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Optimal Control in Two Strain Pneumonia Transmission Dynamics

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  • Mary C. Swai
  • Nyimvua Shaban
  • Theresia Marijani

Abstract

A mathematical model for the transmission dynamics of pneumonia disease in the presence of drug resistance is formulated. Intervention strategies, namely, vaccination, public health education, and treatment are implemented. We compute the effective reproduction numbers and establish the local stability of the equilibria of the model. Global stability of the disease‐free equilibrium is obtained through the comparison method. On the other hand, we apply the Lyapunov method to show that the drug‐resistant equilibrium is globally asymptotically stable under some feasible biological conditions. Furthermore, we apply optimal control theory to the model aiming at minimizing the number of infections from drug‐sensitive and drug‐resistant strains. The necessary conditions for the optimal solutions of the model were derived by using Pontryagin’s Maximum Principle. The optimal controls are characterized in terms of the optimality system, which is solved numerically for several scenarios to investigate the best strategy. The incremental cost‐effectiveness analysis technique is used to find the most cost‐effective strategy, and it is observed that the vaccination program is the most cost‐effective strategy in case of limited resources. However, results show that implementing the three strategies simultaneously provides the best results in controlling the disease.

Suggested Citation

  • Mary C. Swai & Nyimvua Shaban & Theresia Marijani, 2021. "Optimal Control in Two Strain Pneumonia Transmission Dynamics," Journal of Applied Mathematics, John Wiley & Sons, vol. 2021(1).
    • Handle: RePEc:wly:jnljam:v:2021:y:2021:i:1:n:8835918
    DOI: 10.1155/2021/8835918
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    References listed on IDEAS

    as
    1. Mohammed Kizito & Julius Tumwiine, 2018. "A Mathematical Model of Treatment and Vaccination Interventions of Pneumococcal Pneumonia Infection Dynamics," Journal of Applied Mathematics, John Wiley & Sons, vol. 2018(1).
    2. Mohammed Kizito & Julius Tumwiine, 2018. "A Mathematical Model of Treatment and Vaccination Interventions of Pneumococcal Pneumonia Infection Dynamics," Journal of Applied Mathematics, Hindawi, vol. 2018, pages 1-16, March.
    3. Fatmawati & Hengki Tasman, 2016. "An Optimal Treatment Control of TB-HIV Coinfection," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2016, pages 1-11, March.
    4. Furaha Chuma & Gasper Godson Mwanga & Verdiana Grace Masanja, 2019. "Application of Optimal Control Theory to Newcastle Disease Dynamics in Village Chicken by Considering Wild Birds as Reservoir of Disease Virus," Journal of Applied Mathematics, John Wiley & Sons, vol. 2019(1).
    5. Joshua Kiddy K. Asamoah & Francis T. Oduro & Ebenezer Bonyah & Baba Seidu, 2017. "Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis," Journal of Applied Mathematics, John Wiley & Sons, vol. 2017(1).
    6. Furaha Chuma & Gasper Godson Mwanga & Verdiana Grace Masanja, 2019. "Application of Optimal Control Theory to Newcastle Disease Dynamics in Village Chicken by Considering Wild Birds as Reservoir of Disease Virus," Journal of Applied Mathematics, Hindawi, vol. 2019, pages 1-14, March.
    7. Joshua Kiddy K. Asamoah & Francis T. Oduro & Ebenezer Bonyah & Baba Seidu, 2017. "Modelling of Rabies Transmission Dynamics Using Optimal Control Analysis," Journal of Applied Mathematics, Hindawi, vol. 2017, pages 1-23, July.
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    Cited by:

    1. Fikru Shiferaw Tessema & Purnachandra Rao Koya & Boka Kumsa Bole, 2022. "Optimal Control and Cost‐Effectiveness Analysis of Cholera with Vaccination," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    2. Yustina A. Liana & Mary C. Swai, 2024. "Mathematical Modeling of Coccidiosis Dynamics in Chickens with Some Control Strategies," Abstract and Applied Analysis, John Wiley & Sons, vol. 2024(1).

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