Advertisement

Differential Equations and Dynamical Systems

, Volume 23, Issue 4, pp 403–413 | Cite as

Effect of Discretization on Dynamical Behavior in an Epidemiological Model

  • Khalid HattafEmail author
  • Abid Ali Lashari
  • Brahim El Boukari
  • Noura Yousfi
Original Research

Abstract

Dynamical behavior of two discrete epidemic models for disease with nonlinear incidence rate is studied. Both discrete models are derived from the continuous case by applying forward and backward Euler methods. The effect of the two different discretizations on the stability behavior of the disease-free equilibrium and endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results.

Keywords

Discrete epidemic model Forward and backward Euler methods Stability Lyapunov functional 

References

  1. 1.
    Khan, Y., Vazquez-Leal, H., Wu, Q.: An efficient iterated method for mathematical biology model. Neural Comput. Appl. 1–6 (2012).Google Scholar
  2. 2.
    Khan, Y., Vazquez-Leal, H., Faraz, N.: An auxiliary parameter method using adomian polynomials and laplace transformation for nonlinear differential equations. Appl. Math. Model. 37(5), 2702–2708 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Khan, Y., Wu, Q.: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput. Math. Appl. 61(8), 1963–1967 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zhou, Y., Ma, Z., Brauer, F.: A discrete epidemic model for SARS transmission and control in China. Math. Comput. Model. 40, 1491–1506 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Enatsu, Y., Nakata, Y., Muroya, Y., Izzo, G., Vecchio, A.: Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates. J. Differ. Equ. Appl. (2011). doi: 10.1080/10236198.2011.555405 Google Scholar
  6. 6.
    Das, P., Mukherjee, D., Sarkar, A.K.: Study of an S-I epidemic model with nonlinear incidence rate: discrete and stochastic version. Appl. Math. Comput. 218, 2509–2515 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jang, S.: On a discrete West Nile epidemic model. Comput. Appl. Math. 26(3), 397–414 (2007)MathSciNetGoogle Scholar
  8. 8.
    Cruz-Pacheco, G., Esteva, L., Montano-Hirose, J., Vargas, C.: Modelling the dynamics of West Nile virus. Bull. Math. Biol. 67, 1157–1172 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jang, S., Elaydi, S.: Difference equations from discretization of a continuous epidemic model with immigration of infectives. Math. Fac. Res. 32 (2004).Google Scholar
  10. 10.
    Chen, Q., Teng, Z., Wang, L., Jiang, H.: The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence. Nonlinear Dyn. 71, 55–73 (2013). doi: 10.1007/s11071-012-0641-6 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hu, Z., Teng, Z., Jiang, H.: Stability analysis in a class of discrete SIRS epidemic models. Nonlinear Anal.: Real World Appl. 13, 2017–2033 (2012).Google Scholar
  12. 12.
    Xu, R., Ma, R.Z.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal.: Real World Appl. 10, 3175–3189 (2009).Google Scholar
  13. 13.
    Connell McCluskey, C.: Global stability for an SIR epidemic model with delay and nonlinear incidence. Nonlinear Anal.: Real World Appl. 11, 3106–3109 (2010).Google Scholar
  14. 14.
    Hattaf, K., Lashari, A.A., Louartassi, Y., Yousfi, N.: A delayed SIR epidemic model with general incidence rate. Electron. J. Qual. Theory Differ. Equ. 3, 1–9 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jury, E.: Theory and Applications of the Z-transform. Wiley, New York (1964)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2014

Authors and Affiliations

  • Khalid Hattaf
    • 1
    • 3
    Email author
  • Abid Ali Lashari
    • 2
  • Brahim El Boukari
    • 1
  • Noura Yousfi
    • 1
  1. 1.Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sikHassan II UniversityCasablancaMorocco
  2. 2.School of Natural SciencesNational University of Sciences and TechnologyIslamabadPakistan
  3. 3.Centre Régional des Métiers de l’Education et de la Formation (CRMEF)CasablancaMorocco

Personalised recommendations