Global stability of multi?group SAIRS epidemic models

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 disease among groups which population divided. This natural extension homogeneous mixing SAIRS vaccination studied Ottaviano et. al (2021) to network communities. provide global stability analysis for model. determine value basic reproduction number $\mathcal{R}_0$ prove that disease-free equilibrium globally asymptotically stable if $\mathcal{R}_0 < 1$. In case without vaccination, we asymptotic also when $\mathcal{R}_0=1$. Moreover, > 1$, unstable unique endemic exists. First, investigate local subsequently its stability, two variations original Last, numerical simulations compare spreading on different networks topologies.