Return-to-Normality in a Piecewise Deterministic Markov SIR+V Model with Pharmaceutical and Non-pharmaceutical Interventions

This paper studies a two-scales Markov compartmental model (SIRS) with demography, vaccination, reinfections and waning immunity, and with external control inputs of pharmaceutical (vaccination) and non-pharmaceutical (social distancing) type. The demography events are considered to be rare and contribute to a jump mechanism with trajectory depending jump intensity ultimately providing a piecewise deterministic Markov model . We study well-posedness of the system with time-varying total population and the herd immunity zone when intensive care units constraints are enforced. The third contribution describes, via Zubov’s method, the initial configurations that can be driven to the immunity zone. Numeric illustrations are provided using data from INSEE (France) on COVID-19.