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In this study, we developed and analyzed a compartmental epidemic model and a system model that incorporate vaccination, elimination, and quarantine techniques, with a saturated incidence rate, by theoretical and numerical means. The model is described by a system of four nonlinear differential equations. Our analysis included determining the reproduction number Rqand examining equilibrium solutions. The outcomes of the disease are identified through the threshold Rq. When Rq<1, the disease-free equilibrium is globally asymptotically stable, as proved by the LaSalle invariance principle and disease extinction analysis. However, using the Routh–Hurwitz criterion, we have proved that the disease-free equilibrium is not stable, and when Rq>1, the unique endemic equilibrium is asymptotically stable locally. The stability of the endemic and disease-free equilibria globally has been examined using the Routh–Hurwitz and Dulac criteria. Subsequently, numerical simulations were utilized to illustrate the theoretical findings effectively.