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Hopf‐Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

Author

Listed:
  • Fulgensia Kamugisha Mbabazi
  • Joseph Y. T. Mugisha
  • Mark Kimathi

Abstract

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease‐free equilibrium is asymptotically stable if the control reproduction ratio R0 is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf‐bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.

Suggested Citation

  • Fulgensia Kamugisha Mbabazi & Joseph Y. T. Mugisha & Mark Kimathi, 2019. "Hopf‐Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2019(1).
    • Handle: RePEc:wly:jnlaaa:v:2019:y:2019:i:1:n:3757036
    DOI: 10.1155/2019/3757036
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    References listed on IDEAS

    as
    1. Qiming Liu & Baochen Li & Meici Sun, 2017. "Global Dynamics of an SIRS Epidemic Model with Distributed Delay on Heterogeneous Network," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-9, February.
    2. repec:plo:pone00:0175789 is not listed on IDEAS
    3. Mingming Li & Xianning Liu, 2014. "An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    4. Mingming Li & Xianning Liu, 2014. "An SIR Epidemic Model with Time Delay and General Nonlinear Incidence Rate," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-7, February.
    5. Mbabazi, Fulgensia Kamugisha & Mugisha, J.Y.T. & Kimathi, M., 2018. "Modeling the within-host co-infection of influenza A virus and pneumococcus," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 488-506.
    6. Lixia Zuo & Maoxing Liu, 2014. "Effect of Awareness Programs on the Epidemic Outbreaks with Time Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    7. Tongqian Zhang & Xinzhu Meng & Tonghua Zhang, 2014. "SVEIRS: A New Epidemic Disease Model with Time Delays and Impulsive Effects," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-15, May.
    8. Tongqian Zhang & Xinzhu Meng & Tonghua Zhang, 2014. "SVEIRS: A New Epidemic Disease Model with Time Delays and Impulsive Effects," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    9. Lixia Zuo & Maoxing Liu, 2014. "Effect of Awareness Programs on the Epidemic Outbreaks with Time Delay," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-8, July.
    10. Raja Sekhara Rao, P. & Naresh Kumar, M., 2015. "A dynamic model for infectious diseases: The role of vaccination and treatment," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 34-49.
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