In my work in the tourism sector, I am looking for any relevant reference to the following problem statement.
Due to the current outbreak of Coronavirus, many business and leisure events are either being cancelled or an upper limit is being put of on the maximum number of attendants. In some cities, all events whose expected attendants is more than 1,400 people are being cancelled where as in some other cities events with more than 300 people are being called off. I wonder how do the city authorities decide is 1400 is the right cut off or 300 is the right one.
Question: Suppose that an event originally had $N$ expected number of attendants. Is there any mathematical logic/theory similar to the theory of optimal stopping which says what faction of this $N$ expected attendants should be let inside the event venue?
Something that is vaguely similar is the secretary problem which says if there are $N$ candidates who are interviewed in random order than it recommends ignoring the first $37\%$ of the candidates and then selecting next candidate who better than any of the candidates in the first $37\%$.
If you want to minimize the probability of further infection the answer is cancel the event, i.e. do not let anyone in. It depends on the threshold, e.g. you want to keep the probability of further infection below some p and maximize the ppl. Let's say the prob of attendee to be infected is q and the probability that he/she infects someone else is r. Let A denote the event that during the event someone else is infected. For the sake of simplicity assume that r=1. Given the number of attendees is N the distribution of the number of infected is Bin(N,q). The prob that at least someone is infected is $1-(1-q)^N$. To keep this under p we have to solve $1-(1-q)^N < p$ . Hence $ \log (1-p) < N \log (1-q)$, i.e. $ N< \log(1-p)/\log(1-q)$.
If $r<1$ then the probability of some being sick is $q$ and infecting some else is $rq$. The number of infecting sicks is Bin(N,rq), hence $N< \log(1-p)/\log(1-qr)$.