Then these new infections will generate R more cases each – which
translates into R^2 new cases – and so on:
Outbreak size = 1 + R + R^2 + R^3 + ...
We could try and add up all these values to work out the expected
outbreak size. But fortunately there’s an easier option. In the
nineteenth century, mathematicians proved that there’s an elegant
rule we can apply to sequences like the one above. If R is between 0
and 1, the following equation is true:
1 + R + R^2 + R^3 + ... = 1/(1–R)
In other words, as long as the reproduction number is below 1, the
expected outbreak size is equal to 1/(1–R). Even if you’re not
especially interested in nineteenth-century mathematics, it’s worth
taking a moment to appreciate how useful this shortcut is. Rather
than having to simulate how an infection might stutter along from one
generation to the next until it eventually fizzles out, we can instead
estimate the final outbreak size directly from the reproduction
number.[43] If R is 0.8, for example, we’d expect an outbreak with
1/(1–0.8) = 5 cases in total. And that’s not all we can do. We can also
work backwards to estimate the reproduction number from the
average outbreak size. If outbreaks consist of five cases on average,
it means R is 0.8.
In my field, we regularly use this back-of-the-envelope calc ulation
to estimate the reproduction number of new disease threats. During
the early months of 2013, there were 130 human cases of H7N9 bird
flu in China. Although most of these picked up the disease from
contact with poultry, there were four clusters of infection that were
likely to be the result of transmission between humans.[44] Because
most people didn’t infect anyone else, the average size of a human
H7N9 outbreak was 1.04 cases, suggesting that R in humans was a
paltry 0.04.
This idea isn’t only useful for diseases. During the mid-2000s,
Jonah Peretti and Duncan Watts applied the same method to
marketing campaigns. It meant they could get at the underlying
transmissibility of an idea, rather than just describing what a