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

Xiaotian Lv; Weimin Hu; Yongzhen Yun; Youhui Su; Xiaotian Lv; Weimin Hu; Yongzhen Yun; Youhui Su

doi:10.3934/era.2026222

Electronic Research Archive

2026, Volume 34, Issue 7: 5023-5039. doi: 10.3934/era.2026222

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

1.
School of Mathematics and Statistics, Yili Normal University, Yining 835000, China
2.
School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, China
3.
Institute of Applied Mathematics, Yili Normal University, Yining 835000, China

Received: 12 April 2026 Revised: 02 June 2026 Accepted: 03 June 2026 Published: 10 June 2026

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 $.
- SEIR model,
- delay differential equation,
- fear effect,
- behavioral response,
- Hopf bifurcation
Citation: Xiaotian Lv, Weimin Hu, Yongzhen Yun, Youhui Su. Dynamical analysis and Hopf bifurcation of an SEIR epidemic model with nonlinear incidence and time-delayed behavioral response[J]. Electronic Research Archive, 2026, 34(7): 5023-5039. doi: 10.3934/era.2026222

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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 $ 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 $.

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