274 lines
13 KiB
Markdown
274 lines
13 KiB
Markdown
---
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created_at: '2018-02-15T06:38:22.000Z'
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title: 'Your body wasn’t built to last: Math of human mortality (2009)'
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url: https://gravityandlevity.wordpress.com/2009/07/08/your-body-wasnt-built-to-last-a-lesson-from-human-mortality-rates/
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author: lighttower
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points: 87
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story_text:
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comment_text:
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num_comments: 25
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story_id:
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story_title:
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story_url:
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parent_id:
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created_at_i: 1518676702
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_tags:
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- story
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- author_lighttower
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- story_16382322
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objectID: '16382322'
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year: 2009
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---
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# Your body wasn’t built to last: a lesson from human mortality rates
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What do you think are the odds that you will die during the next year?
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Try to put a number to it — 1 in 100? 1 in 10,000? Whatever it is, it
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will be twice as large 8 years from now.
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This startling fact was first noticed by the British actuary Benjamin
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Gompertz in 1825 and is now called the “Gompertz Law of human
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mortality.” Your probability of dying during a given year doubles
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every 8 years. For me, a 25-year-old American, the probability of dying
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during the next year is a fairly minuscule 0.03% — about 1 in 3,000.
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When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be about 1
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in 750, and so on. By the time I reach age 100 (and I do plan on it)
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the probability of living to 101 will only be about 50%. This is
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seriously fast growth — my mortality rate is increasing exponentially
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with age.
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And if my mortality rate (the probability of dying during the next year,
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or during the next second, however you want to phrase it) is rising
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exponentially, that means that the probability of me surviving to a
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particular age is falling super-exponentially. Below are some
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statistics for mortality rates in the United States in 2005, as reported
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by the US Census Bureau (and displayed by [Wolfram
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Alpha](http://www.wolframalpha.com/)):
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This data fits the Gompertz law almost perfectly, with
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death rates doubling every 8 years. The graph on the right also agrees
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with the Gompertz law, and you can see the precipitous fall in survival
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rates starting at age 80 or so. That decline is no joke; the sharp fall
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in survival rates can be expressed mathematically as an exponential
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within an exponential:
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Exponential decay is sharp, but an exponential within an exponential is
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so sharp that I can say with 99.999999% certainty that no human will
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ever live to the age of 130. (Ignoring, of course, the upward shift in
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the lifetime distribution that will result from future medical advances)
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Surprisingly enough, the Gompertz law holds across a large number of
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countries, time periods, and even different species. While the actual
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average lifespan changes quite a bit from country to country and from
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animal to animal, the same general rule that “your probability of dying
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doubles every X years” holds true. It’s an amazing fact, and no one
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understands why it’s true.
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There is one important lesson, however, to be learned from Benjamin
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Gompertz’s mysterious observation. By looking at theories of human
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mortality that are clearly wrong, we can deduce that our fast-rising
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mortality is not the result of a dangerous environment, but of a body
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that has a built-in expiration
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date.
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**The lightning bolt theory**
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If you had never seen any mortality statistics (or known very many old
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people), you might subscribe to what I call the “lightning bolt theory”
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of mortality. In this view, death is the result of a sudden and
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unexpected event over which you have no control. It’s sort of an
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ancient Greek perspective: there are angry gods carousing carelessly
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overhead, and every so often they hurl a lightning bolt toward Earth,
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which kills you if you happen to be in the wrong place at the wrong
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time. These are the “lightning bolts” of disease and cancer and car
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accidents, things that you can escape for a long time if you’re lucky
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but will eventually catch up to you.
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The problem with this theory is that it would produce mortality rates
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that are nothing like what we see. Your probability of dying during a
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given year would be constant, and wouldn’t increase from one year to the
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next. Anyone who paid attention during introductory statistics will
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recognize that your probability of survival to age t would follow a
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Poisson distribution, which means exponential decay (and not
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super-exponential decay).
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Just to make things concrete, imagine a world where every year a
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“lightning bolt” gets hurled in your general direction and has a 1 in
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80 chance of hitting you. Your average life span will be 80 years, just
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like it is in the US today, but the distribution will be very different:
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What a crazy world\! The average lifespan would be the same, but out of
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every 100 people 31 would die before age 30 and 2 of them would live to
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be more than 300 years old. Clearly we do not live in a world where
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mortality is governed by “lightning
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bolts”.
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**The accumulated lightning bolt theory**
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I think most people will see pretty quickly why the “lightning bolt
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theory” is flawed. Our bodies accumulate damage as they get older.
