Periodicity in an epidemic model with a generalized non-linear incidence

Mathematical Biosciences

Volumn 189, Issue 1, 2004, Pages 75-96

  Alexander, M E ^a   Moghadas, S M ^a  

^a NATIONAL RESEARCH COUNCIL CANADA   (Canada)

Author keywords

Bistability; Epidemic models; Hopf bifurcation; Non linear incidence; Poincar index

Indexed keywords

BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; LYAPUNOV METHODS; MATHEMATICAL MODELS; NONLINEAR SYSTEMS; PATIENT TREATMENT;

INFECTIOUS DISEASES; LYAPUNOV COEFFICIENTS;

TUMORS;

DISEASE INCIDENCE; EPIDEMIOLOGY; MODEL;

ARTICLE; DISEASE PREDISPOSITION; DYNAMICS; EPIDEMIC; GENERAL CONDITION; HIGH RISK POPULATION; INCIDENCE; INFECTION SENSITIVITY; MATHEMATICAL PARAMETERS; NONLINEAR SYSTEM; PERIODICITY; SIMIAN IMMUNODEFICIENCY VIRUS; STATISTICAL MODEL; VACCINATION; VIRUS INFECTION;

ALGORITHMS; ANIMALS; COMPUTER SIMULATION; DISEASE OUTBREAKS; INCIDENCE; NONLINEAR DYNAMICS; PERIODICITY; SIMIAN ACQUIRED IMMUNODEFICIENCY SYNDROME; VACCINATION;

SIMIAN IMMUNODEFICIENCY VIRUS;

EID: 1642341634     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2004.01.003     Document Type: Article

References (34)

1. Anderson R.M., May R.M. Infectious Diseases of Humans. 1991;Oxford University, London/New York.
2. Bolker B.M., Grenfell B.T. Chaos and biological complexity in measles dynamics. Philos. Trans. Roy. Soc. Lond. B. 251:1993;75.
3. Capasso V. Mathematical Structure of Epidemic Systems. Lecture Notes in Biomathematics. vol. 97:1993;Springer, Berlin.
4. Capasso V., Serio G. A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42:1978;43.
5. Earn D.J.D., Rohani P., Bolker B.M., Grenfell B.T. A simple model for complex dynamical transitions epidemics. Science. 287:2000;667.
6. Glendinning P. Stability, Instability and Chaos. 1994;Cambridge University, NY.
7. Grenfell B.T., Keczkowski A., Gilligan C.A., Bolker B.M. Spatial heterogeneity, nonlinear dynamics and chaos in infectious diseases. Stat. Meth. Med. Res. 4:1995;160.
8. Greenhalgh D., Doyle M., Lewis F. A mathematical treatment of AIDS and condom use. IMA J. Math. Appl. Med. Biol. 18(3):2001;225.
9. Gumel A.B., Moghadas S.M. A qualitative study of a vaccination model with nonlinear incidence. Appl. Math. Comput. 143:2003;409.
10. Hadeler K.P., Castillo-Chavez C. A core group model for disease transmission. Math. Biosci. 128:1994;41.
11. Hadeler K.P., van den Driessche P. Backward bifurcation in epidemic control. Math. Biosci. 146:1997;15.
12. Hethcote H.W., Lewis M.A., van den Driessche P. Some epidemiological models with nonlinear incidence. J. Math. Biol. 29:1991;271.
13. Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and Applications of Hopf Bifurcation. Lecture Note Series. vol. 41:1981;Cambridge University.
14. Ivlev V.S. Experimental Ecology of the Feeding of Fishes. 1961;Yale University, New Haven, CT.
15. Janaszek W., Gay N.J., Gut W. Measles vaccine efficacy during an epidemic in 1998 in the highly vaccinated population of Poland. Vaccine. 21:2003;473.
16. Jansen H., Twizell E.H. An unconditionally convergent discretization of the SEIR model. Math. Comput. Simulation. 58:2002;147.
17. Keeling M.J., Rohani P., Grenfell B.T. Seasonally forced disease dynamics explored as switching between attractors. Physica D. 148:2001;317.
18. Kribs-Zaleta C.M. Center manifold and normal forms in epidemic models. IMA. 125:2002;269.
19. Kribs-Zaleta C.M., Velasco-Hernández J.X. A simple vaccination model with multiple endemic states. Math. Biosci. 164:2000;183.
20. Kuznetsov Y.A. Elements of Applied Bifurcation Theory. AMS. vol. 112:1995;Springer.
21. Levin S.A., Hallam T.G., Gross L.J. Applied Mathematical Ecology. 1989;Springer, New York.
22. Lin J., Andreasen V., Levin S.A. Dynamics of influenza A drift: the linear three-strain model. Math. Biosci. 162:1999;33.
23. Liu W.M., Hethcote H.W., Levin S.A. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25:1987;359.
24. Liu W.M., Levin S.A., Iwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23:1986;187.
25. Moghadas S.M. Analysis of an epidemic model with bistable equilibria using the Poincaré index. Appl. Math. Comput. 149(3):2004;689.
26. S.M. Moghadas, Modelling the effect of imperfect vaccines on disease epidemiology, Discr. Contin. Dyn. Syst. Ser. B, in press
27. Moghadas S.M., Gumel A.B. Global stability of a two-stage epidemic model with generalized nonlinear incidence. Math. Comput. Simulation. 60:2002;107.
28. Perko L. Differential Equations and Dynamical Systems. 1996;Springer, New York.
29. Rohani P., Keeling M.J., Grenfell B.T. The interplay between determinism and stochasticity in childhood diseases. Am. Nat. 159(4):2002;469.
30. Ruan S., Wang W. Dynamical behavior of an epidemic model with a nonlinear incidence rate. J. Diff. Equ. 188:2003;135.
31. Scherer A., McLean A.R. Mathematical models of vaccination. Brit. Med. Bull. 62:2002;187.
32. van den Driessche P., Watmough J. A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40:2000;525.
33. Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. 1990;Springer, New York.
34. Yorke J.A., London W.P. Recurrent outbreaks of measles, chickenpox and mumps II. Am. J. Epidemiol. 98:1973;469.