Bifurcations in an epidemic model with constant removal rate of the infectives

Journal of Mathematical Analysis and Applications

Volumn 291, Issue 2, 2004, Pages 775-793

  Wang, Wendi ^a   Ruan, Shigui ^b  

^a SOUTHWEST UNIVERSITY   (China)

^b Department of Biology   (United States)

Author keywords

Bifurcation; Constant removal rate; Epidemic; Global analysis; Limit cycle

Indexed keywords

EID: 1542406443     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmaa.2003.11.043     Document Type: Article

References (22)

1. R.M. Anderson, R.M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, 1991.
2. R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Math. Soviet. 1 (1981) 373-388.
3. R. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigen-values, Selecta Math. Soviet. 1 (1981) 389-421.
4. F. Brauer, P. van den Driessche, Models for the transmission of disease with immigration of infectives, Math. Biosci. 171 (2001) 143-154.
5. O. Diekmann, M. Kreztschmar, Patterns in the effects of infectious disease on population growth, J. Math. Biol. 29 (1991) 539-570.
6. D.J.D. Earn, P. Rohani, B.M. Bolker, B.T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science 287 (2000) 667-670.
7. Z. Feng, H.R. Thieme, Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. Biosci. 128 (1995) 93-130.
8. H.W. Hethcote, S.A. Levin, Periodicity in epidemiological models, in: S.A. Levin, T.G. Hallam, L.J. Gross (Eds.), Applied Mathematical Biology, in: Biomath. Texts, vol. 18, Springer-Verlag, New York, 1989, pp. 193-211.
9. H.W. Hethcote, H.W. Stech, P. van den Driessche, Periodicity and stability in epidemic models: a survey, in: S. Busenberg, K.L. Cooke (Eds.), Differential Equations and Applications in Ecology, Epidemics and Populations Problems, Academic Press, New York, 1981, pp. 65-82.
10. H.W. Hethcote, H.W. Stech, P. van den Driessche, Nonlinear oscillations in epidemic models, SIAM J. Appl. Math. 40 (1981) 1-9.
11. H.W. Hethcote, P. van den Driessche, Two SIS epidemiologic models with delays, J. Math. Biol. 40 (2000) 3-26.
12. J.M. Hyman, J. Li, Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. Math. 58 (1998) 912-925.
13. J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1996.
14. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998.
15. W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987) 359-380.
16. W.M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23 (1986) 187-204.
17. M. Lizana, J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol. 35 (1996) 21-36.
18. S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations 188 (2003) 135-163.
19. F. Takens, Forced oscillations and bifurcation, in: Applications of Global Analysis I, in: Comm. Math. Inst. Rijksuniversitat Utrecht, vol. 3, 1974, pp. 1-59.
20. W. Wang, Z. Ma, Global dynamics of an epidemic model with time delay, Nonlinear Anal. RWA 3 (2002) 365-373.
21. L. Wu, Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differential Equations 168 (2000) 150-167.
22. Y.Q. Ye, et al., Theory of Limit Cycles, in: Trans. Math. Monogr., vol. 66, AMS, Providence, 1986.