# Why do epidemiologists fit the number of transmission cases with negative binomial distribution?

In the recent paper on coronavirus spread in Hong Kong, the number of individuals infected by one person (relative to all transmission cases) is fit by a negative binomial distribution.

Why is that? It seems like a regular binomial fit would work here, just treat infection as a success.

• Binomial might be appropriate if the number of contacts was known a priori. When we don't know how many people cases have come in contact with, the random variable is supported on the entire integers, Oct 2 '20 at 16:57
• Following up on @DemetriPananos comment, the negative binomial is more specifically the model for an over-dispersed Poisson/Binomial distribution (binomial approximates Poisson when p/n is small) where the rate of occurances follows a gamma distribution (I'd say it's not about the varying number of contacts, which is large anyway, and instead about the varying rate/probability of contacting those contacts). A binomial/Poisson distribution wouldn't make sense (in that case the assumption is that every person has an equal p/rate to make others sick, which isn't likely; people are different). Oct 2 '20 at 18:54

A Binomial random variable, $$X$$, counts the number of successes in $$n$$ IID Bernoulli trials. If you call the number of secondary cases $$X$$, then what would you call $$n$$? There is no way to know $$n$$, the potential number of people who could become secondary cases. That is why a binomial distribution does not make sense.
The solution to this is the negative binomial distribution. A Negative Binomial random variable, $$Y$$, counts the number of the trial in a sequence of IID Bernoulli trials on which the $$r^{th}$$ success occurs. A trial, in this case, would be testing a secondary person for COVID. A success would be if that person is positive for COVID. I'm guessing that they are looking at the number of the trial on which the first person tests positive among all the possible people exposed from the initial case. Because there is an unknown number of trials (as Demetri Pananos mentioned in his comment), they use a Negative Binomial distribution.
By the way, in the paper, $$R$$ stands for "reproductive number" and $$k$$ stands for "dispersion parameter".