  We present a compartmental mathematical model with demography for the spread
of the COVID-19 disease, considering also asymptomatic infectious individuals.
We compute the basic reproductive ratio of the model and study the local and
global stability for it. We solve the model numerically based on the case of
Italy. We propose a vaccination model and we derive threshold conditions for
preventing infection spread in the case of imperfect vaccines.

Populations and Evolution;Dynamical Systems;Physics and Society