Dynamics of seis epidemic models with varying population size

International Journal of Bifurcation and Chaos

Volumn 17, Issue 5, 2007, Pages 1513-1529

  Hui, Jing ^a,b   Zhu, De Ming ^b  

^a GUANGXI UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b EAST CHINA NORMAL UNIVERSITY   (China)

Author keywords

Compound matrix; Density dependent birth pulse; Global stability; Periodicity; Varying population size

Indexed keywords

CHAOTIC SYSTEMS; DISEASES; MATHEMATICAL MODELS; POPULATION DYNAMICS; POPULATION STATISTICS;

COMPOUND MATRIX; DENSITY DEPENDENT BIRTH PULSE; GLOBAL STABILITY; PERIODICITY; POPULATION SIZE;

EPIDEMIOLOGY;

EID: 34547736424     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127407017902     Document Type: Article

References (46)

1. Anderson, R. M. & May, R. M. [1979] "Population biology of infectious disease I," Nature 180, 361-367.
2. Anderson, R. M. σ May, R. M. [1981] "The population dynamics of microparasites and their invertebrares host," Philos. Trans. R. Soc. Lond. B 291, 451-524.
3. Anderson, R. M., May, R. M. & Mclean, A. R. [1988] "Possible demographic consequences of AIDS in developing countries," Nature 332, 228-234.
4. Anderson, R. M. [1989] "Mathematical and statistical studies of the epidemiology of HIV," J. AIDS 3, 333-346.
5. Anderson, R. M. & May, R. M. [1992] Infectious Disease of Humans, Dynamics and Control (Oxford University Press, Oxford).
6. Aron, J. L. & Schwartz, I. B. [1984] "Seasonality and period-doubling biburcations in an epidemic model," J. Theor. Biol. 110, 665-679.
7. Bainov, D. & Simeonov, P. [1989] System with Impulsive Effect: Stability, Theory and Applications (John Wiley, NY).
8. Bainov, D. & Simeonov, P. [1993] Impulsive Differential Equations: Periodic Solutions and Applications (Longman Scientific & Technical, England).
9. Bellenir, K. & Dresser, P. [1996] Contagious and Non-Contagious Infectious Disease Sourcebook, Health Science Series, Vol. 8 (Omnigraphics Inc., Detroit).
10. Brauer, F. [1990] "Models for the spread of universally fatal diseases," J. Math. Biol. 28, 451-462.
11. Busenberg, S. N. & Hadeler, K. P. [1990] "Demography and epidemics," Math. Biosci. 101, 41-62.
12. Busenberg, S. & Cooke, K. [1993] Vertically Transmitted Diseases, Models and Dynamics, Biomathematics, Vol. 23 (Springer-Verlag, Berlin).
13. Butler, G., Freedman, H. I. & Waltman, P. [1986] "Uniformly persistent systems," Proc. Amer. Math. Soc. 96, 425-430.
14. Butler, G. J. & Waltman, P. [1986] "Persistence in dynamical systems," J. Diff. Eqs. 63, 255-163.
15. Castillo-Chavez, C. C., Cooke, K. L., Huang, L. & Levin, S. A. [1989] "On the role of long periods of infectiousness in the dynamics of AIDS, part 1, single population models," J. Math. Biol. 27, 373-398.
16. Goppel, W. A. [1965] Stability and Asymptotic Behavior of Differential Equations (Heath, Boston).
17. De Jong, M. C. M., Diekmann, O. & Heesterbeek, J. A. P. [1995] How Does Transmission Depend on Population Size?, in Human Infectious Diseases, Proc. March 1993 Conf. Isaac Newton Institute for Mathematical Sciences, eds. Isham, V. & Medley, G. (Cambridge University Press, UK).
18. Fan, M., Li, M. Y. & Wang, K. [2001] "Global stability of an SEIS epidemic model with recruitment and a varying total population size," Math. Biosci. 170, 199-208.
19. Freedman, H. I., Ruan, S. G. & Tang, M. X. [1994] "Uniform persistence and flows near a closed positively invariant set," J. Dyn. Diff. Eqs. 6, 583-600.
