The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

DYNAMICS OF AN IMPROVED SIS EPIDEMIC MODEL

  • Received : 2023.02.07
  • Accepted : 2023.04.25
  • Published : 2023.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

A new modification of the SIS epidemic model incorporating the adaptive host behavior is proposed. Unlike the common situation in most epidemic models, this system has two disease-free equilibrium points, and we were able to prove that as the basic reproduction number approaches the threshold of 1, these two points merge and a Bogdanov-Takens bifurcation of codimension three occurs. The occurrence of this bifurcation is a sign of the complexity of the dynamics of the system near the value 1 of basic reproduction number. Both local and global stability of disease-free and endemic equilibrium point are studied.

Keywords

References

  1. F. Arrigoni & A. Pugliese: Limits of a multipatch SIS epidemic model. J. Math. Biol. 45 (2002), 419-440. doi:10.1007/s002850200155
  2. S.M. Baer, et. al.: Multiparametric bifurcation analysis of a basic two-stage population model. SIAM J. Appl. Math. 66 (2006), 1339-1365. doi:10.1137/050627757
  3. S. Bentout, et. al.: Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior. International Journal of Modeling, Simulation, and Scientific Computing. doi:10.1142/s1793962321500239
  4. R.I. Bogdanov: Versal deformations of a singular point on the plane in the case of zero eigenvalues. Func. Anal. Appl. 9 (1975), 144-145. doi:10.1007/bf01075453
  5. F. Brauer & C. Castillo-Chavez: Mathematical Models in Population Biology and Epidemiology: Second Edition. Springer, New York, 2012. doi:10.1007/978-1-4164-1686-9
  6. C. Castillo-Chavez & H.R. Thieme: Asymptotically autonomous epidemic models. In: Mathematical Population Dynamics, Analysis of Heterogeneity, Vol. 1. Theory of Epidemics, 1995, 33-50. doi:10.1016/0022-247x(87)90211-3
  7. J. Cui, et. al.: The impact of media on the control of infectious diseases. Journal of Dynamics and Differential Equations 20 (2008), 31-53. doi:10.1007/s10884-007-9075-0
  8. A. D'Onfrio & P. Manfredi: Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious disease. J. Theor. Biol. 256 (2009), 473-478. doi:10.1016/j.jtbi.2008.10.005
  9. B. Dubeya, et. al.: Role of media and treatment on an SIR model. Nonlinear Analysis, Modeling and Control 21 (2016), no. 2, 185-200. doi:10.15388/na.2016.2.3
  10. J. Guckenheimer & P.J. Holmes: Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, New York, 1983. doi:10.1007/978-1-4612-1140-2
  11. H.W. Hethcote & J.A. Yorke: Gonorrhea: transmission dynamics and control. volume 56 of Lecture Notes in Biomathematics. Springer-Verlag, Berlin, 1984. doi:10.1007/978-3-662-07544-9
  12. J. Joo & J.L. Lebowitz: Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation. Phys. Rev. E 69 (2004), 066105. doi:10.1103/physreve.69.066105
  13. M.J. Keeling & P. Rohani: Modeling Infectious Diseases In Humans and Animals. Princeton Univ. Press, New Jersey, 2008. doi:10.1515/9781400841035
  14. I.Z. Kiss, et. al.: The impact of information transmission on epidemic outbreaks. Math. Biosciences 225 (2010), 1-10. doi:10.1016/j.mbs.2009.11.009
  15. Y.A. Kuznetsov, Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcation. International J. Bif. Chaos. 15(11) (2005),3535-3546. doi:10.1142/s0218127405014209
  16. Y.A. Kuznetsov: Elements of applied bifurcation theory. Third Edition, Springer, NewYork, 2004. doi:10.1007/978-1-4757-3978-7
  17. W.M. Liu, et. al.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23 (1986), 187-204. doi:10.1007/bf00276956
  18. M. Lizana & J. Rivero: Multiparametric bifurcations for a model in epidemiology. J. Math. Biol. 35 (1996), 21-36. doi:10.1007/s002850050040
  19. M. Martcheva: An introduction to mathematical epidemiology. Springer, New York, 2015. doi:10.1007/978-1-4899-7612-3
  20. R. Memarbashi & E. Sorouri: Modeling the effect of information transmission on the drug dynamic. Eur. Phys. J. Plus, 135, 54 (2020). doi:10.1140/epjp/s13360-019-00064-5
  21. Z. Mukandavire, et. al.: Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics. Applied Mathematical Modelling 33 (2009), no. 4, 2084-2095. doi:10.1016/j.apm.2008.05.017
  22. R.K. Naji & A.A. Thirthar: Stability and bifurcation of an SIS epidemic model with saturated incidence rate and treatment function. Iranian J. Math. Sci. Inf. 15 (2020), no. 2, 129-146. doi:10.1016/j.amc.2013.10.020
  23. F. Takens: Forced oscillations and bifurcations. Comm. Math. Inst. Rijkoniversiteit. 2 (1974), 1-111. doi:10.1887/0750308036/b1058c1
  24. Y. Takeuchi, X. Liu & J. Cui: Global dynamics of SIS models with transport-related infection. J. Math. Anal. Appl. 329 (2007), 1460-1471. doi:10.1016/j.jmaa.2006.07.057
  25. P. Van den Driessche & J. Watmough J.: A simple SIS epidemic model with a backward bifurcation. J. Math. Bio. 40, (2000), 525-540. doi:10.1007/s002850000032
  26. P. Van den Driessche & J. Watmough J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 (2002), 29-48. doi:10.1016/s0025-5564(02)00108-6
  27. Y.N. Xiao, et. al.: Dynamics of an infectious disease with media/psychology induced non-smooth incidence. Math. Biosci. Eng. 10 (2013), 445-461. doi:10.3934/mbe.2013.10.445