×

zbMATH — the first resource for mathematics

Oscillations and chaos in epidemics: A nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark. (English) Zbl 0639.92012
Summary: Using traditional spectral analysis and recently developed non-linear methods, we analyze the incidence of six childhood diseases in Copenhagen, Denmark. In three cases, measles, mumps, rubella, the dynamics suggest low dimensional chaos. Outbreaks of chicken pox, on the other hand, conform to an annual cycle with noise superimposed. The remaining diseases, pertussis and scarlet fever, remain problematic. The real epidemics are compared with the output of a Monte Carlo analog of the SEIR model for childhood infections. For measles, mumps, rubella, and chicken pox, we find substantial agreement between the model simulations and the data.

MSC:
92D25 Population dynamics (general)
65C20 Probabilistic models, generic numerical methods in probability and statistics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, R.M.; May, R.M., Directly transmitted infectious diseases: control by vaccination, Science, 215, 1053-1060, (1982) · Zbl 1225.37099
[2] Anderson, R.M.; May, R.M., Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes, J. hyg, 94, 365-436, (1985)
[3] Aron, J.L.; Schwartz, I.B., Seasonality and period-doubling bifurcations in an epidemic model, J. theor. biol, 110, 665-679, (1984)
[4] Bailey, N.T.J., The mathematical theory of infectious diseases ad its applications, (1975), Griffin London
[5] Bartlett, M.S., Stochastic population models in ecology and epidemiology, (1960), Methuen London · Zbl 0096.13702
[6] Bennetin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.M., