Dynamical analysis and Hopf bifurcation of an SEIR epidemic model with nonlinear incidence and time-delayed behavioral response

In this paper, an SEIR epidemic dynamical model incorporating a fear effect, a nonlinear incidence rate, and a time-delayed behavioral response was established. The basic reproduction number $R_0$ was derived using the next-generation matrix method. Theoretical analysis demonstrates that if $R_0<1$, the disease-free equilibrium is globally asymptotically stable; if $R_0>1$, there exists a unique endemic equilibrium, which is also globally asymptotically stable. By performing stability analysis, the critical time delay threshold ${\tau }_0$ for triggering a Hopf bifurcation was derived. Moreover theoretical analysis and numerical simulations consistently showed that when the time delay exceeds this threshold, the endemic equilibrium loses its stability and a stable limit cycle is generated, leading to periodic epidemic outbreaks. Furthermore, sensitivity analysis revealed an inverse relationship between fear coefficient $\alpha$ and the critical time delay threshold ${\tau }_0$.