Sunday, August 2, 2009

Exponentials and superexponentials

What do you think are the odds that you will die during the next year? Try to put a number to it — 1 in 100? 1 in 10,000? Whatever it is, it will be twice as large 8 years from now.

This startling fact was first noticed by the British actuary Benjamin Gompertz in 1825 and is now called the “Gompertz Law of human mortality.” Your probability of dying during a given year doubles every 8 years. For me, a 25-year-old American, the probability of dying during the next year is a fairly miniscule 0.03% — about 1 in 3,000. When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be about 1 in 750, and so on. By the time I reach age 100 (and I do plan on it) the probability of living to 101 will only be about 50%. This is seriously fast growth — my mortality rate is increasing exponentially with age.

And if my mortality rate (the probability of dying during the next year, or during the next second, however you want to phrase it) is rising exponentially, that means that the probability of me surviving to a particular age is falling super-exponentially. Below are some statistics for mortality rates in the United States in 2005, as reported by the US Census Bureau (and displayed by Wolfram Alpha):

This data fits the Gompertz law almost perfectly, with death rates doubling every 8 years. The graph on the right also agrees with the Gompertz law, and you can see the precipitous fall in survival rates starting at age 80 or so. That decline is no joke; the sharp fall in survival rates can be expressed mathematically as an exponential within an exponential:

Surprisingly enough, the Gompertz law holds across a large number of countries, time periods, and even different species. While the actual average lifespan changes quite a bit from country to country and from animal to animal, the same general rule that “your probability of dying doubles every X years” holds true. It’s an amazing fact, and no one understands why it’s true.

There is one important lesson, however, to be learned from Benjamin Gompertz’s mysterious observation. By looking at theories of human mortality that are clearly wrong, we can deduce that our fast-rising mortality is not the result of a dangerous environment, but of a body that has a built-in expiration date.

The excerpt above comes from a post in Gravity and Levity, a blog full of "big crazy ideas behind the equations" in physics. I highly recommend reading the entire post, which talks about how the curves would be shaped under different models of mortality--the author shows that the so-called "lightning bolt" and "accumulated lightning bolt" models of mortality are not consistent with the observed graphs above, but a "cops and criminals inside your body" model, rooted in cell biology, is consistent with the graphs above.

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