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Epidemics are 98% Below Average
A 1997 Proc. Royal Society paper by Rhodes, Jensen, and Anderson found that the frequency of cases of measles and whooping cough on the Faroe Islands from 1866-1970 each varied according to a power law with an amazingly long tail. (Watts et al. 2005 find similar results in Iceland from 1888-1990.) Here is what Rhodes’ power law looks like (up to a rate constant) extended to a larger population.
Imagine a disease that infects different numbers of people each year. In 16% of years it infects only one or two people, in 15% of years it infects three or four, in 50% of years it infects 16 or less, in 25% of years it infects 256 or more, in 12.5% of years it infects 4096 or more, and so on according to the power law P(>s infected) = s^-0.25.
The average number of infections per year would be infinite were it not for the fact that no disease can infect more people than there are. Given a world population of ten billion, average infections per year would be 42 million, even though epidemics this large or larger happen in only 1.25% of years. Most of this average comes from the 0.3% of years when the entire world is infected; if you worry at all about this sort of epidemic, worry most about the very largest ones.
Looking at recent track records of these sort of problems could easily bias us to pay too little attention to them. I learned all this while researching a new paper on catastrophes, social collapse, and human extinction, and it turns out that many other types of disasters, like wars and earthquakes, are distributed with such long tails.