Rich dynamics of an SIR epidemic model

Nonlinear Analysis: Modelling and Control

Volumn 15, Issue 1, 2010, Pages 71-81

  Pathak, S ^a   Maiti, A ^b   Samanta, G P ^c  

^a Belur Girls' High School (HS)   (India)

^b PRESIDENCY UNIVERSITY   (India)

^c INDIAN INSTITUTE OF ENGINEERING SCIENCE AND TECHNOLOGY   (India)

Author keywords

Basic reproduction number; Disease free equilibrium; Endemic equilibrium; SIR model; Stability; Transmission function

Indexed keywords

EID: 77949346441     PISSN: 13925113     EISSN: 23358963     Source Type: Journal    
DOI: 10.15388/na.2010.15.1.14365     Document Type: Article

References (33)

1. W.O. Kermack, A.G. McKendrick, Contribution to mathematical theory of epidemics, P. Roy. Soc. Lond. A Mat., 115, pp. 700-721, 1927.
2. N.T.J. Bailey, The Mathematical Theory of Infectious Diseases, Griffin, London, 1975.
3. J.D. Murray, Mathematical Biology, Springer-Verlag, New York, 1993.
4. R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1998.
5. V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42, pp. 43-61, 1978.
6. H.W. Hethcote, D.W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9, pp. 37-47, 1980.
7. W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, pp.359-380, 1987.
8. W. M. Liu, S. A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23, pp. 187-204, 1986.
9. H.W. Hethcote, M.A. Lewis, P. van den Driessche, An epidemiological model with delay and a nonlinear incidence rate, J. Math. Biol., 27, pp. 49-64, 1989.
10. H.W. Hethcote, P. van den Driessche, Some epidemiological model with nonlinear incidence, J. Math. Biol., 29, pp. 271-287, 1991.
11. W.R. Derrick, P. van den Driessche, A disease transmission model in nonconstant population, J. Math. Biol., 31, pp. 495-512, 1993.
12. E. Beretta, Y. Takeuchi, Global stability of a SIR epidemic model with time delay, J. Math. Biol., 33, pp.250-260, 1995.
13. E. Beretta, Y. Takeuchi, Convergence results in SIR epidemic model with varying population sizes, Nonlinear Anal., 28, pp. 1909-1921, 1997.
14. E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability of a SIR epidemic model with distributed time delay, Nonlinear Anal., 47, pp. 4107-4115, 2001.
15. W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time-delay, Appl. Math. Lett., 17, pp. 1141-1145, 2004.
16. W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54, pp. 581-591, 2002.
17. S. Ruan, W. Wang, Dynamical behaviour of an epidemic model with nonlinear incidence rate, J. Differ. Equations, 188, pp.135-163, 2003.
18. M. Song, W. Ma, Asymptotic properties of a revised SIR epidemic model with density dependent birth rate and tie delay, Dynamic of Continuous, Discrete and Impulsive Systems, 13, pp. 199-208, 2006.
19. M. Song, W. Ma, Y. Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comput. Appl. Math., 201, pp. 389-394, 2007.
20. A. D'Onofrio, P. Manfredi, E. Salinelli, Bifurcation thresholds in an SIR model with information-dependent vaccination, Mathematical Modelling of Natural Phenomena, 2, pp. 23-38, 2007.
21. D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208, pp. 419-429, 2007.
22. H. McCallum, N. Barlow, J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol., 16, pp. 295-300, 2001.
23. S. Busenberg, K. Cooke, Vertically Transmitted Disease-Model and Dynamics, Springer-Verlag, New York, 1993.
24. A.B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M.E. Alexander, S. Ardal, J. Wu, B.M. Sahai, Modelling strategies for controlling SARS outbreaks, P. Roy. Soc. Lond. B Bio., 271, pp. 2223-2232, 2004.
25. W. Wang, S. Ruan, Simulating the SARS outbreak in Beijing with limited data, J. Theor. Biol., 227, pp. 369-379, 2004.
26. F. Brauer, Some simple epidemic models, Math. Biosci. Eng., 3(1), pp.1-15, 2006.
27. R.B. Wallace, Public Health and Preventive Medicine, McGraw-Hill Medical, 2007.
28. Prevention of plague: recommendations of the Advisory Committee on Immunization Practices, Morbidity and Mortality Weekly Report, 45(RR-14), pp.1-15, 1996.
29. O. Diekmann, M. Kretzschmar, Patterns in the effects of infectious diseases on population growth, J. Math. Biol., 29, pp. 539-570, 1991.
30. A. Zegeling, R.E. Kooij, Uniqueness of limit cycles in models for micropaasitic and macroparasitic diseases, J. Math. Biol., 36, pp. 407-417, 1998.
31. L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Vol. 7, Springer-Verlag, New York, 1991.
32. S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Studies in Nonlinearity, Perseus Books Group, Massachusetts, 2001.
33. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, Vol. 2, Springer-Verlag, New York, 2003.