We assume that the total number of cases produced by those on day i, X i? , are Poisson distributed with parameter N i R i , where R i is the reproductive number for cases on day i. We further assume that X i = {X i,i+1 , X i,i+2 ,…,X i,i+k } follows a multinomial distribution with parameters X i? , p, k, where p = {p 1, p 2,…, p k } represent the distribution of the serial interval. Using these assumptions we can construct a likelihood function (see details in the Supplemental Information), which, when simplified, yields the following convenient form