra. Exp.Gerontol.33, 217}225.
MCLAREN, A. (1998). Genetics and human reproduction.
¹rends Genet.14, 427}431.
MEDAWAR, P. B. (1946). Old age and natural death. Mod.Q.
2, 30}49. [Reprinted in ¹he ;niqueness of the Individual
(Medawar, P. B., ed.), pp. 17}43. New York: Basic Books,
1958.
MEDAWAR, P. B. (1952). An ;nsolved Problem in Biology.
London: H. K. Lewis. [Reprinted in ¹he ;niqueness of the
Individual (Medawar, P. B., ed.), pp. 44 }70. New York:
Basic Books, 1958.
MILDVAN,A.&STREHLER, B. L. (1960). A critique of the-
ories of mortality. In: ¹he Biology of Aging (Strehler, B. L.,
Ebert, J. D., Glass, H. B. & Shock, N. W., eds),
pp. 216}235. Washington, D.C.: American Institute of Bio-
logical Sciences.
MILLER, A. R. (1989). The distribution of wearout over
evolved reliability structures. J.theor.Biol.136, 27}46.
MOFFETT, D. F., MOFFETT,S.B.&SCHAUF, C. L. (1993).
Human Physiology:Foundations and Frontiers, 2nd Edn.
Dubuque, etc.: Wm. C. Brown Publishers.
542 L. A. GAVRILOV AND N. S. GAVRILOVA
MUELLER,L.&ROSE, M. R. (1996). Evolutionary theory
predicts late-life mortality plateaus. Proc.Natl Acad.Sci.
;.S.A.93, 15 249}15 253.
MURPHY, E. A. (1978). Genetics of longevity in man. In:
¹he Genetics of Aging (Schneider, E. L., ed.), pp. 261}301.
New York: Plenum Press.
OLSHANSKY, S. J. (1998). On the biodemography of aging:
a review essay. Popul.Dev.Rev.24, 381}393.
OLSHANSKY,S.J.&CARNES, B. A. (1997). Ever since Gom-
pertz. Demography 34, 1}15.
PARTRIDGE,L.&MANGEL, M. (1999). Messages from mor-
tality: the evolution of death rates in the old. ¹rends Ecol.
Evol.14, 438}442.
PERKS, W. (1932). On some experiments in the graduation of
mortality statistics. J.Inst.Actuaries 63, 12}57.
PLETCHER,S.D.&CURTSINGER, J. W. (1998). Mortality
plateaus and the evolution of senescence: why are old-age
mortality rates so low? Evolution 52, 454}464.
PLETCHER,S.D.&NEUHAUSER, C. (2000). Biological ag-
ing*criteria for modeling and a new mechanistic model.
Int.J.Mod.Phys.C11, 525}546.
PLETCHER,S.D.KHAZAELI,A.A.&CURTSINGER,J.W.
(2000). Why do life spans di!er? partitioning mean longe-
vity di!erences in terms of age-speci"c mortality para-
meters. J.Gerontol.55A, B381}B389.
RIGDON,S.E.&BASU, A. P. (2000). Statistical Methods for
the Reliability of Repairable Systems. New York: John
Wiley & Sons, Inc.
ROSE, M. R. (1991). ¹he Evolutionary Biology of Aging.
Oxford: Oxford University Press.
SKURNICK,I.D.&KEMENY, G. (1978). Stochastic studies of
aging and mortality in multicellular organisms. I. The
asymptotic theory. Mech.Ageing Dev.7, 65}80.
STREHLER, B. L. (1960). Fluctuating energy demands as
determinants of the death process (A parsimonious theory
of the Gompertz function). In: ¹he Biology of Aging (Streh-
ler, B. L., Ebert, J. D., Glass, H. B. & Shock, N. W., eds),
pp. 309}314. Washington, D.C.: American Institute of Bio-
logical Sciences.
STREHLER, B. L. (1978). ¹ime,Cells,and Aging, 2nd Edn.
New York and London: Academic Press.
STREHLER,B.L.&MILDVAN, A. S. (1960). General theory of
mortality and aging. Science 132, 14}21.
TOWER, J. (1996). Aging mechanisms in fruit #ies. BioEssays
18, 799}807.
VANFLETEREN, J. R., DEVREESE,A.&BRAECKMAN,B.P.
(1998). Two-parameter logistic and Weibull equations pro-
vide better "ts to survival data from isogenic populations
of Caenorhabditis elegans in axenic culture than does the
Gompertz model. J.Gerontol.Ser.A53, B393}403.
VAUPEL, J. W., CAREY, J. R., CHRISTENSEN, K., JOHNSON,
T., YASHIN, A. I., HOLM, N. V., IACHINE, I. A., KANNISTO,
V., KHAZAELI, A. A., LIEDO, P., LONGO, V. D., ZENG, Y.,
MANTON,K.&CURTSINGER, J. W. (1998). Biodemo-
graphic trajectories of longevity. Science 280, 855}860.
WACHTER, K. W. (1999). Evolutionary demographic models
for mortality plateaus. Proc.Natl Acad.Sci.;.S.A.96,
10 544}10 547.
WILLIAMS, G. C. (1957). Pleiotropy, natural selection and
the evolution of senescence. Evolution 11, 398}411.
WILLIAMS, G. C. (1966). Natural selection, the costs of
reproduction, and a re"nement of Lack's principle. Am.
