And%20one%20man%20in%20his%20time%20plays%20many%20parts,

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Stochastic models of aging and mortality

• And one man in his time plays many parts,
• His acts being seven ages

• For his shrunk shank and his big manly voice,
• Turning again toward childish treble, pipes
• And whistles in his sound. Last scene of all,
• That ends this strange eventful history,
• Is second childishness and mere oblivion,
• Sans teeth, sans eyes, sans taste, san
  everything.

The sixth age shifts Into the
lean and slipper'd pantaloon, With spectacles on
nose and pouch on side, His youthful hose, well
saved, a world too wide
A nonstochastic model of aging and mortality by
W. Shakespeare and Titian
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What is aging?

• J. M. Smith (1962) Aging processes are those
  which render individuals more susceptible as they
  grow older to the various factors, intrinsic or
  extrinsic, which may cause death.
• P. T. Costa and R. R. McCrae (1995) What
  happens to an organism over time.

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• Force of mortality (hazard rate) increases with
  age

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Increasing mortality as a proxy for aging

• We cant measure aging processes directly,
  particularly since we cant define them.
• Mortality rates are easy to measure. In most
  metazoans, mortality rates increase with age.

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Increasing mortality as a proxy for aging

• We cant measure aging processes directly,
  particularly since we cant define them.
• Mortality rates are easy to measure. In most
  metazoans, mortality rates increase with age.
• This includes us.

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This is not a trivial observation!

• Implicit in Roman annuity rates.
• 17th C. annuity and life-insurance rates were
  generally age-independent.
• Annuity as gamble Who is the best bet?
• Very old (for whom the increased hazard would be
  most clear) rare, uncertain age.
• Extreme haphazardness of plagues, wars.

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Some questions about aging

1. Why do creatures age?

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Some questions about aging

1. Why do creatures age?

Old (and recurrent) idea Improper nutrition.

• Unto the woman he said, I will greatly multiply
  thy sorrow and thy conception in sorrow thou
  shalt bring forth children and thy desire shall
  be to thy husband In the sweat of thy face shalt
  thou eat bread, till thou return unto the ground
  for out of it wast thou taken for dust thou art,
  and unto dust shalt thou return.

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Some questions about aging

1. Why do creatures age?

THINGS FALL APART
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Some questions about aging

1. Why do creatures age?

Problems with this naïve answer

• Repair.
• Not universal.

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Aging not universal.

• Negligible senescence prokaryotes, bristlecone
  pine, tortoises, lobster
• Gradual senescence mammals, birds, fish, yeast
• Rapid senescence flies, bees (workers), nematodes

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Some questions about aging

1. Why do creatures age?
2. Why does aging have the particular age-patterns
   that it does?

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Some questions about aging

1. Why do creatures age?
2. Why does aging have the particular age-patterns
   that it does?
3. Why do different species have characteristic
   patterns of aging?

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Some questions about aging

1. Why do creatures age?
2. Why does aging have the particular age-patterns
   that it does?
3. Why do different species have characteristic
   patterns of aging?
4. Why is aging so variable?

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Some questions about aging

1. Why do creatures age?
2. Why does aging have the particular age-patterns
   that it does?
3. Why do different species have characteristic
   patterns of aging?
4. Why is aging so variable?
5. Why is aging so constant?

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The Gompertz-Makeham mortality law

• Gompertz (1825) we observe that in those tables
  the numbers of living in each yearly increase of
  age are from 25 to 45 nearly, in geometrical
  progression.
• Makeham (1867) Theory of partial forces of
  mortality. Diseases of lungs, heart, kidneys,
  stomach, liver, brain associated with diminution
  of the vital power.

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log hazard rate in Japan 1981-90
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log infectious disease hazard rate in Japan
1981-90
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log cancer hazard rate in Japan 1981-90
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log suicide hazard rate in Japan 1981-90
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log auto accident hazard rate in Japan 1981-90
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log breast cancer hazard rate in Japan 1981-90
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log homicide hazard rate in Japan 1981-90
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Species Initial mort. MRDT (yrs) Max life (yrs)

Human (US F, 1980 .0002 8.9 122
Herring Gull .0032 5.4 49
Horse .0002 4 46
Rhesus monkey .02 15 gt35
Starling .5 gt8 20
Lake sturgeon .013 10 gt150
House fly 4-12 .02-.04 .3
Soil nematode 2 .02 .15
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MRDT seems to be species-characteristic
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Mortality plateaus
Female mortality at ages 80 (Japan 13 W.
European countries (1980-92)
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Mediterranean fruit fly mortality
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Is this about biology?
Lifetimes of electrical relays
Force of junking for automobiles in
various periods
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What is a Markov process?

