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This paper presents a new hybrid compartmental model for studying the COVID-19 epidemic evolution in Italy since the beginning of the vaccination campaign started on 2020/12/27 and shows forecasts of the epidemic evolution in Italy. The proposed compartmental model subdivides the population into six compartments and extends the SEIRD model proposed in [E.L.Piccolomini and F.Zama, PLOS ONE, 15(8):1–17, 08 2020] by adding the Vaccinated population and framing the global model as a hybrid-switched dynamical system. Aiming to represent the quantities that characterize the epidemic behaviour from an accurate fit to the observed data, we partition the observation time interval into sub-intervals. The model parameters change according to a switching rule depending on the data behaviour and the infection rate continuity condition. In particular, we study the representation of the infection rate both as linear and exponential piecewise continuous functions. We choose the length of sub-intervals balancing the data fit with the model complexity through the Bayesian Information Criterion. The calibration of the model shows an excellent representation of the epidemic behaviour and thirty days forecasts have proven to reproduce the infection spread reliably. Finally, we discuss different possible forecast scenarios obtained by simulating an increased vaccination rate.