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.
Feng et al. (Sat,) studied this question.