Abstract and Applied Analysis

Bifurcation Analysis of an SIR Epidemic Model with the Contact Transmission Function

Guihua Li and Gaofeng Li

Full-text: Open access

Abstract

We consider an SIR endemic model in which the contact transmission function is related to the number of infected population. By theoretical analysis, it is shown that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. Furthermore, we find that the threshold value of disease spreading will be increased, when the half-saturation coefficient is more than zero, which means that it is an effective intervention policy adopted for disease spreading. However, when the endemic equilibria exist, we find that the disease can be controlled as long as we let the initial values lie in the certain range by intervention policy. This will provide a theoretical basis for the prevention and control of disease.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 930541, 7 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858479

Digital Object Identifier
doi:10.1155/2014/930541

Mathematical Reviews number (MathSciNet)
MR3166666

Citation

Li, Guihua; Li, Gaofeng. Bifurcation Analysis of an SIR Epidemic Model with the Contact Transmission Function. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 930541, 7 pages. doi:10.1155/2014/930541. https://projecteuclid.org/euclid.aaa/1395858479


Export citation

References

  • V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1-2, pp. 43–61, 1978.
  • S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135–163, 2003.
  • Y. Tang, D. Huang, S. Ruan, and W. Zhang, “Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate,” SIAM Journal on Applied Mathematics, vol. 69, no. 2, pp. 621–639, 2008.
  • D. Xiao and H. Zhu, “Multiple focus and Hopf bifurcations in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 802–819, 2006.
  • W. Wang, “Epidemic models with nonlinear infection forces,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 267–279, 2006.
  • D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419–429, 2007.
  • A. B. Gumel and S. M. Moghadas, “A qualitative study of a vaccination model with non-linear incidence,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 409–419, 2003.
  • W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–380, 1987.
  • W. M. Liu, S. A. Levin, and Y. Iwasa, “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,” Journal of Mathematical Biology, vol. 23, no. 2, pp. 187–204, 1986.
  • R. R. Regoes, D. Ebert, and S. Bonhoeffer, “Dose-dependent infection rates of parasites produce the Allee effect in epidemiology,” Proceedings of the Royal Society B, vol. 269, no. 1488, pp. 271–279, 2002.
  • G. Li and W. Wang, “Bifurcation analysis of an epidemic model with nonlinear incidence,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 411–423, 2009.
  • R. M. Anderson and R. M. May, “Population biology of infectious diseases: Part I,” Nature, vol. 280, no. 5721, pp. 361–367, 1979.
  • R. M. May and R. M. Anderson, “Population biology of infectious diseases: Part II,” Nature, vol. 280, no. 5722, pp. 455–461, 1979.
  • L. Perko, Differential Equations and Dynamical Systems, vol. 7, Springer, New York, NY, USA, 2nd edition, 1996.
  • R. Bogdanov, “Bifurcations of a limit cycle for a family of vector fields on the plan,” Selecta Mathematica, vol. 1, pp. 373–388, 1981.
  • R. Bogdanov, “Versal deformations of a singular point on the plan in the case of zero eigenvalues,” Selecta Mathematica, vol. 1, pp. 389–421, 1981.
  • F. Takens, “Forced oscillations and bifurcations,” in Applications of Global Analysis I, pp. 1–59, Rijksuniversitat Utrecht, 1974.
  • A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, “Limit cycles and theirčommentComment on ref. [18?]: Please note that the URL is not accessible. bifucations in MatCont,” http://www.matcont.ugent.be/ Tutorial2.pdf. \endinput