A continuously time-varying transmission rate is suggested by many control-theoretic investigations on non-pharmaceutical interventions for mitigating the COVID-19 pandemic. However, such a continuously varying rate is impossible to implement in any human society. Here, we significantly extend a preliminary work (M. Fliess, C. Join, A. d’Onofrio, Feedback control of social distancing for COVID-19 via elementary formulae, MATHMOD, Vienna, 2022), based on the combination of flatness-based and model-free controls with respect to the classic parsimonious SIR model. Indeed, to take into account severe uncertainties and perturbations, we propose a feedback control where the transmission rate, i.e., the control variable, is piecewise constant. More precisely, the transmission rate remains constant during an appreciable time interval, which is not too large. Strict extended lockdowns may therefore be avoided. The poor knowledge of fundamental quantities such as the rate of infection hinders a precise calibration of the transmission rate. Thus, the results of our approach ought therefore not to be regarded as rules of action to follow accurately but as a guideline for a wise behaviour.
