Exciting chaos with noise: Unexpected dynamics in epidemic outbreaks

Journal of Mathematical Biology

Volumn 44, Issue 1, 2002, Pages 31-48

  Billings, L ^a   Schwartz, I B ^b  

^a MONTCLAIR STATE UNIVERSITY   (United States)

^b NAVAL RESEARCH LABORATORY   (United States)

Author keywords

Bi instability; Chaos; Noise; Population models

Indexed keywords

ARTICLE; BIOLOGICAL MODEL; CHILD; EPIDEMIC; HUMAN; MEASLES; NONLINEAR SYSTEM; PRESCHOOL CHILD;

CHILD; CHILD, PRESCHOOL; DISEASE OUTBREAKS; HUMANS; MEASLES; MODELS, BIOLOGICAL; NONLINEAR DYNAMICS;

EID: 0036369855     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s002850100110     Document Type: Article

References (23)

1. Alligood, K.T., Sauer, T.D., Yorke, J.A., Chaos. New York: Springer-Verlag, (1997)
2. Billings, L., Schwartz, I.B., Intermittency vs. stochastic chaos: noise dominated dynamics, (in preparation).
3. Blarer, A., Doebeli, M., Resonance effects and outbreaks in ecological time series, Ecol. Lett. 2, 167-177 (1999)
4. Bolker, B.M., Chaos and complexity in measles models - a comparative numerical study, IMA J. Math. Appl. Med., 10, 83-95 (1993)
5. Bolker, B.M., Grenfell, B.T., Chaos and biological complexity in measles dynamics, P. Roy. Soc. Lond. B Bio., 251, 75-81 (1993)
6. Broomhead, D., Huke, J., Muldoon, M., Delay embedding of iterated function systems, in preparation.
7. Carr, T.W., Billings, L., Schwartz, I.B., Triandaf, I., Bi-instability and the global role of unstable resonant orbits in a driven laser, Physica D., 147, 59-82 (2000)
8. Crutchfield, J.P., Farmer, J.D., Huberman, B.A., Fluctuations and simple chaotic dynamics. Phys. Rep. 92, 45-82 (1982)
9. Crutchfield, J.P., Huberman, B.A., Fluctuations and the onset of chaos, Phys. Lett. A., 77, 407-410 (1980)
10. Earn, D.J.D., Rohani, P., Bolker, B.M., Grenfell, B., A simple model for complex dynamical transitions in epidemics. Science., 287, 667-670 (2000)
11. Gao, J.B., Hwang, S.K., Liu, J.M., When can noise induce chaos?, Phys. Rev. Lett., 82, 1132-1135 (1999)
12. Glendinning, P., Perry, L.P., Melnikov analysis of chaos in a simple epidemiological model, J. Math. Biol., 35, 359-373 (1997)
13. Grenfell, B.T., Bolker, B.M., Kleczkowski, A., Seasonality and extinction in chaotic metapopulations, P. Roy. Soc. Lond. B Bio., 259, 97-103 (1995)
14. Lloyd, A.L., May, R.M., Spatial heterogeneity in epidemic models, J. Theor. Biol., 179, 1-11 (1996)
15. Rand, D. A., Wilson, H.B., Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics, Proc. R. Soc. Lond. B., 246, 179-184 (1991)
16. Schaffer, W.M., Kendall, B.E., Tidd, C.W., Olsen, L.F., Transient periodicity and episodic predictability in biological dynamics, IMA J. Math. Appl. Med., 10, 227-247 (1993)
17. Schwartz, I.B., Multiple stable recurrent outbreaks an predictability in seasonally forced nonlinear epidemic models, J. Math. Biol., 21, 347-361 (1985)
18. Schwartz, I.B., Small amplitude, long period outbreaks in seasonally driven epidemics, J. Math. Biol., 30, 473-491 (1992)
19. Schwartz, I.B., Carr, T.W., Bi-instability as a precursor to global mixed-mode chaos, Phys. Rev. E., 59, 6658-6661 (1999)
20. Schwartz, I.B., Smith, H.L., Infinite subharmonic bifurcations in an SEIR epidemic model, S. Math. Biol., 18, 233-253 (1983)
21. Smale, S., Differentiate dynamical systems, Bull. Amer. Math. Soc., 73, 747-817 (1967)
22. Tidd, C.W., Olsen, L.F., Schaffer, W.M., The case for chaos in childhood epidemics 2. Predicting historical epidemics from mathematical models, P. Roy. Soc. Lond. B Bio., 254, 257-23 (1993)
23. Zyczkowski, K., Bollt, E.M., On the entropy devil's staircase in a family of gap-tent maps, Physica D., 132, 392-410 (1999)