Dynamics of A delayed epidemiological model with nonlinear incidence: The role of infected incidence fraction

Journal of Biological Systems

Volumn 13, Issue 4, 2005, Pages 341-361

  Mukhopadhyay, B ^a   Bhattacharyya, R ^a  

^a UNIVERSITY OF CALCUTTA   (India)

Author keywords

Discrete Time Delay; Distributed Time Delay; Global Stability; Hopf Bifurcation; Infected Incidence Fraction; Nonlinear Incidence; SIS Model

Indexed keywords

EID: 28844486756     PISSN: 02183390     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218339005001562     Document Type: Article

References (37)

1. Hethcote HW, Stech HW, van den Diressche P, Periodicity and stability in epidemic models: a survey, in Busenberg SN, Cooke KL (eds.), Differential Equations and Applications in Ecology, Epidemics and Population Problems, Academic Press, pp. 65-82, 1981.
2. Aron J, Schwartz I, Seasonality and periodic-doubling bifurcation in an epidemic model, J Theor Biol 110:665-679, 1984.
3. London WP, Yorke JA, Recurrent outbreak of measles, chickenpox and mumps: I, Seasonal variation in contact rates, Am J Epidemiology 98:458-468, 1973.
4. Schwartz IB, Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models, J Math Biol 21:347-362, 1985.
5. Hethcote HW, Stech HW, van den Diressche P, Non-linear oscillations in epidemic models, SIAM J Appl Math 40:1-9, 1981.
6. Schenzle D, An age-structure model of pre- and post-vaccination measles transmission, IMA J Math Appl Med Biol 1:169-191, 1984.
7. Liu WM, Levin SA, Iwasa, Y, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J Math Biol 23:187-204, 1984.
8. Liu WM, Hethcote HW, Levin SA, Dynamical behavior of epidemiological models with nonlinear incidence rates, J Math Biol 25:359-380, 1987.
9. Hethcote HW, The mathematics of infectious diseases, SIAM Rev 42:599-653, 2000.
10. Wilson EB, Worcester J, The law of mass action in epidemiology, Proc NAS 31:24-34, 1945.
11. Wilson EB, Worcester J, The law of mass action in epidemiology II, Proc NAS 31: 109-116, 1945.
12. Severo NC, Generalizations of some stochastic epidemic models, Math Biosci 4: 395-402, 1969.
13. Yorke JA, London WP, Recurrent outbreak of measles, chickenpox and mumps: II, Am J Epidemiol 98:469-482, 1973.
14. Capasso V, Serio G, A generalization of the Kermack-McKendrick deterministic epidemic model, Math Biosci 42:41-61, 1978.
15. May RM, Anderson RM, Population biology of infectious diseases: II, Nature 280: 455-461, 1979.
16. Gani J, Some problems in epidemic theory, J R Stat Soc Ser A140:323-347, 1977.
17. Wang FJS, Asymptotic behavior of some deterministic epidemic models, SIAM J Math Anal 9:529-534, 1978.
18. Cunningham J, A deterministic model for measles, Z Naturforsch 34C:647-648, 1978.
19. Gabriel JP, Hanisch H, Hirsch WM, Dynamic equilibria of helminthic infections, in Chapman DG, Gallucci VF (eds.), Quantitative Population Dynamics, International Cooperative Publishing House, Maryland, Stat Ecol Ser 13:83-104, 1981.
20. Lizana M, Rivero J, Multiparametric bifurcations for a model in epidemiology, J Math Biol 35:21-36, 1996.
21. Wu L, Feng Z, Homoclinic bifurcation in an SIQR model for childhood diseases, J Diff Equal 168:150-167, 2000.
22. Derrick WR, van den Driessche P, A disease transmission model in a non-constant population, J Math Biol 31:495-512, 1993.
23. Busenberg S, Cooke KL, Vertically Transmitted Diseases, Springer, Berlin, 1993.
24. Cushing JM, Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Berlin, 1993.
25. Gopalsamy K, Stability and Oscillations in Delay-Differential Equations of Population Dynamics, Kluwer, Dordrechet, 1992.
26. Kuang Y, Delay-Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
27. MacDonald N, Time Delays in Biological Models, Springer, Heidelberg, 1978.
28. Stepan G, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman, UK, 1989.
29. Hethcote HW, Lewis MA, van den Diressche P, An epidemiological model with a delay and a nonlinear incidence rate, J Math Biol 27:49-64, 1989.
30. van den Driessche P, Watmough J, A simple SIS epidemic model with a backward bifurcation, J Math Biol 40:525-540, 2000.
31. Hadeler K, van den Driessche P, Backward bifurcation in epidemic models, Math Biosci 146:15-35, 1997.
32. Dushoff J, Huang W, Castillo-Chavez C, Backward bifurcations and catastrophe in simple models of fatal diseases, J Math Biol 36:227-248, 1998.
33. Busenberg S, Travis C, On the use of reducible functional differential equations in biological models, J Math Anal Appl 89:46-66, 1992.
34. Murray JD, Mathematical Biology, Springer, Berlin, 2002.
35. THE MATHWORKS Inc., MATLAB User's Guide, Natick, Massachusetts, 2000.
36. Hale HK, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
37. Hassard BD, Kazarinoff ND, Wan YN, Theory and Applications of Hopf-Bifurcation, Cambridge University Press, 1981.