Title: Bifurcation and Stability Analysis of a Discrete Time Sir Epidemic Model with Vaccination
Author(s): Özlem Ak Gümüş, A. George Maria Selvam, D. Abraham Vianny
Pages: 809-820
Cite as:
Özlem Ak Gümüş, A. George Maria Selvam, D. Abraham Vianny, Bifurcation and Stability Analysis of a Discrete Time Sir Epidemic Model with Vaccination, Int. J. Anal. Appl., 17 (5) (2019), 809-820.


In this paper, we study the qualitative behavior of a discrete-time epidemic model with vaccination. Analysis of the model shows forth that the Disease Free Equilibrium (DFE) point is asymptotically stable if the basic reproduction number R0 is less than one, while the Endemic Equilibrium (EE) point is asymptotically stable if the basic reproduction number R0 is greater than one. The results are reinforced with numerical simulations and enhanced with graphical representations like time trajectories, phase portraits and bifurcation diagrams for different sets of parameter values.

Full Text: PDF



  1. L.J.S. Allen, An Introduction to Mathematical Biology, Pearson, New Jersey, 2007. Google Scholar

  2. Q. Din, Qualitative behavior of a discrete SIR epidemic model, Int. J. Biomath. 9 (6) (2016), 1650092. Google Scholar

  3. A. George Maria Selvam and D. Abraham Vianny, Behavior of a Discrete Fractional Order SIR Epidemic model, Int. J. Eng. Technol. 7 (2018), 675-680. Google Scholar

  4. A. George Maria Selvam, D. Abraham Vianny and Mary Jacintha, Stability in a fractional order SIR epidemic model of childhood diseases with discretization, IOP Conf. Ser., J. Phys., Conf. Ser., 1139 (2018), 012009. Google Scholar

  5. A.George Maria Selvam and D.Abraham Vianny, Discrete Fractional Order SIR Epidemic Model of Childhood Diseases with Constant Vaccination and its Stability, Int. J. Techn. Innov. Modern Eng. Sci., 4 (11) (2018), 405-410. Google Scholar

  6. E.A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. Google Scholar

  7. X. Liu, D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system. Chaos Solution Fract., 32 (2007), 80-94. Google Scholar

  8. X. Liu, C. Mou, W. Niu, D. Wang, Stability analysis for discrete biological models using algebraic methods. Math. Comput Sci. 5 (2011), 247-262. Google Scholar

  9. R M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467. Google Scholar

  10. H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, Netherlands, 2003. Google Scholar

  11. M. J. Keeling and L. Danon, Mathematical modelling of infectious diseases, Br. Med. Bull. 92 (1) (2009), 33-42. Google Scholar

  12. Kermack, W. O. McKendrick, A. G, A Contribution to the Mathematical Theory of Epidemics, Proc. Royal Soc. Ser A, 115 (772) (1927), 700-721. Google Scholar

  13. W. Hamer, II. Epidemic disease in England. Lancet, 1 (1906), 733-739. Google Scholar

  14. R. Ross, The Prevention of Malaria. 2nd ea. John Murray, London, 1911. Google Scholar

  15. O. Ak Gumus, Global and local stability analysis in a nonlinear discretetime population model, Adv. Difference Equ. 2014 (2014), 299. Google Scholar

  16. H Merdan, O. Ak Gumus, Stability analysis of a general discrete-time population model involving delay and Allee effects. Appl. Math. Comput. 219 (2012), 1821-1832. Google Scholar

  17. Q Din, OA G¨um¨u¸s and H Khalil, Neimark-Sacker Bifurcation and Chaotic Behavior of a Modified Host-Parasitoid Model, ¨ Z. Naturforsch., A, 72 (1) (2017), 25-37. Google Scholar

  18. H. Merdan, O. Ak G¨um¨u¸s and G. Karahisarli, Global Stability Analysis of a General Scalar Difference Equation, Discon- ¨ tinuity, Nonlinear. Complex. 7 (3) (2018), 225-232. Google Scholar

  19. I. Longili, The generalized discrete-time epidemic model with immunity: a synthesis, Math. Biosci. 82(1986), 19-41. Google Scholar

  20. S. Jang, S. Elaydi, Difference equations from discretization of a continuous epidemic model with immigration of infectives, Can. Appl. Math. Q. 11 (2003), 93-105. Google Scholar

  21. X. Ma, Y. Zhou and H. Cao, Global stability of the endemic equilibrium of a discrete SIR epidemic model, Adv. Difference Equ. 2013 (2013), 42. Google Scholar

  22. L. Allen, An Introduction to Deterministic Models in Biology, Prentice-Hall, 2004. Google Scholar

  23. S. Elaydi, Discrete Chaos, Chapman and Hall/CRC, 2007. Google Scholar

  24. May, R. M., Parasitic infections as regulators of animal populations. Amer. Scientist, 71 (1983), 36-45. Google Scholar


Copyright © 2020 IJAA, unless otherwise stated.