Differential Equations and Dynamical Systems

, Volume 17, Issue 4, pp 393–417 | Cite as

Bifurcation and chaos in S-I-S epidemic model

  • Prasenjit Das
  • Zindoga Mukandavire
  • Christinah Chiyaka
  • Ayan Sen
  • Debasis Mukherjee
Original Research


We present a Susceptible-Infective-Susceptible (S-I-S) model with two distinct discrete time delays representing a period of temporary immunity of newborns and a disease incubation period with randomly fluctuating environment. The stability of the equilibria is robustly investigated for the case with and without delay. Conditions for supercritical and subcritical Hopf bifurcation are derived. Comprehensive numerical simulations show that adding delay to an epidemic model could change the asymptotic stability of the system, altering the location of (stable or unstable) endemic equilibrium, or even leading to chaotic behavior. Further, simulation results illustrate that, in some cases where the disease becomes endemic in the model system without delay, addition of delays for temporary immunity and incubation period facilitates smaller final infective population sizes, even if endemicity is still maintained. Effects of randomness of the environment in terms of white noise are thoroughly investigated jointly with delay. The results demonstrate that there are no significant differences in dynamical behaviour of the system when considering delay solely or jointly with stochasticity.


SIS model Delay Equilibria Stability Stochasticity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Busenberg S., Cooke K. L. and Pozio M. A., Analysis of a Model of a Vertically Transmitted Disease, Journal of Mathematical Biology, 17, 305–329, (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Franke J. E. and Yakubu A. A., Discrete-time SIS epidemic model in a seasonal environment, SIAM Journal on Applied Mathematics, 66(5), 1563–1587, (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hassard B. D., Kazarinoff N. D. and Wan Y. H., Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge, England, (1981)zbMATHGoogle Scholar
  4. 4.
    Hethcote H. W., Wang W., Han L. and Ma Z., A predator-prey model with infected prey, Theor. Popul. Biol., 66, 259–268, (2004)CrossRefGoogle Scholar
  5. 5.
    Hofbauer J. and Sigmund K., The Theory of Evolution and Dynamical System. Cambridge University Press, Cambridge, (1998)Google Scholar
  6. 6.
    Hsieh Y. H. and Hsiao C. K., Predator-prey Model with Disease Infection in Both Populations, Mathematical Medicine and Biology, 25(3), 247–266, (2008)CrossRefGoogle Scholar
  7. 7.
    Marsden J. E. and McCracken M., The Hopf bifurcation and its Applications. Springer-Verlag, New York, (1976)zbMATHGoogle Scholar
  8. 8.
    Nisbet R. M. and Gurney W. S. C., Modelling Fluctuating Populations, Wiley-Interscience, (1982)Google Scholar
  9. 9.
    Woods C. R., Syphilis in children: congenital and Acquired. Seminars in Pediatric Infectious Diseases, 16, 245–257, (2005)CrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2009

Authors and Affiliations

  • Prasenjit Das
    • 1
  • Zindoga Mukandavire
    • 2
  • Christinah Chiyaka
    • 2
  • Ayan Sen
    • 3
  • Debasis Mukherjee
    • 4
  1. 1.The Kidderpore AcademyKolkataIndia
  2. 2.Department of Public Health and Institute of BiostatisticsChina Medical UniversityTaichungTaiwan
  3. 3.Sarat Chandra Sur InstitutionKolkataIndia
  4. 4.Department of MathematicsVivekananda CollegeThakurpukur, KolkataIndia

Personalised recommendations