Before revision:  Using general branching processes and assuming that epidemic parameters (including the basic reproductive number) are constant over time, we characterize the distribution of the first hospitalization time and of the epidemic size at this random time. We find that previous estimates either overestimate or largely underestimate the actual epidemic size. In addition, we provide upper and lower bounds for the number of infectious individuals of the local outbreak over time. The upper bound is the cumulative epidemic size, and the lower bound is a constant fraction of it. Lastly, we compute the number of detectable cases if one were to test the whole local outbreak at a single point in time.
After revision:  Using general branching processes and assuming that epidemic parameters (including the basic reproductive number) are constant over time, we characterize the distribution of the first detection time of an infected individual in an infection cluster depending on the testing effort (assumed to be constant in time), and estimate that the SARS-CoV-2 variant of concern B.1.1.7, detected in September 2020, first appeared in the United Kingdom early August 2020. We estimate a minimal testing frequency to detect clusters before they exceed a certain threshold size, and we compute the number of detectable cases if one were to test the whole local outbreak at a single point in time.