A time-delayed mathematical framework for analyzing pertussis transmission dynamics

Pertussis remains one of the most important public health challenges worldwide, with 5,436 cases reported in the United States in 2023, representing an 8% increase from the previous year, and up to 18,000 cases reported globally in recent outbreaks. This study is important because delayed epidemiological processes, including waning immunity and delayed behavioral responses, may help explain recurrent pertussis persistence despite vaccination, but their combined role is not fully captured in many classical instantaneous models. This paper examines the delayed transmission dynamics of pertussis and analyzes the epidemiological impact of time delays associated with vaccination, waning immunity, and delayed behavioral responses within a deterministic modeling framework. We present a deterministic, time-delayed compartmental model that divides the population into susceptible, infected, vaccinated, recovered, waned immunity, and secondary susceptible classes. Parameters representing temporary mitigation effects and behavioral responses are incorporated into the model. We establish basic mathematical properties, including positivity and boundedness of solutions. The disease-free and endemic equilibria are derived and analyzed using the Routh-Hurwitz criterion and Lyapunov stability techniques to assess their local and global stability with respect to the basic reproduction number. Numerical simulations are performed to confirm the analytical results and to illustrate the dynamics of each subpopulation under different scenarios. The model shows that the basic reproduction number determines the transition from pertussis elimination to persistence. Time delays significantly affect system stability at certain thresholds, leading to oscillatory or damped epidemic patterns. Simulation results indicate that delay, in combination with vaccination, reduces infection prevalence and can shift the system toward disease eradication when parameters are appropriately chosen. The proposed time-delayed mathematical framework provides insight into how vaccination, waning immunity, and delayed behavioral responses interact in shaping pertussis dynamics. The results highlight the influence of delay mechanisms on transmission dynamics and epidemic persistence, and provide a mathematical basis for improving public health strategies targeting pertussis transmission.