Global dynamics of a staged progression model for whooping cough
Keywords:
Whooping cough disease, Mathematical model, Equilibria, Global Stability
Abstract
This paper introduces a comprehensive stability analysis of a pertussis (whooping cough) model that incorporates staged disease progression. To examine the asymptotic and symptomatic behavior of the disease with respect the model’s equilibria, a qualitative analysis of the model is conducted. The local stability of the disease-free equilibrium is demonstrated using the Jacobian stability method, demonstrating its local asymptotic stability. Additionally, the comparison method is utilized to demonstrate the global stability of the model, establishing that the disease-free equilibrium achieves global asymptotic stability when the basic reproduction number falls below one. Numerical simulations based on baseline data further validate the analytical results. Furthermore, the research explores the influence of varying some of the model parameters.
References
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Ji, Y. (2014). Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 12(3), 525-536.
Joshi, H. K., Sawant, S., and Rasaria, S. (2019). To assess the knowledge of Mothers in Relation to Selected Aspect of Immunization. Innovational: Journal of Nursing and Healthcare, 7-12.
Keeling, M. J., and Rohani, P. (2011). Modeling infectious diseases in humans and animals. Princeton University Press.
Koenig, K. L., Farah, J., McDonald, E. C., Thihalolipavan, S. and Burns, M. J. (2019). Pertussis: The identify, isolate, inform tool applied to a re-emerging respiratory illness. Western Journal of Emergency Medicine, 20: 191-191. DOI: 10.5811/westjem.2018.11.40023
Aisha A.Y, Farah A. A, Ahmad I. I, and Yazariah M. Y. (2020). Dynamical Analysis on the Transmission of pertussis with Maternally Derived Immunity. Journal of mathematics and Statistics. Vol.16: 104. 112. DOI: 10.3844/jmssp.2020.104.112.
Akinyemi, S. T., Ibrahim, M. O., Usman, I. G., & Odetunde, O. (2016). Global stability analysis of sir epidemic model with relapse and immunity loss.
Akinyemi, S. T., Yemisi, O., Olarenwaju, I. M., and Bala, A. (2023). Approximate Solution of a Fractional-Order Ebola Virus Disease Model with Contact Tracing and Quarantine. Applied Mathematics and Computational Intelligence (AMCI), 12(1), 30-42.
Alqarni, M. M., Nasir, A., Alyami, M.A., Raza, A., Awrejcewicz, J., Rafiq, M., … and Mahmoud, E. E. (2022). A SEIR Epidemic Model of whooping cough-Like Infeciouss and its Dynamically Consistent Approximation. Complexity, 2022(1), 3642444.
Ayoade, A., Akinyemi, S., and Oyedepo, T. (2024). Minimization of HIV Infection among Nigerian Women through the Use of Microbicides: An Insight from Mathematical Modeling. Journal of Advanced Mathematical Modeling, 13(5), 1-22.
Azizi, T. (2024). ‘Mathematical Modeling of Cancer Progression. AppliedMath, 4(3), 1065-1079.
Bokhari, H., bilal, I., and Zafar, S. (2012). BapC autotransporter protein of Bordetella pertussis is an adhesion factor. Journal of basic microbiology, 52(4), 390-396.
Brauer, F., Castillo-Chavez, C. and Feng, Z. (2019). Mathematical Models in Epidemiology; Springer: New York, NY, USA; Volume 32.
Carbonetti, N. H. (2015). Contribution of pertussis toxin to the pathogenesis of pertussis disease. Pathogens and disease, 73(8), ftv073.
Carrasquilla, G., Porras, A., Martinez, S., DeAntonio, R., Devadiga, R., Caceres, D. C., and Juliao, P. (2020). Incidence and mortality of pertussis disease in infants< 12 months of age following introduction of pertussis maternal universal mass vaccination in Bogotá, Colombia. Vaccine, 38(46), 7384-7392.
Chahine, M., Tzenios, N., & poh, O. B. J. (2024). How can mathematical models be used to predict the progression and impact of chronic diseases in the medical field?. Internalation journal of artificial intelligence, 4(07), 54-69.
Dago, M. M., Ibrahim, M. O., and Tosin, A. S. (2015). Stability Analysis of a Deterministic Mathematical Model for Transmission Dynamics of Tuberculosis. 5. International Journal of Advances in Science Engineering and Technology, ISSN, 2, 2321-9009.
De Cellès, M.D., Magpantay, F.M., King, A.A. and Rohani, P. (2018). The impact of past vaccination coverage and immunity on pertussis resurgence. Sci. Translat. Med., 10: 1748-1748. DOI: 10.1126/scitranslmed. aaj1748.
Fabricius, G., Aispuro P.M., Bergero P., Bottero D. and Gabrielli, M. (2018). Pertussis epidemiology in Argentina: TRENDS after the introduction of maternal immunisation. Epidemiol. Infect., 146: 858-866. DOI: 10.1017/S0950268818000808
Gharahasanlou, T. K., Roomi, V., and Hemmatzadeh, Z. (2022). Global stability analysis of viral infection model with logistic growth rate, general incidence function and cellular immunity. Mathematics and Computers in Simulation, 194, 64-79.
Hefferman, J., Smith, R., and Wahl, L. (2005). Perspectives on the basic reproduction ratio. Journal of the Royal Society interface 2, 281-293.
Hethcote, H.W. (2000). The mathematics of infectious diseases. SIAM Re-view. 42: 599-653
Huang, G., Beretta, E., and Takeuchi, Y. (2012). Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 9(2),297-312.
Ibragimov, R.N., Bakhtiyarov, S., and Zweigart, P. (2017). Correlation between three-dimensional sine sweep dynamics vibration testing and resonantly interacting internal gravity wave field. Mathematical Modelling of Natural Phenomena, 12(2), 62-93.
Ibrahim, M. O., Akinyemi, S. T., Dago, M. M., and Bakare, N. G. (2015). Mathematical Modelling of a Staged Progression HIV/AIDS Model with Control Measures. Journal of the Nigerian Association of Mathematical Physics, 29, 163-166.
Jamal, A., Jahan, S., Choudhry, H., Rather, I. A., and Khan, M.I. (2022). “A subtraction genomics-based approach to identify and characterize new drug targets in Bordetella pertussis: Whooping cough”. Vaccine, 10(11), 1915. 1-10.
Jenkins, V. A., Savic, M., and Kandeil, W. (2020). Pertussis in high risk groups: an overview of the past quarter century. Human vaccines & immune therapeutics, 16(11), 2609-2617.
Ji, Y. (2014). Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 12(3), 525-536.
Joshi, H. K., Sawant, S., and Rasaria, S. (2019). To assess the knowledge of Mothers in Relation to Selected Aspect of Immunization. Innovational: Journal of Nursing and Healthcare, 7-12.
Keeling, M. J., and Rohani, P. (2011). Modeling infectious diseases in humans and animals. Princeton University Press.
Koenig, K. L., Farah, J., McDonald, E. C., Thihalolipavan, S. and Burns, M. J. (2019). Pertussis: The identify, isolate, inform tool applied to a re-emerging respiratory illness. Western Journal of Emergency Medicine, 20: 191-191. DOI: 10.5811/westjem.2018.11.40023
Published
2025-12-03
How to Cite
Akinyemi, S., Ogbuagu, B., Maigemu, M., & Abubakar, S. (2025). Global dynamics of a staged progression model for whooping cough. International Journal of Artificial Intelligence & Mathematical Sciences, 4(1), 1-18. https://doi.org/10.58921/ijaims.v4i1.134
