Did you solve it? Mathematics of a hypothetical new variant of Covid

Earlier today I posed the following puzzle to you about a hypothetical new variant of Covid.

Just at in case this is the first article you've ever read about viruses: R is the reproduction number, i.e. the average number of infections caused by any infected person.

R's Riddle

Suppose a hypothetical new variant of COVID emerges and everyone is initially susceptible to infection (but not necessarily severe disease).

At the start of this new wave, each infected person exposes the variant to two other people ( i.e. R=2). Anyone exposed to the virus will become infected unless they have already had it, in which case they are immune.

As and when as more and more people become infected, immunity builds, which gradually reduces R until the epidemic peaks and wanes. By the end of the wave of variants, 75% of the population had been infected with that variant.

On average, how many times each person was she in the population exposed to the infection during this wave? What is surprising about this result?

I also suggested you use the following equation:

R = R0 x S

R0 (R null) is the base reproduction number, that is- ie the reproduction number when everyone is susceptible. S is a number between 0 and 1 representing the proportion of the susceptible population.

Response Each person is exposed on average 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 had 1.5x the total population size, each person must have been exposed 1.5 times on average.

Discussion At first there seems to be a kind of a contradiction: on average, everyone is exposed more than once, but 25% don't expose yourself! The reason for this discrepancy is that the exposures are not evenly distributed. They occur more or less randomly in a population. Some people will have been exposed more than once, and because they are immune the second (or third or fourth) time, this slows transmission, which means some people are not exposed at all.

It's kind of like picking marbles out of a bag - if you randomly select a marble each time, there will be marbles you pick up more than once and marbles that you never pick up.

This is the fundamental concept behind "herd immunity" or "indirect protection", and explains why epidemics end without any the world is infected, even if the average number of exposures is greater than the size of the population.

Thanks to Professor Adam Kucharski from the London School of Hygiene and Tropical Medicine, which posed this problem. Adam is the author of the fantastic Rules of Contagion: Why Things Spread and Why They Stop

I put a puzzle here every two weeks on a Monday. I'm always on the lookout for great puzzles. If you would like to suggest one, email me.

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