Nat.100, 687}690.
WITTEN, M. (1985). A return to time, cells, systems, and
aging: III. Gompertzian models of biological aging and
some possible roles for critical elements. Mech.Ageing Dev.
32, 141}177.
APPENDIX A
Consider the simplest case of the model when
all the elements of the system are initially func-
tional (which is typical for technical devices) and
have a constant failure rate k. If these non-aging
elements are organized into blocks of nmutually
substitutable elements so that the failure of
a block occurs only when all the elements of the
block fail (parallel construction in the reliability
theory context), the cumulative distribution func-
tion for block failure, F@(n,k,x), depends on age
xin the following way:
F@(n,k,x)"P(X)x)"(1!e\IV)L. (A.1)
Therefore, the reliability function of a block,
S@(n,k,x) can be represented as
S@(n,k,x)"1!F@(n,k,x)"1!(1!e\IV)L. (A.2)
The probability density function, f@(n,k,x)is
de"ned as
f@(n,k,x)"dF@(n,k,x)
dx"!dS@(n,k,x)
dx
"nke\IV(1!e\IV)L\. (A.3)
Consequently, the failure rate of a block k@(n,k,x)
can be written as follows:
k@(n,k,x)"! dS@(n,k,x)
S@(n,k,x)dx"f@(n,k,x)
S@(n,k,x)
"nke\IV(1!e\IV)L\
1!(1!e\IV)L. (A.4)
APPENDIX B
Consider the case when the distribution of the
blocks in the organism according to the number
iof initially functional elements they contain is
described by the Poisson law with parameter
j"nq, corresponding to the mean number of
initially functional elements in a block. Strictly
RELIABILITY THEORY 543
speaking, this distribution ought to be truncated
on the right, since the number of functional ele-
ments (i), cannot exceed the total number (n)of
elements in a block. In addition, for initially liv-
ing organisms the distribution ought also to be
truncated on the left, since according to the
model, an organism which contains a block with-
out any functional elements (i"0) cannot be
alive. Therefore, the distribution of blocks ac-
cording to the number iof initially functional
elements within initially living organisms is deter-
mined by the following probabilities PG:
PG"0 for i"0, n#1, n#2, n#3,2,
instead of PG"e\H(j)G
i!, (B.1)
PG"ce\H(j)G
i!for i"1, 2, 3,2,n; (B.2)
where c"1
1!e\H!e\H
GL>(j)G/i!. (B.3)
Parameter cis a normalizing factor that ensures
the sum of the probabilities of all possible out-
comes is equal to unity:
L
G
PG"1. (B.4)
For su$ciently high values of nand j, the nor-
malizing factor turns out to be hardly greater
than unity.
As has already been noted [see eqn (8)], the
failure rate of a system constructed out of
mblocks connected in series is equal to the sum
of the failure rates of these blocks k@(i):
kQ"K
H
k@(i,j)"L
G
mPGk@(i)
"mce\HL
G
(j)Gk@(i)
i!. (B.5)
In its turn, the failure rate of blocks with iini-
tially functional elements is given by
k@(i)"ike\IV(1!e\IV)G\
1!(1!e\IV)G(B.6)
+ik(kx)G\ when x;1/k(early life period
approximation, when 1!e\IV+kx)
+kwhen x<1/k(late-life period approxima-
tion, when 1!e\IV+1).
By putting together these two formulae, we
obtain
kQ"kjmce\He\IV L
G
jG\ (1!e\IV)G\
(i!1)!(1!(1!e\IV)G).
(B.7)
APPENDIX C
Consider the case when the distribution of
blocks in the organism according to the number
of initially functional elements is described by the
binomial distribution. For an initially living or-
ganism, this distribution has to be truncated on
the left, since according to the model, an organ-
ism which contains a block without any func-
tional elements (i"0) cannot be alive.
Therefore, the distribution of blocks according
to the number iof initially functional elements
within initially living organisms is given by the
following probabilities:
PG"0 for i"0, instead of P"(1!q)L, (C.1)
PG"cn
iqG(1!q)L\Gfor i"1, 2, 3,2,n, (C.2)
where
n
i"n!
i!(n!i)!"n
in!1
i!1
for i"1, 2, 3,2,n, (C.3)
c"1
1!(1!q)L*1, (C.4)
Parameter cis a normalizing factor that
ensures that the sum of the probabilities of all
544 L. A. GAVRILOV AND N. S. GAVRILOVA
outcomes is unity
L
G
PG"1. (C.5)
The failure rate of a system constructed of
mseries-connected blocks is equal to the sum of
the failure rates of the blocks:
kQ"K
H
k@(i,j)"L
G
mPGk@(i)
"cm L
G n
iqG(1!q)L\Gk@(i). (C.6)
As has been already noted [see eqn (B.6)], at
the initial moment in time, when x;1/k, the
failure rate of a block with iinitially functional
elements is given by
k@(i)+ik(kx)G\. (C.7)
Putting together these two formulae, we obtain
kQ+cm L
G n
iqG(1!q)L\Gik(kx)G\. (C.8)
Since in
i"nn!1
i!1for i"1, 2, 3,2,n,we
obtain
kQ+cmknq L
G n!1
i!1(qkx)G\ (1!q)L\\G\.
(C.9)
RELIABILITY THEORY 545
