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An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns

Author

Listed:
  • Santiago Alonso-Quesada

    (Department of Electricity and Electronics, Campus of Leioa, University of the Basque Country, UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain)

  • Manuel De la Sen

    (Department of Electricity and Electronics, Campus of Leioa, University of the Basque Country, UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain)

  • Raúl Nistal

    (Department of Electricity and Electronics, Campus of Leioa, University of the Basque Country, UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain)

Abstract

This paper analyses an SIRS epidemic model with the vaccination of susceptible individuals and treatment of infectious ones. Both actions are governed by a designed control system whose inputs are the subpopulations of the epidemic model. In addition, the vaccination of a proportion of newborns is considered. The control reproduction number R c of the controlled epidemic model is calculated, and its influence in the existence and stability of equilibrium points is studied. If such a number is smaller than a threshold value R ¯ c , then the model has a unique equilibrium point: the so-called disease-free equilibrium point at which there are not infectious individuals. Furthermore, such an equilibrium point is locally and globally asymptotically stable. On the contrary, if R c > R ¯ c , then the model has two equilibrium points: the referred disease-free one, which is unstable, and an endemic one at which there are infectious individuals. The proposed control strategy provides several free-design parameters that influence both values R c and R ¯ c . Then, such parameters can be appropriately adjusted for guaranteeing the non-existence of the endemic equilibrium point and, in this way, eradicating the persistence of the infectious disease.

Suggested Citation

  • Santiago Alonso-Quesada & Manuel De la Sen & Raúl Nistal, 2021. "An SIRS Epidemic Model Supervised by a Control System for Vaccination and Treatment Actions Which Involve First-Order Dynamics and Vaccination of Newborns," Mathematics, MDPI, vol. 10(1), pages 1-32, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2021:i:1:p:36-:d:709343
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    References listed on IDEAS

    as
    1. De la Sen, M. & Alonso-Quesada, S. & Ibeas, A., 2015. "On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 953-976.
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    Cited by:

    1. Javier Cifuentes-Faura & Ursula Faura-Martínez & Matilde Lafuente-Lechuga, 2022. "Mathematical Modeling and the Use of Network Models as Epidemiological Tools," Mathematics, MDPI, vol. 10(18), pages 1-14, September.

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