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Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity

Author

Listed:
  • Imane Abouelkheir

    (Department of Mathematics and Computer Sciences, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Casablanca 20000, Morocco)

  • Fadwa El Kihal

    (Department of Mathematics and Computer Sciences, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Casablanca 20000, Morocco)

  • Mostafa Rachik

    (Department of Mathematics and Computer Sciences, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Casablanca 20000, Morocco)

  • Ilias Elmouki

    (Department of Mathematics and Computer Sciences, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca, Casablanca 20000, Morocco)

Abstract

Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when it comes to vaccination, most optimization approaches in the literature and those that have been subject to epidemic models have focused on treating problems that led to continuous vaccination schedules but their applicability remains debatable. In search of a more realistic methodology to resolve this issue, a control modeling design, where the control can be characterized analytically and then optimized, can definitely help to find an optimal regimen of pulsed vaccinations. Therefore, we propose a susceptible-infected-removed (SIR) hybrid epidemic model with impulse vaccination control and a compartment that represents the number of vaccinated individuals supposed to not acquire sufficient immunity to become permanently recovered due to the short-term effect of vaccines. A basic reproduction number, when the control is defined as a constant parameter, is calculated. Since we also need to find the optimal values of this impulse control when it is defined as a function of time, we start by stating a general form of an impulse version of Pontryagin’s maximum principle that can be adapted to our case, and then we apply it to our model. Finally, we provide our numerical simulations that are obtained via an impulse progressive-regressive iterative scheme with fixed intervals between impulse times (theoretical example of an impulse at each week), and we conclude with a discussion of our results.

Suggested Citation

  • Imane Abouelkheir & Fadwa El Kihal & Mostafa Rachik & Ilias Elmouki, 2019. "Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity," Mathematics, MDPI, vol. 7(5), pages 1-21, May.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:420-:d:230077
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    References listed on IDEAS

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    1. Zeng, Guang Zhao & Chen, Lan Sun & Sun, Li Hua, 2005. "Complexity of an SIR epidemic dynamics model with impulsive vaccination control," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 495-505.
    2. Gakkhar, Sunita & Negi, Kuldeep, 2008. "Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 626-638.
    3. Chahim, Mohammed & Hartl, Richard F. & Kort, Peter M., 2012. "A tutorial on the deterministic Impulse Control Maximum Principle: Necessary and sufficient optimality conditions," European Journal of Operational Research, Elsevier, vol. 219(1), pages 18-26.
    4. Gao, Shujing & Teng, Zhidong & Xie, Dehui, 2009. "Analysis of a delayed SIR epidemic model with pulse vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 1004-1011.
    5. Oluwaseun Sharomi & Tufail Malik, 2017. "Optimal control in epidemiology," Annals of Operations Research, Springer, vol. 251(1), pages 55-71, April.
    6. Guang-Zhao Zeng & Lan-Sun Chen, 2005. "Complexity And Asymptotical Behavior Of A Sirs Epidemic Model With Proportional Impulsive Vaccination," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 8(04), pages 419-431.
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    Cited by:

    1. Angelo Alessandri & Patrizia Bagnerini & Roberto Cianci & Mauro Gaggero, 2019. "Optimal Propagating Fronts Using Hamilton-Jacobi Equations," Mathematics, MDPI, vol. 7(11), pages 1-10, November.
    2. Gamboa, M. & López-García, M. & Lopez-Herrero, M.J., 2024. "On the exact and population bi-dimensional reproduction numbers in a stochastic SVIR model with imperfect vaccine," Applied Mathematics and Computation, Elsevier, vol. 468(C).
    3. Maria Gamboa & Maria Jesus Lopez-Herrero, 2020. "The Effect of Setting a Warning Vaccination Level on a Stochastic SIVS Model with Imperfect Vaccine," Mathematics, MDPI, vol. 8(7), pages 1-23, July.
    4. Yunhan Huang & Quanyan Zhu, 2022. "Game-Theoretic Frameworks for Epidemic Spreading and Human Decision-Making: A Review," Dynamic Games and Applications, Springer, vol. 12(1), pages 7-48, March.

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