  We have further extended our compartmental model describing the spread of the
infection in Italy. The model is based on the assumption that the time
evolution of all of the observable quantities (number of people still positive
to the infection, hospitalized and fatalities cases, healed people, and total
number of people that has contracted the infection) depend on average
parameters, namely people diffusion coefficient, infection cross-section, and
population density. The model provides precious information on the tight
relationship between the variation of the reported infection cases and a well
defined observable physical quantity: the average number of people that lie
within the daily displacement area of any single person. The extension of the
model now includes self-consistent evaluation of the reproduction index, effect
of immunization due to vaccination, and potential impact of virus variants on
the dynamical evolution of the outbreak. The model fits very well the epidemic
data, and allows us to strictly relate the time evolution of the number of
hospitalized case and fatalities to the change of people mobility, vaccination
rate, and appearance of an initial concentration of people positives for new
variants of the virus.
