We provide a new method to analyze the COVID-19 cumulative reported case data based on a two-step process: first, we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. This allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. Our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. The number of parameters required by the method is parsimonious: for the French epidemic between February 2020 and January 2021 we use only 11 parameters in total.