https://doi.org/10.1140/epjp/s13360-025-07189-w
Regular Article
Mathematical analysis and optimal control of a pertussis transmission model with stability and sensitivity analysis
1
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
2
Department of Anatomy and Physiology, College of Medicine, Imam Mohammad Ibn Saud Islamic University (IMSIU), 13317, Riyadh, Saudi Arabia
3
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
4
Mathematics Department, Faculty of Science, Taibah University, 41411, Al-Madinah Al-Munawarah, Saudi Arabia
a
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Received:
6
November
2025
Accepted:
5
December
2025
Published online:
16
December
2025
Abstract
This paper presents a comprehensive mathematical and control theoretic analysis of a pertussis (whooping cough) transmission model with five compartments: susceptible, latent immunity, exposed, infectious, and recovered populations. The model incorporates key biological features of pertussis, including waning vaccine-induced immunity, reinfection, and disease-induced mortality, and is shown to possess fundamental properties such as positivity, boundedness, and invariance of the feasible region. The basic reproduction number 
is derived via the next-generation matrix method, and sensitivity analysis identifies transmission, progression to infectivity, recovery, and vaccination rates as the most influential parameters governing disease spread and persistence. The local and global stability of both the disease-free and endemic equilibria are rigorously established, and a nonstandard finite difference scheme is constructed to generate numerical simulations that confirm the analytical results and illustrate how variations in key parameters affect the epidemic dynamics. To investigate the impact of active interventions, the model is further extended by incorporating an integral sliding mode control (ISMC) strategy acting on the vaccination rate; numerical experiments demonstrate that ISMC robustly suppresses exposed and infectious populations and can steer the system toward disease elimination despite parameter uncertainties and external disturbances. Overall, the study highlights how combining rigorous stability analysis, sensitivity insights, and robust control design can inform evidence-based public health policies for the effective management of pertussis.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
