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The existence of a universal law which maps the bell curve of daily cases to a sigmoid curve for cumulative ones is used for making robust estimations about the final outcome of a disease. Computations of real time effective reproduction rate are presented and its limited usefulness is derived. After using methods ESE and EDE we are able to find the inflection point of the cumulative curve under consideration and study its time evolution. Since mortality processes tend to follow a Gompertz distribution, we apply the properties of it and introduce novel estimations for both the time remaining after inflection time and the capacity of the curve. Special properties of sigmoid curves are used for assessing the quality of estimation and as indices for the cycle completion. Application is presented for COVID-19 evolution for most affected countries and the World.

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We are introducing two methods for revealing the true inflection point of data that contains or not error. The starting point is a set of geometrical properties that follow the existence of an inflection point p for a smooth function. These properties connect the concept of convexity/concavity before and after p respectively with three chords defined properly. Finally a set of experiments is presented for the class of sigmoid curves and for the third order polynomials.

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