Bifurcation and Stability of an Discrete-time SIS Epidemic Model with Treatment

Abstract

Mathematical models are useful in examining the effect of an infection on populations. Conditions involving the spread and control of the disease are calculated by analyzing mathematical models, so that it is possible to have information about the behavior of the infection. This article includes the dynamic behavior of a discrete-time SIS epidemic model with treatment. Existence conditions of the fixed points of the model are obtained, and stability analysis is performed for these fixed points. The stability and bifurcation conditions of the obtained endemic fixed point are investigated. Depending on the infection coefficient, the flip bifurcation condition is obtained. At the same time, it is determined in which situation Neimark Sacker bifurcation may occur depending on the step size, and bifurcation is controlled. Rich dynamic behaviors are given to support our theoretical results.

Keywords

• Stability
• Neimark-Sacker Bifurcation
• Flip Bifurcation
• Treatment
• Epidemic Model

References

1. Brauer. F., Castillo-Chavez. C., “Mathematical models in population biology and epidemiology”, Texts in Applied Mathematics, 40, Springer, New York, (2001).
2. [2] Britton. N. F., “Essential Mathematical Biology”, Springer, London, (2003).
3. Wang. W., Ruan. S., “Bifurcation in an epidemic model with constant removal rate of the infectives”, J. Math. Anal. Appl., 291: 775, (2004).
4. Feng. Z, Thieme. H. R, “Recurrent outbreaks of childhood diseases revisited: the impact of isolation”, Math. Biosci., 128: 93, (1995).
5. Hyman. J.M., Li. J., “Modeling the effectiveness of isolation strategies in preventing STD epidemics”, SIAM J. Appl. Math., 58:912, (1998).
6. Wu. L., Feng. Z., “Homoclinic bifurcation in an SIQR model for childhood diseases”, J. Differ. Equat., 168:150, (2000).
7. Wang. W., “Backward bifurcation of an epidemic model with treatment”, Math Biosci., 201, 58–71, (2006).
8. Hethcote. H. W., “The mathematics of infectious disease”, SIAM Rev., 42:599, (2000).

