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Related Experiment Video

Updated: Jun 11, 2026

Evaluation of Host-Pathogen Responses and Vaccine Efficacy in Mice

Evaluation of Host-Pathogen Responses and Vaccine Efficacy in Mice

Published on: February 22, 2019

A time-delayed mathematical framework for analyzing pertussis transmission dynamics.

Ali Raza^1,2, Marek Lampart³, Eugenio M Rocha⁴

¹IT4Innovations, VSB-Technical University of Ostrava, 17 listopadu 2172/15, Ostrava, 708 33, Czech Republic. alimustasamcheema@gmail.com.

Scientific Reports

|June 9, 2026

Summary

This summary is machine-generated.

Pertussis (whooping cough) outbreaks persist due to waning immunity and delayed responses. Mathematical modeling shows that incorporating time delays into vaccination strategies can reduce pertussis prevalence and aid eradication efforts.

Keywords:

Delay differential equations Delayed modeling Numerical results Pertussis dynamics Stability analysis Time delay effects

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Area of Science:

Epidemiology
Mathematical Biology
Public Health

Background:

Pertussis (whooping cough) remains a significant global public health concern, with increasing case numbers reported.
Classical epidemiological models often overlook the impact of time delays, such as waning immunity and behavioral responses, on disease dynamics.

Purpose of the Study:

To investigate the role of time delays in pertussis transmission dynamics.
To analyze the epidemiological impact of vaccination, waning immunity, and delayed behavioral responses using a deterministic modeling framework.

Main Methods:

Development of a deterministic, time-delayed compartmental model for pertussis transmission.
Analysis of model properties, equilibria, and stability using Routh-Hurwitz criterion and Lyapunov techniques.
Numerical simulations to explore subpopulation dynamics under various scenarios.

Main Results:

The basic reproduction number dictates the transition between pertussis elimination and persistence.
Time delays significantly influence epidemic patterns, leading to oscillations or damping.
Incorporating delays with vaccination strategies can reduce infection prevalence and promote disease eradication.

Conclusions:

Time-delayed mathematical models offer crucial insights into pertussis transmission dynamics.
Vaccination, waning immunity, and behavioral response delays interact to shape epidemic persistence.
The findings provide a mathematical foundation for enhancing public health strategies against pertussis.