  We study an SAIRS-type epidemic model with vaccination, where the role of
asymptomatic and symptomatic infectious individuals are explicitly considered
in the transmission patterns of the disease. 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 $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$,
condition under which a positive endemic equilibrium exists. We investigate the
global stability of the endemic equilibrium for some variations of the original
model under study and answer to an open problem proposed in Ansumali et al.
\cite{ansumali2020modelling}. In the case of the SAIRS model without
vaccination, we prove the global asymptotic stability of the disease-free
equilibrium also when $\mathcal{R}_0=1$. We provide a thorough numerical
exploration of our model, to validate our analytical results.
