Mathematical Relationship between Effective Reproduction Number Rt and Epidemic Curve of Daily Cases -- Demonstration and Details

Caudana, B.

The strict mathematical relationship between Rt and the curve of daily cases f(t) is shown. Up-to-date and statistically robust Rt from the curve of daily cases can be estimated as soon as new cases are added to the curve. That is equivalent to estimating Rt by averaging all detected cases of infection, without any distortion induced by the difficulty of following and weighting trees of secondary cases from original ones, and without needing to wait for secondary cases to manifest infection. With this method, if Rt scaled numbers are of interest, only the average duration of infectivity of subjects has to be estimated directly, but independently of linking secondary cases to primary ones. A new index, instantaneous reproduction number Rist is introduced, which does not depend on the duration of infectivity of subjects. Rist, Rt and the doubling/halving time of the epidemics may be estimated by simple computations at the very detection time of new daily cases. Any smoothed curve of daily cases gives smooth Rt and Rist. No phase lag on Rt estimate is introduced by this method.