Global stability for a discrete SIR epidemic model with delay in the general incidence function

  • Authors

    • Guiro Aboudramane Département de Mathématique, UFR/ST,Université Nazi Boni, Bobo-Dioulasso, Burkina Faso
    • Dramane Ouedraogo
    • Harouna Ouedraogo
    2019-09-14
    https://doi.org/10.14419/ijamr.v8i2.29528
  • Discrete SIR Epidemic Model, General Incidence, Lyapunov Function, Backward Difference Scheme, Local Stability, Global Stability.

  • Abstract

    In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.

    Author Biography

    • Guiro Aboudramane, Département de Mathématique, UFR/ST,Université Nazi Boni, Bobo-Dioulasso, Burkina Faso

      Département de Mathématiques,

      UFR Sciences et Techniques

  • References

    1. [1] C. Connell McCuskey, Global stability for an SEIR epidemiological with varying infectivity and infinite delay, Math. Biosci. Eng. 6 (2009) 603-610.

      [2] C. Connell McCuskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA 10(2009) 3175-3189.

      [3] C. Connell McCuskey, Complete global stability for an SIR epidemic model with delay distributed or discrete, Nonlinear Anal. RWA 11(2010) 55-59.

      [4] Y. Enatsu, Y. Nakata, Y. Muroya, Global stability for class of discrete SIR epidemic models, Mathematical Biosciences and Engineering 7 (2010)347-361.

      [5] A. Guiro, D. Ngom, D. Ouedraogo, Stability analysis for a class of discrete Schistosomiasis models with general incidence, Advances in Difference

      Equation, (2017); 1-16.

      [6] A. Guiro, D. Ou´edraogo, H. Ouedraogo, Global stability for a delay SIR epidemic model with general incidence function, observers design, Submitted

      to Applied mathematics.

      [7] J. P. Lasalle, The stability of Dynamical Systems, SIAM, Philadelphia, 1976.

      [8] V. Lakshmikantham, S. Leela, A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York, 1989.

      [9] Z. Teng, Y. Wang, M. Rehim, On the backward difference scheme for a class of SIRS epidemic models with nonlinear incidence. J. Computation

      Analysis and Applications, vol. 20, No 7, 2016, pp. (1268-1289).

      [10] P. Van den Driesche and J. Watmough, Reproduction Numbers and Substhreshold Endemic Equilibria for the compartmental Model of Disease

      Transmission, Mathematical Biosciences, 180(2002)No:1-2; pp:29-48.

      [11] M. Sekiguchi, E. Ishiwata, Global dynamics of a discrete SIRS epidemic model with time delay, J. Math. Anal. Appl., 371;195-202(2010).

      [12] M. Sekiguchi, Permanence of a discrete SIRS epidemic model with time delays, Appl. Math. Letters, 23;1280-1285(2010).

      [13] Z. Hu, Z. Teng, H. Jiang, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Anal.: RWA, 13;2017-2033(2012).

      [14] R. Xu, Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA 10(2009)3175-3189.

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  • How to Cite

    Aboudramane, G., Ouedraogo, D., & Ouedraogo, H. (2019). Global stability for a discrete SIR epidemic model with delay in the general incidence function. International Journal of Applied Mathematical Research, 8(2), 32-45. https://doi.org/10.14419/ijamr.v8i2.29528

    Received date: 2019-06-20

    Accepted date: 2019-08-23

    Published date: 2019-09-14