To mitigate the harmful effects of the COVID-19 pandemic, world countries have resorted – though with different timing and intensities – to a range of interventions. These interventions and their relaxation have shaped the epidemic into a multi-phase form, namely an early invasion phase often followed by a lockdown phase, whose unlocking triggered a second epidemic wave, and so on. In this article, we provide a kinematic description of an epidemic whose time course is subdivided by mitigation interventions into a sequence of phases, on the assumption that interventions are effective enough to prevent the susceptible proportion to largely depart from 100% (or from any other relevant level). By applying this hypothesis to a general SIR epidemic model with age-since-infection and piece-wise constant contact and recovery rates, we supply a unified treatment of this multi-phase epidemic showing how the different phases unfold over time. Subsequently, by exploiting a wide class of infectiousness and recovery kernels allowing reducibility (either to ordinary or delayed differential equations), we investigate in depth a low-dimensional case allowing a non-trivial full analytical treatment also of the transient dynamics connecting the different phases of the epidemic. Finally, we illustrate our theoretical results by a fit to the overall Italian COVID-19 epidemic since March 2020 till February 2021 i.e., before the mass vaccination campaign. This show the abilities of the proposed model in effectively describing the entire course of an observed multi-phasic epidemic with a minimal set of data and parameters, and in providing useful insight on a number of aspects including e.g., the inertial phenomena surrounding the switch between different phases.
