A compartmental model for the dynamic simulation of pandemics with a multi-phase vaccination and its application to Italian COVID-19 data

We introduce a generalization of the 4 compartments SVIR epidemic model discussed in [1] for the first time. Our model has K+4 compartments. K-1 of these compartments represent additional subsequent vaccination stages not considered in the original SVIR model, while a further compartment accounts for dead people. We analyze the equilibrium points of the model. A time-varying parameters version of it, having $K=3$ vaccination compartments, is then calibrated to Italian COVID-19 dataset. This analysis is carried out for three specific sub-periods: the first one, ranging from February 24th, 2020, up to December 26th 2020, when no vaccines were available; the second one, from the December 27th, 2020 up to December 31st, 2021, during which the Delta variant of the virus prevailed and Delta-targeted vaccination doses were administered to the population for the first time; finally, the last considered period is ranging from January 10th, 2022 up to June 3rd, 2022, and it was characterized by the diffusion of the Omicron variant. To tackle the problem of undetected infected or undetected recovered people we adopt an approach relying on different scenarios. The calibration of the model uses the property that the discrete-time version of it turns out to be explicitly solvable with respect to the parameters, hence providing a daily estimate of the involved parameters. This produces meaningful evolution patterns of the COVID-19 epidemic which allow a better understanding of the diffusive behavior of the pandemic along time. Lastly a statistical analysis of the epidemiological parameters estimators supports the non stationarity of their time series.