Noise and nonlinearity in measles epidemics: Combining mechanistic and statistical approaches to population modeling

American Naturalist

Volumn 151, Issue 5, 1998, Pages 425-440

  Ellner, S P ^a   Bailey, B A ^a,d   Bobashev, G V ^a,e   Gallant, A R ^b   Grenfell, B T ^c   Nychka, D W ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

^b UNIVERSITY OF NORTH CAROLINA   (United States)

^c UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^d NATIONAL CENTER FOR ATMOSPHERIC RESEARCH   (United States)

^e JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

Local Lyapunov exponents; Measles; Modeling; Population dynamics; Time series analysis

Indexed keywords

EID: 0031895048     PISSN: 00030147     EISSN: None     Source Type: Journal    
DOI: 10.1086/286130     Document Type: Article

References (49)

1. Abarbanel, H. D. I., R. Brown, and M. B. Kennel. 1991. Variation of Lyapunov exponents on a Strange attractor. Journal of Nonlinear Science 1:175-199.
2. _. 1992. Local Lyapunov exponents computed from observed data. Journal of Nonlinear Science 2: 343-365.
3. Anderson, R. M., and R. M. May. 1991. Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford.
4. Bailey, B. A. 1996. Local Lyapunov exponents: predict- ability depends on where you are. Pages 343-357 in W. Barnett, A. Kirman, and M. Salmond, eds. Nonlinear dynamics in economics: proceedings of the Tenth International Symposium in Economic Theory and Econometrics. Cambridge University Press, Cambridge.
5. Bailey, B. A., S. Ellner, and D. W. Nychka. 1997. Chaos with confidence: asymptotics and applications of local Lyapunov exponents. Pages 115-134 in C. D. Cutler and D. T. Kaplan, eds. Nonlinear dynamics and time series: building a bridge between the natural and statistical sciences. American Mathematical Society, Providence, R.I.
6. Bartlett, M. S. 1957. Measles periodicity and community size. Journal of the Royal Statistical Society A, General 120:48-70.
7. Bobashev, G. 1997. Spatial modeling of measles epidemic dynamics. Ph.D. thesis. North Carolina State University, Raleigh.
8. Bolker, B. M., and B. T. Grenfell. 1993. Chaos and biological complexity in measles dynamics. Proceedings of the Royal Society of London B, Biological Sciences 251:75-81.
9. Chambers, J. M., and T. J. Hastie, eds. 1992. Statistical models in S. Wadsworth & Brooks/Cole, Pacific Grove, Calif.
10. Cheng, B., and H. Tong. 1992. On consistent nonparametric order determination and chaos. Journal of the Royal Statistical Society B, Methodological 54: 427-450.
11. Costantino, R. F., J. M. Gushing, and R. A. Desharnais. 1995. Experimentally induced transitions in the dynamics behavior of insect populations. Nature (London) 375:227-229.
12. Daniels, S., K. Simkiss, and R. H. Smith. 1991. A simple larval diet for population studies on the blowfly Lucilia sericata (Diptera: Calliphoridae). Medical and Veterinary Entomology 5:283-292.
13. Deissler, R. J., and J. D. Farmer. 1992. Deterministic noise amplifiers. Physica D 55:155-165.
14. Dennis, B., R. A. Desharnais, J. M. Gushing, and R. F. Costantino. 1995. Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecological Monographs 65:261-281.
15. Ellner, S. 1991. Detecting low-dimensional chaos in population dynamics data: a critical review. Pages 63-90 in J. Logan and F. Hain, eds. Chaos and insect ecology. Virginia Agricultural Experimental Station, Virginia Polytechnic Institute and State University, Charlottesville.
16. Ellner, S., and P. Turchin. 1995. Chaos in a noisy world: new methods and evidence from time-series analysis. American Naturalist 145:343-375.
17. Ellner S., D. W. Nychka, and A. R. Gallant. 1992. LENNS, a program to estimate the dominant Lyapunov exponent of noisy nonlinear systems from time series data. Institute of Statistics Mimeo Series no. 2235. Statistics Department, North Carolina State University, Raleigh.
18. Ellner, S., A. R. Gallant, and J. Theiler. 1995. Detecting nonlinearity and chaos in epidemic data. Pages 229-247 in D. Mollison, ed. Epidemic models: their structure and relation to data. Cambridge University Press, Cambridge.
19. Ellner, S. P., B. E. Kendall, S. M. Wood, E. McCauley, and C. J. Briggs. 1997. Inferring mechanism from time-series data: delay differential equations. Physica D: Nonlinear Phenomena 110:182-194.
20. Gallant, A. R., and G. Tauchen. 1992. A nonparametric approach to nonlinear time series analysis: estimation and simulation. Pages 77-92 in D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt, and M. S. Taqqu, eds. New directions in time series analysis. Pt. 2. Springer, New York.
21. Grenfell, B. T., A. Kleczkowski, S. Ellner, and B. M. Bolker. 1995a. Non-linear forecasting and chaos in ecology and epidemiology: measles as a case study. Pages 345-371 in H. Tong, ed. Chaos and forecasting: proceedings of the Royal Society Discussion Meeting. World Scientific, Singapore.
