Photograph: Edgloris Marys/Alamy
Earlier today I set you the following puzzle about a hypothetical new Covid variant.
Just in case this is the first article you have ever read on viruses: R is the reproduction number, meaning the average number of infections caused by any infected person.
The riddle of R
Suppose a hypothetical new COVID variant emerges, and everyone is initially susceptible to infection (but not necessarily severe disease).
During the early stages of this new wave, each infected person exposes the variant to two other people (i.e. R=2). Every person exposed to the virus will get infected unless they have already had it, in which case they are immune.
As more people get infected, immunity builds, which gradually reduces R until the epidemic peaks and declines. By the end of the variant wave, 75% of the population have been infected with this variant.
On average, how many times was each person in the population exposed to infection during this wave? What is surprising about this result?
I also suggested you use the following equation:
R = R0 x S
R0 (R naught) is the basic reproduction number, meaning the reproduction number when everyone is susceptible. S is a number between 0 and 1 representing the proportion of the population susceptible.
Answer Each person is exposed an average of 1.5 times
Solution If R = 2 at the beginning (i.e. when S = 1), then R0 = 2.
The total number of exposures is equal to the total number of infections multiplied by R0. (Since each infected person exposes the virus to 2 people). Thus the total number of exposures is 0.75 x population size x 2 = 1.5 x population size.
If there have been 1.5 x population size exposures in total, each person must have been exposed 1.5 times on average.
Discussion At first, it seems like there is some kind of contradiction: on average, everyone is exposed more than once, but a full 25 per cent don’t get exposed at all! The reason for this discrepancy is that exposures are not distributed evenly. They happen more or less randomly throughout a population. Some people will have been exposed more than once, and because they’re immune the second (or third or fourth) time, it slows transmission, meaning some people don’t get exposed at all.
Its a bit like picking marbles from a bag – if you select a marble randomly each time, there will be marbles you pick up more than once, and marbles that you never pick.
This is the fundamental concept behind ‘herd immunity’ or ‘indirect protection’, and explains why epidemics end without everyone getting infected, even if the average number of exposures is larger than the population size.
Thanks to Professor Adam Kucharski of the London School of Hygiene and Tropical Medicine, who set this problem. Adam is the author of the fantastic Rules of Contagion: Why Things Spread – and Why They Stop
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me..
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
