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Dynamics of a SIR Epidemic Model of Childhood Diseases with a Saturated Incidence Rate: Continuous Model and Its Nonstandard Finite Difference Discretization

Author

Listed:
  • Isnani Darti

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia)

  • Agus Suryanto

    (Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang 65145, Indonesia)

Abstract

A SIR epidemic model that describes the dynamics of childhood disease with a saturated incidence rate and vaccination program at a constant rate was investigated. For the continuous model we first show its basic properties, namely, the non-negativity and boundedness of solutions. Then we investigate the existence and both local and global stability of the equilibrium points. It was found that the existence and stability properties of equilibrium points fully determined the basic reproduction number. We also propose and analyze a discrete-time analogue of the continuous childhood diseases by applying a nonstandard finite difference method. It is shown that our discrete model preserves the dynamical properties of the corresponding continuous model, such as the positivity solutions, the population conservation law, the existence of equilibrium points and their global stability properties.

Suggested Citation

  • Isnani Darti & Agus Suryanto, 2020. "Dynamics of a SIR Epidemic Model of Childhood Diseases with a Saturated Incidence Rate: Continuous Model and Its Nonstandard Finite Difference Discretization," Mathematics, MDPI, vol. 8(9), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1459-:d:406561
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    Cited by:

    1. Akhtar, S. & Ahmed, R. & Batool, M. & Shah, Nehad Ali & Chung, Jae Dong, 2021. "Stability, bifurcation and chaos control of a discretized Leslie prey-predator model," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Manuel De la Sen & Asier Ibeas & Santiago Alonso-Quesada, 2022. "On the Supervision of a Saturated SIR Epidemic Model with Four Joint Control Actions for a Drastic Reduction in the Infection and the Susceptibility through Time," IJERPH, MDPI, vol. 19(3), pages 1-26, January.

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