9. Ak Gümüş . Ö., George Maria Selvam. A., Abraham Vianny. D., “Bifurcation and stability analysis of a discrete time SIR epidemic model with vaccination”, Int. J. Anal. Appl., 17(5):809–820, (2019), doi: 10.28924/2291-8639.
10. Ak Gümüş . Ö., Acer. S., “Period-doubling bifurcation analysis and stability of epidemic model”, Journal of Science and Arts Year 19, 4(49): 905–914, (2019).
11. Li. X, Mou. C., Niu. W., Wang. D., “Stability analysis for discrete biological models using algebraic methods”, Math. Comput. Sci., 5: 247–262, (2011).
12. Li. X. Z., Li. W. S., Ghosh. M., “Stability and bifurcation of an SIS epidemic model with treatment”, Chaos, Solitons & Fractals, 42 (5): 2822–2832, (2009).
13. Kuznetsov. Y. A., “Elements of Applied Bifurcation Theory”, Springer-Verlag, New York, NY, USA, 2nd edition, (1998).
14. Guckenheimer. J., Holmes. P., “Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields”, Springer-Verlag, New York, NY, USA, (1983).
15. Robinson. C., “Dynamical Systems: Stability, Symbolic Dynamics and Chaos”, CRC Press, Boca Raton, Fla, USA, 2nd edition, (1999).
16. Wiggins. S., “Introduction to Applied Nonlinear Dynamical Systems and Chaos”, 2, Springer-Verlag, New York, NY, USA, (2003).
17. Ak Gümüş . Ö., Feckan. M., “Stability, Neimark-Sacker Bifurcation and Chaos control for a prey-predator system with harvesting effect on predator”, Miskolc Mathematical Notes, (Accepted).
18. Jang. S., Elaydi. N., “Difference equations from discretization of a continuous epidemic model with immigration of infectives”, Can. Appl. Math. Q., 11:93-105, (2003).
19. Din. Q., “Bifurcation analysis and chaos control in discrete-time glycolysis models”, J Math Chem, 56: 904–931, (2018), https://doi.org/10.1007/s10910-017-0839-4
20. Rana. M. S., Kulsum. U., “Bifurcation analysis and chaos control in a discrete-time predator-prey system of Leslie type with simplified Holling type IV Functional Response”, Discrete Dynamics in Nature and Society, 2017:1-17, 9705985, (2017), https://doi.org/10.1155/2017/9705985.
21. Wang. J., Feckan. M., “Dynamics of a discrete nonlinear prey-predator model”, Int. J. Bifurcation and Chaos, 30:1-15, 2050055, (2020).
22. Baydemir. P., Merdan. H., Karaoglu. E., Sucu. G., “Complex dynamics of a discrete-time prey-predator system with Leslie type: stability, bifurcation analysis and chaos”, Journal of Bifurcation and Chaos, 3(10), 2050149, (2020).
23. George Maria Selvam. A., Dhineshbabu. R., Ak Gümü. Ö., “Stability and neimark-sacker bifurcation for a discrete system of one-scroll chaotic attractor with fractional order”, Journal of Physics: Conference Series, 1597, 012009, (2020).
24. Ak Gümüş. Ö., Yalcin. Y., “Stability and Hopf Bifurcation analysis of delay prey-predator model”, Journal of Science and Arts, 20(2): 277–282, (2020).
25. Ak Gümüş. Ö., “Neimark-Sacker bifurcation and stability of a prey-predator system”, Miskolc Mathematical Notes, 21(2): 873–885, (2020).
26. Din. Q., Ak Gümüş. Ö., Khalil. H., “Neimark-Sacker bifurcation and chaotic behaviour of a modified Host Parasitoid model”, Z. Naturforsch. A, 72(1):25-37, (2017).
27. Ak Gümüş. Ö., George Maria Selvam. A., Janagaraj. R., “Stability of modified Host-Parasitoid model with Allee effect”, Applications and Applied mathematics: An International Journal, 15(2):1032-1045, (2020).
28. George Maria Selvam. A., Janagaraj. R., Alaa Hlafta, “Bifurcation behaviour of a discrete differential algebraic prey-predator system with Holling type II functional response and prey refuge”, AIP Conference Proceedings, 2282:1-13, 020011, (2020).
29. George Maria Selvam. A., Abraham Vianny. D., Mary Jacintha, “Stability in a fractional order SIR epidemic model of childhood diseases with discretization”, J. Phys., Conf. Ser., 1139, 012009, (2018).
30. George Maria Selvam. A., Janagaraj. R., “Bifurcation analysis in a discrete time square root response function of predator-prey system with fractional order”, IOP Conference Series: Journal of Physics, 1597:1-10, 012004, (2020).
31. Din. Q., “Qualitative behavior of a discrete SIR epidemic model”, Int. J. Biomath., 9(6), 1650092, (2016).
32. Karahisarli. G., Merdan. H., Tridane. A., “Stability and zero-Hopf bifurcation analysis of a tumour and T-helper cells interaction model in the case of HIV infection”, Miskolc Mathematical Notes, 2: 911-937, (2020), DOI: 10.18514/MMN.2020.3412
33. Merdan. H., Ak Gümüş. Ö., “Stability analysis of a general discrete-time population model involving delay and Allee effects”, Appl. Math. Comput., 219:1821-1832, (2012).
34. Isik S., “A study of stability and bifurcation analysis in discrete-time predator-prey system involving the Allee effect”, International Journal of Biomathematics, 12(1): 1–15, (2019).
35. Kapçak. S., Elaydi. S., Ufuktepe. U., “Stability of a predator-prey model with refuge effect”, J. Diff. Equ. Appl., 22: 989–1004, (2016). Elaydi. S. N., “An Introduction to Difference Equations”, Springer-Verlag, New York, NY, USA, (2005).
36. Kartal. Ş., “Flip and Neimark-Sacker bifurcation in a differential equation with piecewise constant arguments model”, Journal of Difference Equations and Applications, 23(4): 763–778, (2017). Ak Gümüş . Ö., “Generalized stability for a class of nonlinear difference population model”, AKU J. Sci. Eng., 16: 585–591, 031303, (2016).
37. Ak Gümüş . Ö., “Dynamical consequences and stability analysis of a new host-parasitoid model”, General Mathematical Notes, 27(1): 9–15, (2015).
38. Ak Gümüş. Ö., “Global and local stability analysis in a nonlinear discretetime population model”, Advances in Difference Equations, 299: 1687–1847, (2014).
39. Ak Gümüş. Ö., Kangalgil. F., “Dynamics of a host-parasite model connected with immigration”, New Trends in Mathematical Sciences, 5(3): 332–339, (2017).
40. Liu. X., Xiao. D., “Complex dynamic behaviors of a discrete-time predator prey system”, Chaos, Solitons & Fractals, 32:80-94, (2007).