• A stochastic process Xt (where t is time, usually
  taken to be the positive reals) such that if you
  know the process up to a given time t, the
  behavior after time t depends only on the state
  at time t.
• The process may be killed, either randomly (at a
  rate depending on the current position) or
  instantaneously when it hits a certain part of
  the state space.

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Lessons for young scientists from reviewing the
Markov mortality model literature

• Its easy to get your work published if your
  model reproduces known phenomena

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Lessons for young scientists from reviewing the
Markov mortality model literature

• Its easy to get your work published if your
  model reproduces known phenomena
• even if you put them in (decently concealed)
  with your assumptions

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Lessons for young scientists from reviewing the
Markov mortality model literature

• Its easy to get your work published if your
  model reproduces known phenomena
• even if you put them in (decently concealed)
  with your assumptions
• and even if the mathematics is wrong.

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Challenge to homeostasis (B. Strehler, A.
Mildvan 1960)

• The rate of decrease of most physiologic
  functions of human beings is between 0.5 and 1.3
  percent per year after age 30, and is fit as well
  by a straight line as by any other simple
  mathematical function.
• Challenges come at constant rate.
• Death occurs when a challenge exceeds the
  organisms vital capacity.
• Challenges follow the Maxwell-Boltzmann
  distribution Exponentially distributed.

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Challenge to homeostasis (B. Strehler, A.
Mildvan 1960)

• Predicts the Gompertz curve.
• Predicts the nonintuitive fact that initial
  mortality and rate of aging are inversely
  related.
• Problem The exponential rate was built into the
  assumptions, for which there is no external basis.

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Extreme-value theory

• J. D. Abernethy (J. Theor. Biol. 1979) Model
  organisms by independent systems, which all
  fais at the same random rate. Death is the
  time of the first failure.
• Proves that such an organism could have
  exponentially increasing death rates.
• Claims (in the nonmathematical introduction and
  conclusion) that this is the generic situation,
  which will arise whenever the hazard rates of the
  components are nondecreasing, which is untrue.
  (In fact, the individual components would also
  have to have exponential hazard rates.)
• Still gets cited.

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Reliability theory

• M. Witten (1985) large number (m) of critical
  components death comes when all fail.
• Components are independent. Fail with constant
  rate.
• Derives hazard rate approximately mexp(?t).

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Reliability theory

• M. Witten (1985) large number (m) of critical
  components death comes when all fail.
• Components are independent. Fail with constant
  rate.
• Derives hazard rate approximately mexp(?t).
• Unmentioned ? is negative, so the hazard rate
  decreases exponentially

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More reliability theory Gavrilov Gavrilova
(1990)

• m critical organs, each with n redundant
  components.
• Organ fails when all components fail.
• Death comes when any organ fails.
• Components fail independently with constant
  exponential rate.
• Derive Weibull hazard rates (power law).
• Want Gompertz hazard rates.
• Declare that biological systems have most of
  their components nonfunctioning from the
  beginning Number of functioning components in
  each organ is Poisson.
• The exponential of the Poisson then provides the
  exponential hazard rates.

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Biological problems with GG

• Arbitrary.
• Where are the missing components?
• What are the missing components?
• Theory seems to predict that nearly all organisms
  should be born dead, with Gompertz mortality only
  conditioned on the rare survivors.

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Small mathematical problem with GG model
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Big mathematical problem with GG model
The computation is wrong.
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Hazard rate for the G G series-parallel
process with k1 and ??1 (solid) or ?2, ?3
(dotted).
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H. Le Brass cascading failures model

• Start at senescence state Xt1.
• Rate of jumps to next higher state is ?Xt.
• Rate of dying is ?Xt.
• Le Bras (1976) pointed out that when ?gtgt?, the
  mortality rate is about ?e?t for small t.
• True but a little cheap. When ?gtgt?, the system
  behaves like a deterministic system d Xt/dt ?Xt.
  State is Xt? e?t.

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H. Le Brass cascading failures model

• In fact, as Gavrilov Gavrilova pointed out
  (1990) the result is even better when you dont
  assume ?gtgt?. By this time, mortality plateaus
  had been recognized. The general hazard rate is
• (??)?e(??)t(??e(??)t)-1,
• which is a nice logistic Gompertz curve, with
  plateau at ??.

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H. Le Brass cascading failures model

• But the exponential is still in the assumptions.
• Also, the assumptions are quite specific and
  arbitrary.

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Attempts to explain the Gompertz curve with
Markov models have been successful only when

• The exponential increase was built into the
  assumptions in a fairly transparent way or
• The computations were wrong.