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With each misfortune our defenses are weakened — a car accident might
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leave me paralyzed, or a knee injury could give me arthritis, or a
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childhood bout with pneumonia could leave me with a compromised immune
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system. Maybe dying is a matter of accumulating a number of “lightning
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strikes”; none of them individually will do you in, but the accumulated
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effect leads to death. I think of it something like [Monty Python’s
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Black Knight](http://www.youtube.com/watch?v=zKhEw7nD9C4): the first
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four blows are just flesh wounds, but the fifth is the end of the
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line.
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Fortunately, this theory is also completely testable. And, as it turns
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out, completely wrong. Shown above are the results from a simulated
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world where “lightning bolts” of misfortune hit people on average every
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16 years, and death occurs at the fifth hit. This world also has an
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average lifespan of 80 years (16\*5 = 80), and its distribution is a
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little less ridiculous than the previous case. Still, it’s no Gompertz
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Law: look at all those 160-year-olds\! You can try playing around with
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different “lightning strike rates” and different number of hits required
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for death, but nothing will reproduce the Gompertz Law. No explanation
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based on careless gods, no matter how plentiful or how strong their
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blows are, will reproduce the strong upper limit to human lifespan that
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we actually
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observe.
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**The cops and criminals inside your body
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**
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Like I said before, no one knows why our lifespans follow the Gompertz
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law. But it isn’t impossible to come up with a theoretical world that
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follows the same law. The following argument comes from [this short
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paper](http://arxiv.org/PS_cache/q-bio/pdf/0411/0411019v3.pdf), produced
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by the Theoretical Physics Institute at the University of Minnesota
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\[update: also published
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[here](http://www.springerlink.com/content/lp63258564432853/) in the
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journal Theory in Biosciences\].
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Imagine that within your body is an ongoing battle between cops and
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criminals. And, in general, the cops are winning. They patrol randomly
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through your body, and when they happen to come across a criminal he is
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promptly removed. The cops can always defeat a criminal they come
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across, unless the criminal has been allowed to sit in the same spot for
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a long time. A criminal that remains in one place for long enough (say,
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one day) can build a “fortress” which is too strong to be assailed by
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the police. If this happens, you die.
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Lucky for you, the cops are plentiful, and on average they pass by every
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spot 14 times a day. The likelihood of them missing a particular spot
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for an entire day is given (as you’ve learned by now) by the Poisson
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distribution: it is a mere .
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But what happens if your internal police force starts to dwindle?
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Suppose that as you age the police force suffers a slight reduction, so
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that they can only cover every spot 12 times a day. Then the
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probability of them missing a criminal for an entire day increases to
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. The difference between 14 and 12
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doesn’t seem like a big deal, but the result was that your chance of
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dying during a given day jumped by more than 10 7 times. And if the
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strength of your police force drops linearly in time, your mortality
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rate will rise exponentially.
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This is the Gompertz law, in cartoon form: your body is deteriorating
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over time at a particular rate. When its “internal policemen” are good
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enough to patrol every spot that might contain a criminal 14 times a
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day, then you have the body of a 25-year-old and a 0.03% chance of dying
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this year. But by the time your police force can only patrol every spot
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7 times per day, you have the body of a 95-year-old with only a 2-in-3
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chance of making it through the
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year.
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**More questions than answers**
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The example above is tantalizing. The language of “cops and criminals”
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lends itself very easily to a discussion of the immune system fighting
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infection and random mutation. Particularly heartening is the fact that
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rates of cancer incidence also follow the Gompertz law, doubling every 8
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years or so. Maybe something in the immune system is degrading over
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time, becoming worse at finding and destroying mutated and potentially
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dangerous cells.
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Unfortunately, the full complexity of human biology does not lend itself
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readily to cartoons about cops and criminals. There are a lot of
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difficult questions for anyone who tries to put together a serious
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theory of human aging. Who are the criminals and who are the cops that
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kill them? What is the “incubation time” for a criminal, and why does
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it give “him” enough strength to fight off the immune response? Why is
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the police force dwindling over time? For that matter, what kind of
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“clock” does your body have that measures time at all?
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There have been attempts to describe DNA degradation (through the
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shortening of your[telomeres](http://en.wikipedia.org/wiki/Telomere) or
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through [methylation](http://en.wikipedia.org/wiki/DNA_methylation)) as
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an increase in “criminals” that slowly overwhelm the body’s DNA-repair
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mechanisms, but nothing has come of it so far. I can only hope that
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someday some brilliant biologist will be charmed by the simplistic
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physicist’s language of cops and criminals and provide us with real
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insight into why we age the way we
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do.
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**UPDATE:** G\&L reader Michael has made a cool-looking (if slightly
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morbid) [web
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calculator](http://forio.com/simulate/simulation/mbean/death-probability-calculator/)
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to evaluate the Gompertz law prediction for different ages. If you want
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to know what the law implies for you in particular, and are not terribly
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handy with a calculator, then you might want to check it
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out.
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