 Solution

One analysis can treat a reported symptom-onset date as the exact $H\to I$ transition time. That adds an exactly observed infection-time density to the likelihood, but assumes symptoms begin immediately at infection, every relevant episode is symptomatic, and dates are recalled and reported without error.

A more realistic analysis treats true infection as a latent transition and symptom onset as a noisy observation. A reporting-delay distribution, and possibly probabilities of asymptomatic infection and non-reporting, can be added to a Hidden Markov model. Weekly tests then interval-censor the state transition while the symptom date refines its distribution. This approach uses more information but requires an identifiable and correctly specified symptom-delay and reporting model.