We propose a population dynamical model for SARS-CoV-2 that takes into account mask compliance and effectiveness, in the context of saturated treatment. This model also considers reinfection and relapse among individuals with comorbidities. Our findings indicate that global mask usage, in conjunction with other public health measures, effectively reduces the basic reproduction number ( \(\texttt{R}_0\) ). We establish the local and conditional global stability of the disease-free equilibrium point. Notably, the model exhibits intriguing behavior due to saturated treatment and reinfection. Under specific parameter conditions, it demonstrates multiple endemic equilibria when \(\texttt{R}_0<1\) resulting and backward and forward bifurcation. We conduct sensitivity analysis to pinpoint the key factors influencing disease spread. The existence of multiple equilibria contributes to intricate and diverse dynamics, showcasing a variety of bifurcations and oscillations through Hopf bifurcation. Under specific conditions, global asymptotic stability for the unique endemic equilibrium, when it exists, is established. Among further nonlinear dynamics exhibited by the proposed model, we establish backward Hopf bifurcation, Hopf–Hopf bifurcation and saddle-node bifurcation. Bistability of the equilibrium points is also observed through forward hysteresis. Additionally we provide the impact of parameters most effective in reducing in COVID-19 spread. Numerical simulations of the theoretical findings are offered to validate the results.
