Mathematical Relationship between Reproduction Number and Epidemic Curve of Daily Cases

The strict mathematical relationship between R t and the curve of daily cases f ( t ) is shown. Up to date and statistically robust R t from the curve of daily cases can be estimated as soon as new cases are added to the curve. That is equivalent to estimating R t 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 R t scaled numbers are of interest, only the average duration of subjects’ infectivity has to be estimated directly, but independently of linking secondary cases to primary ones. A new index, Instantaneous Reproduction Number R ist is introduced, which is mathematically tied to R t and does not depend on the duration of infectivity of subjects. R ist , R t and the doubling/halving time of the epidemic may be estimated by simple computations at the very detection time of new daily cases. Any smoothed curve of daily cases gives smooth R t and R ist . No phase lag on R t estimate is introduced by this method.