Delay-induced dynamics and Hopf bifurcation in a quarantined SEIQR epidemic model with nonlinear incidence

Mathematical modeling provides a rigorous framework for analyzing infectious disease transmission and assessing control strategies, among which quarantine plays a central role in limiting disease spread. In this study, we propose a delayed SEIQR epidemic model incorporating quarantine measures and a nonlinear incidence rate, where the population is partitioned into five compartments: susceptible, exposed, infected, quarantined, and recovered. Initially, the positivity and boundedness of solutions are established to ensure the biological feasibility of the model. The basic reproduction number \(\:{R}_{0}\) is derived using the next-generation matrix approach, and the local stability of the equilibrium points is rigorously analyzed. In particular, the healthy equilibrium point (HEP) is shown to be locally asymptotically stable when \(\:{R}_{0}\:<\:1\) , while the unhealthy equilibrium point (Un-HEP) becomes locally asymptotically stable when \(\:{R}_{0}\:>\:1\) , based on the Routh–Hurwitz criterion. Furthermore, by considering the time delay \(\:\tau\:\) as a bifurcation parameter, sufficient conditions for the occurrence of Hopf bifurcation are established, and the direction and stability of the emerging periodic solutions are determined. Unlike existing studies, this work introduces a non-monotonic saturated incidence with delay within a quarantine-based SEIQR framework, and rigorously derives explicit analytical conditions governing the onset, direction, and stability of Hopf bifurcation, thereby uncovering critical delay-induced oscillatory dynamics that are often overlooked in classical epidemic models. Finally, numerical simulations with appropriately selected parameter values are provided to validate the theoretical findings and to illustrate the dynamical behavior of the proposed model.