20. Gao, L. Q. & Hethcote, H. W. [1992] "Disease transmission models with density-dependent demographics," J. Math. Biol. 30, 717-731.
21. Gao, L. Q., Mena-Lorca, J. & Hethcote, H. W. [1995] "Four SEI endemic models with periodicity and separatrices," Math. Biosci. 128, 157-184.
22. Greenhalgh, D. & Das, R. [1995] "Modelling epidemics with variable contact rates," Theoret. Popul. Biol. 47, 129-179.
23. Hale, J. K. [1969] Ordinary Differential Equations (Wiley, NY).
24. Hethcote, H. W. [1976] "Qualitative analyses of communicable disease models," Math. Biosci. 28, 335-356.
25. Hethcote, H. W. & Levin, S. A. [1989] Periodicity in Epidemiological Models, in Applied Mathematical Ecology, eds. Gross, I. & Levin, S. A. (Springer, NY).
26. Hyman, J. M. & Stanley, E. A. [1988] "Using mathematical models to understand the AIDS epidemic," Math. Biosci. 90, 415-473.
27. Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L. & Perry, T. [1988] "Modelling and analyzing HIV transmission: The effect of contact patterns," Math. Biosci. 92, 119-199.
28. LaSalle, J. P. [1976] The Stability of Dynamical Systems, Regional Conf. Series in Applied Mathematics (SIAM, Philadelphia).
29. Li, M. Y. & Muldowney, J. S. [1993] "On Bendixson's criterion," J. Diff. Eqs. 106, 27-39.
30. Li, M. Y. & Muldowney, J. S. [1996] "A geometric approach to global-stability problems," SIAM J. Math. Anal. 27, 1070-1083.
31. Li, M. Y., Graef, J. R., Wang, L. & Karsai, J. [1999] "Global dynamics of a SEIR model with varying total population size," Math. Biosci. 160, 191-213.
32. Li, M. Y., Smith, H. L. & Wang, L. [2001] "Global dynamics of an SEIR epidemic model with vertical transmission," SIAM J. Appl. Math. 62, 58-69.
33. Lund, O., Mosekilde, E. & Hansen, J. [1993] "Period-doubling route to chaos in a model of HIV infection of the immune system," Simul. Prac. Th. 1, 49-55.
34. May, R. M. [1976] "Simple mathematical models with very complicated dynamics," Nature (London) 261, 459-467.
35. Mena-Lorca, J. & Hethcote, H. W. [1992] "Dynamic models of infectious diseases as regulations of population sizes," J. Math. Biol. 30, 693-716.
36. Olsen, L. F., Truty, G. L. & Schaffer, W. M. [1988] "Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark," Theoret. Popul. Biol. 33, 344-370.
37. Pugh, C. C. [1967] "The closing lemma," Amer. J. Math. 89, 956-1009.
38. 1 closing lemma including Hamiltonians," Erg. Th. Dyn. Syst. 3, 261-313.
39. Pugliese, A. [1990] "Population models for disease with no recovery," J. Math. Biol. 28, 65-82.
40. Rössler, O. E. [1976] "Chaotic behaviour in simple reaction systems," Z. Natiirforsch 31a, 259-264.
41. Schreiber, S. J. [2001] "Chaos and population disappearances in simple ecological models," J. Math. Biol. 42, 239-260.
42. Smith, R. A. [1986] "Some applications of Hausdorff dimension inequalities for ordinary differential equations," Proc. Roy. Soc. Edinb. Sect. A 104, 235-259.
43. Thieme, H. R. [1992] "Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations," Math. Biosci. 111, 99-130.
44. Waltman, P. [1991] "A brief survey of persistence in dynamical systems," in Delay Differential Equations and Dynamical Systems, eds. Busenberg, S. & Martelli, M. (Springer-Verlag, NY), pp. 31-40.
45. Zhou, J. [1994] "An epidemiological model with population size dependent incidence," Rocky Mt. J. Math. 24, 429-445.
46. Zhou, J. & Hethcote, H. W. [1994] "Population size dependent incidence in models for disease without immunity," J. Math. Biol. 32, 809-834.