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

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Abstract

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 R₀ was derived using the next-generation matrix method. Theoretical analysis demonstrates that if R₀ < 1, the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, there exists a unique endemic equilibrium, which is also globally asymptotically stable. By performing stability analysis, the critical time delay threshold τ₀ 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 α and the critical time delay threshold τ₀.

Keywords

SEIR model / delay differential equation / fear effect / behavioral response / Hopf bifurcation

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