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. 

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  • This is boring self-promotion, Robin, but if you have not done so, you really ought to check out my _From Catastrophe to Chaos: A General Theory of Economic Discontinuities_, first edn., 1991, Kluwer, Vol. I of 2nd edn, 2000, Kluwer. ‘Nuff said.

  • rcriii

    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.

    To quibble, this should read “in 29% of years it infects 4 or less” (sounds better than 13% of years 3-4, which is 4^-.25 – 2^-.25).

    Also should there be a correction to the formula based on the number of persons? I have a hard time believing that the probability of 2 infections out of 6 billion is the same as the probability of 2 out of the 48000 residents of the Faroe Islands. Using a correction of 10B/48000, I get infections of 42M or 26.5% of the time.

  • rcrii (did your mommy really call you that?), I was trying crudely to adapt the distribution I gave for continuous s to discrete s. And I said my example was only correct up to a rate constant. That is, P(>s) = k s^-0.25.

  • Carl Shulman

    One unfortunate result of this effect is that taking appropriate precautions for such risks can produce a ‘boy who cried wolf’ syndrome. Consider the 1976 swine flu affair, (in which the threat of deadly flu pandemic lead to mass vaccinations that ended up killing more people than the relatively mild actual pandemic in the United States. Support for vaccines and political will to prepare for pandemic threats was substantially reduced thereafter, even though the measures were well justified in expected value terms, given the scale of casualties in early plagues such as the 1918-1919 flu pandemic.


    On the other hand, biased disease researchers will tend to exaggerate the risk from their particular pathogen of study. People who are more concerned about a disease are more likely to select it as a research topic, and hyping a threat attracts funding and creates status or a sense of self-importance.

  • TGGP

    Robin, you mentioned that nuclear weapons might prevent small wars but cause big ones. Empirically it seems that very large wars between great powers decreased with the threat of M.A.D, but that small proxy wars increased. Granted, nuclear war has yet to occur, it would certainly qualify as a “large war” and it would not be possible without nuclear weapons. However, most of your other measurements were based on disasters that actually happened and it would seem odd to treat nuclear weapons differently (supernovas were also discussed, but only as an extremity on the power scale).

  • TGGP, yes we have no data on the rate of extremely large wars with and without nuclear weapons. Nevertheless, it seems plausible to me that nuclear weapons increase the chances of extremely damaging wars, whatever else they do to smaller wars.

  • rcriii

    You are correct Robin. I missed your qualification in the first paragraph.

    My mommy called me Robin, actually. When I was 12 or 13 I insisted on being called Robert. I now half-wish I hadn’t.