A widely used tool for analysing the Covid-19 pandemic is the standard SIR model. It seems often to be used as a black box, not taking into account that this model was derived as a special case of the seminal Kermack-McKendrick theory from 1927. This is our starting point. We explain the setup of the Kermack-McKendrick theory (passing to a discrete approach) and use medical information for specializing to a model which we call {\em adapted K-McK-model}. This includes effects of vaccination, mass testing and mutants. We demonstrate the use of the model by applying it to the development in Germany. As a striking application we demonstrate that a comparatively mild intervention reducing the time until quarantine by one day leads to a drastic improvement. A similar effect can be obtained by certain mass testings as we will demonstrate. We discuss possibilities to apply the model both for predictions and as an analysis tool. We compare the adapted K-McK-model to the standard SIR model and observe condiderable differences if the contact rates are not constant. Finally we compare the model reproduction rate with the empirical reproduction rate determined by the Robert Koch-Institut.