Novel mathematical approach to covid-19 studies with reproduction-like number and acceleration parameter



To study the progression of a pandemic, the cumulative case numbers Zn, assigned to discrete time steps tn, are transferred into a continuously differentiable function Z(t) by means of a best least squares fit. In this setting, a reproduction- like number is introduced which is easily applicable to many different situations due to its handy analytical form. It can be understood as a cross between a volatile in- stantaneous reproduction number and the more robust effective reproduction number commonly used. Starting from it, a further quantity, termed acceleration parameter, is introduced, which facilitates a more differentiated characterization of the infection dynamics. In particular, it allows to determine precisely when the limit to exponential growth is reached and exceeded. In this context, the frequently encountered equaliza- tion of dangerous and exponential growth of the infection numbers is shown to be rather misleading. Hence it is extensively studied how different kinds of growth can be correctly described. In many situations exponential growth is just a timeless tran- sient between sub- and super-exponential growth, and, quite generally, it proves to be much more complex than commonly assumed. It turned out useful to incorporate the incidence as a further epidemiological indicator, and for comparison purposes the effective reproduction number is also included. A close relationship is found between the two. In addition, the incidence is used for calculating the trace that the progres- sion of the pandemic leaves behind on a plain spanned by itself and the acceleration parameter. This plane can be divided into a dangerous area, where the pandemic is  uncontrollable, and a safer area that must be the target of mitigation efforts. At present, many countries and the world as a whole are mired in the dangerous area. The latter was chosen as an example for all applications to the covid-19 pandemic.

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