Stability and Hopf Bifurcation of a Time-Delayed SVIR Epidemic Model with Saturated Incidence Rates and Virus Mutation

In this paper, we propose a time-delayed SVIR epidemic model incorporating saturated incidence rates and virus mutation. First, with the help of the next-generation matrix method, we compute the basic reproduction number [Formula: see text] of the model, which is determined entirely by the basic reproduction numbers of the original strain and the variant strain. Meanwhile, we prove that the model contains three equilibrium points: a disease-free equilibrium [Formula: see text], a boundary equilibrium [Formula: see text] and an endemic equilibrium [Formula: see text]. Second, by means of eigenvalue theory and the Routh–Hurwitz criterion, we analyze the local stability of the equilibria. The analysis demonstrates that the stability of [Formula: see text] and [Formula: see text] is unaffected by the time delay [Formula: see text], while [Formula: see text] undergoes a switch in stability as [Formula: see text] crosses a critical threshold. Furthermore, by constructing a Lyapunov function, we verify that [Formula: see text] is globally asymptotically stable when [Formula: see text]. Third, taking [Formula: see text] as a bifurcation parameter, we derive sufficient conditions for the occurrence of Hopf bifurcation at the endemic equilibrium [Formula: see text], while the direction and stability of the bifurcating periodic solutions are determined using normal form theory and the center manifold theorem. Finally, we perform numerical simulations to validate and complement our theoretical results. Especially, we also investigate the effect of changes in saturation coefficients on disease transmission dynamics.