22. Grenfell, B., B. M. Bolker, and A. Kleczkowski. 1995b Seasonality and extinction in chaotic metapopulations. Proceedings of the Royal Society of London B, Biological Sciences 259:97-103.
23. _. 1995c. Seasonality, demography and the dynamics of measles in developed countries. Pages 248-268 in D. Mollison, ed. Epidemic models: their structure and relation to data. Cambridge University Press, Cambridge.
24. Gurney, W. S. C., R. M. Nisbet, and J. H. Lawton. 1983. The systematic formulation of tractable single-species population models incorporating age structure. Journal of Animal Ecology 52:479-495.
25. Hanski, I., P. Turchin, E. Korpimäki, and H. Henttonen. 1993. Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos. Nature (London) 364:232-235.
26. Härdie, W. 1990. Applied nonparametric regression. Cambridge University Press, Cambridge.
27. Hastings, A., C. L. Horn, S. Ellner, P. Turchin, and H. C. J. Godfray. 1993. Chaos in ecology: is Mother Nature a strange attractor? Annual Reviews of Ecology and Systematics 24:1-33.
28. Higgins, K., A. Hastings, J. N. Sorvela, and L. W. Botsford. 1997. Stochastic dynamics and deterministic skeletons: population behavior of Dungeness crab. Science (Washington, D.C.) 276:1431-1435.
29. Keeling, M. J., and B. T. Grenfell. 1997. Disease extinction and community size: modeling the persistence of measles. Science (Washington, D.C.) 275:65-67.
30. Kendall, B. E., W. M. Schaffer, L. F. Olsen, C. W. Tidd, and B. L. Jorgensen. 1994. Using chaos to understand biological dynamics. Pages 184-203 in J. Grasman and G. van Straten, eds. Predictability and nonlinear modeling in natural sciences and economics. Kluwer Academic, Dordrecht.
31. Kot, M., W. M. Schaffer, G. L. Truty, D. J. Graser, and L. F. Olsen. 1988. Changing criteria for imposing order. Ecological Modelling 43:75-110.
32. Levin, S. A., B. Grenfell, A. Hastings, and A. S. Perelson. 1997. Mathematical and computational challenges in population biology and ecosystems science. Science (Washington, D.C.) 275:334-343.
33. McCaffrey, D., S. Ellner, A. R. Gallant, and D. Nychka. 1992. Estimating the Lyapunov exponent of a chaotic system with nonparametric regression. Journal of the American Statistical Association 87:682-695.
34. McCauley, E., R. M. Nisbet, A. M. de Roos, W. W. Murdoch, and W. S. C. Gurney. 1996. Structured population models of herbivorous Zooplankton. Ecological Monographs 66:479-501.
35. Nychka, D. W., S. Ellner, A. R. Gallant, and D. McCaffrey. 1992. Finding chaos in noisy systems (with discussion). Journal of the Royal Statistical Society Series B 54:399-426.
36. Nychka, D. W., B. A. Bailey, S. P. Ellner, P. Haaland, and M. O'Connell. 1996. FUNFITS: data analysis and statistical tools for estimating functions, http://www.stat. ncsu.edu/∼nychka.
37. Rand, D. A., and H. Wilson. 1991. Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proceedings of the Royal Society of London B, Biological Sciences 246:179-184.
38. Rhodes, C. J., and R. M. Anderson. 1996. A scaling analysis of measles epidemics in a small population. Philosophical Transactions of the Royal Society of London B, Biological Sciences 351:1679-1688.
39. Schaffer, W. M., L. M. Olsen, G. L. Truty, and S. L. Fulmer. 1990. The case for chaos in childhood epidemics. Pages 138-166 in S. Krasner, ed. The ubiquity of chaos. American Association for the Advancement of Science, Washington, D.C.
40. Schenzle, E. 1984. An age-structured model of pre- and post-vaccination measles transmission. IMA Journal of Mathematics Applied in Medicine and Biology 1:169-191.
41. Simkiss, K., S. Daniels, and R. H. Smith. 1993. Effects of population density and cadmium toxicity on growth and survival of blowflies. Environmental Pollution 81:41-45.
42. StatSci. 1995. S-PLUS guide to statistical and mathematical analysis. Mathsoft, Inc., Seattle, Wash.
43. Sugihara, G., and R. M. May. 1990. Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature (London) 344:734-741.
44. Tidd, C. W., L. F. Olsen, and W. M. Schaffer. 1993. The case for chaos in childhood epidemics. II. Predicting historical epidemics from mathematical models. Proceedings of the Royal Society of London B, Biological Sciences 254:257-273.
45. Turchin, P. 1993. Chaos and stability in rodent population dynamics: evidence from nonlinear time-series analysis. Oikos 68:167-172.
46. Weigend, A. S., and N. A. Gershenfeld, eds. 1994. Time series prediction: forecasting the future and understanding the past. Addison-Wesley, New York.
47. Wolff, R. C. W. 1992. Local Lyapunov exponents: looking closely at chaos. Journal of the Royal Statistical Society B, Methodological 54:353-371.
48. Wood, S. N. 1994. Obtaining birth and mortality patterns from structured population trajectories. Ecological Monographs 64:23-44.
49. Yao, Q. W., and H. Tong. 1995. On prediction and chaos in stochastic systems. Pages 57-86 in H. Tong, ed. Chaos and forecasting: proceedings of the Royal Society Discussion Meeting. World Scientific, Singapore.