  We study a multi-group SAIRS-type epidemic model with vaccination. The role
of asymptomatic and symptomatic infectious individuals is explicitly considered
in the transmission pattern of the disease among the groups in which the
population is divided. This is a natural extension of the homogeneous mixing
SAIRS model with vaccination studied in Ottaviano et. al (2021) to a network of
communities. We provide a global stability analysis for the model. We determine
the value of the basic reproduction number $\mathcal{R}_0$ and prove that the
disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0 <
1$. In the case of the SAIRS model without vaccination, we prove the global
asymptotic stability of the disease-free equilibrium also when
$\mathcal{R}_0=1$. Moreover, if $\mathcal{R}_0 > 1$, the disease-free
equilibrium is unstable and a unique endemic equilibrium exists. First, we
investigate the local asymptotic stability of the endemic equilibrium and
subsequently its global stability, for two variations of the original model.
Last, we provide numerical simulations to compare the epidemic spreading on
different networks topologies.
