## Wednesday, September 16, 2009

The math of death...

Oddly the statistics of dying are pretty straightforward, even if the messiness of actually living is not.
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.
But even though the numbers fit neatly to expectations, we don't really know why.
Like I said before, no one knows why our lifespans follow the Gompertz law.  But it isn’t impossible to come up with a theoretical world that follows the same law.  The following argument comes from this short paper, produced by the Theoretical Physics Institute at the University of Minnesota.
Imagine that within your body is an ongoing battle between cops and criminals.  And, in general, the cops are winning.  They patrol randomly through your body, and when they happen to come across a criminal he is promptly removed.  The cops can always defeat a criminal they come across, unless the criminal has been allowed to sit in the same spot for a long time.  A criminal that remains in one place for long enough (say, one day) can build a “fortress” which is too strong to be assailed by the police.  If this happens, you die.
Lucky for you, the cops are plentiful, and on average they pass by every spot 14 times a day.  The likelihood of them missing a particular spot for an entire day is given (as you’ve learned by now) by the Poisson distribution: it is a mere $e^{-14} \approx 8 \times 10^{-7}$.
But what happens if your internal police force starts to dwindle?  Suppose that as you age the police force suffers a slight reduction, so that they can only cover every spot 12 times a day.  Then the probability of them missing a criminal for an entire day decreases to $e^{-12} \approx 6 \times 10^{-6}$.  The difference between 14 and 12 doesn’t seem like a big deal, but the result was that your chance of dying during a given day jumped by more than 10 times.  And if the strength of your police force drops linearly in time, your mortality rate will rise exponentially.
This is the Gompertz law, in cartoon form: your body is deteriorating over time at a particular rate.  When its “internal policemen” are good enough to patrol every spot that might contain a criminal 14 times a day, then you have the body of a 25-year-old and a 0.03% chance of dying this year.  But by the time your police force can only patrol every spot 7 times per day, you have the body of a 95-year-old with only a 2-in-3 chance of making it through the year.
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