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What about mortality plateaus?
Suggested explanations

1. Heterogeneity in the population selection.
   (Analyzed in DS Estimating mortality rate
   doubling time doubling times. Available as
   preprint.)
2. Individuals actually deteriorate more slowly at
   advanced ages.

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What about mortality plateaus?

• J. Weitz and H. Fraser (PNAS 2001) did explicit
  computation for Brownian motion with constant
  downward drift, killed at 0. It shows
  senescence and plateaus -- hazard rates
  increase rapidly (though not exponentially) at
  first, but eventually converge to a finite
  nonzero constant.

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Criticisms of Weitz-Fraser

• The assumptions (constant diffusion rate,
  constant downward drift, killing only at 0) are
  very specific.
• The assumptions have little empirical
  justification.
• The methods cannot be generalized to any other
  case.

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Criticisms of Weitz-Fraser

• The assumptions (constant diffusion rate,
  constant downward drift, killing only at 0) are
  very specific.
• The assumptions have little empirical
  justification.
• The methods cannot be generalized to any other
  case.
• For understanding anything other than the
  plateaus, the model is too general By choosing
  the starting distribution, the hazard rate can
  become almost anything, as long as it eventually
  converges to a rate b2/2. (Here b is the rate
  of negative drift.)

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In fact, the result is very general.

• Killed markov processes (which also may be
  thought of as submarkov processes) converge under
  fairly general conditions to quasistationary
  distributions. That is,

The hazard rate will also converge to the rate of
killing in this distribution.
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Some Theorems

• Let Xt be a diffusion on a one-dimensional
  interval with drift b and variance ?2, so
  satisfying the SDE
• dXt ?(Xt)dWt b(Xt)dt,
• and killed at the rate k(Xt). Let ? be the
  smallest such that there is a positive solution ?
  to
• (??)"(b?)'k???.

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Some Theorems

• If both boundaries are regular then the
  distribution of Xt,conditioned on survival
  converges to density ?, and the mortality rate
  converges to ?.
• If 8 is a natural or entrance boundary and r1
  regular, then this convergence still holds if

()

• In the above, condition () may be replaced
• by lim k(z)gt ?.

z?8
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Interpretation

• Mortality rates level off, not because the
  process is changing, or because of selection on
  populations with heterogeneous frailties. The
  individuals who happen to survive a long time
  simply tend to have a distribution of vitality
  states which are somewhat spread out, not
  arbitrarily piled up near points of killing.

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Why are there mortality-rate plateaus?
Suggested explanations

1. Heterogeneity in the population selection.
2. Individuals actually deteriorate more slowly at
   advanced ages.
3. Those alive at advanced ages are, purely by
   chance, inclined to be more fit than the bare
   minimum required to stay alive.

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For more information, see Markov mortality
models implications of quasistationarity and
varying initial distributions by DS and Steven
Evans. (Available as a preprint.)
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Evolutionary theories

• Is mortality a trait shaped by natural selection?
  
• No overdesigned parts.
• Medawar, Hamilton Mutation-selection
  equilibrium.
• Antagonistic pleiotropy.
• Kirkwood Disposable soma.

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Can evolutionary theory explain Gompertz curve?
Mortality plateaus?

• Charlesworth says yes to Gompertz.
• Mueller and Rose say yes to mortality plateaus.

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Charlesworth Mutation-selection equilibrium

• Not an evolutionary optimum.
• B. Charlesworth for rare deleterious genes,
  equilibrium frequency should be inversely
  proportional to evolutionary cost.
• In state of nature, most mortality is exogenous,
  so should come with constant rate ?.
  Evolutionary cost of a mutation that kills at age
  x should be like e-?x.

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Problems with Charlesworths model

• Requires most aging to depend on genes with
  age-specific effects.
• What happens when you iterate? (So far, only
  simulations.)
• Doesnt allow for gene interactions.
• What is exogenous mortality really?

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Mueller and Rose on mortality plateaus.

• State space of age-specific mortality rates for
  100 days.
• Random mutations act over 2 randomly chosen
  ranges of ? days. (?1 or 40)
• In beneficial range
• In harmful range
• Mutant either vanishes or become fixed.
  Probability depends on change in fitness.

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Mueller and Rose on mortality plateaus.
Mortality rates after simulating 105 mutations,
with several thousand going to fixation. MR
say Evolutionary theory predicts late-life
mortality plateaus.
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Ken Wachter pointed out that the graphs on MRs
paper show transient states of the population.
They didnt run the simulations long enough to
reach equilibrium. Selection reduces mortality
first at low ages, and only later starts to
transfer mortality from intermediate to higher
ages. If you stop early, it looks like a
plateau. Lesson Simulate, but verify.
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http//www.demog.berkeley.edu/dstein/agingpage.ht
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