Modified SIR-model applied to covid-19, similarity solutions and projections to further development

Rebhan, E.

The SIR-model is adapted to the covid-19 pandemic through a modification that consists in making the basic reproduction number variable. Independent of it, another reproduction number is introduced, which is defined similarly to the usual net reproduction number. Due to its simple analytic form, it enables a clear interpretation for all values. A further parameter, provisionally called acceleration parameter, is introduced and applied, which enables a more differentiated characterization of the infection number dynamics. By a variable transformation the 3 equations of the modified SIR-model can be reduced to 2. The latter are solved up to ordinary integrations. The solutions are evaluated for current situations, yielding a pretty good match with the data reported. Encouraged by this, a variety of possible future developments is examined, including linear and exponential growth of the infection numbers as well as sub- and super-exponential growth. In particular, the behavior of the two reproduction numbers and the acceleration parameter is studied, which in some cases leads to surprising results. With regard to the number of unreported infections it is shown, that from the solution for a special one solutions for others can be derived by similarity transformations.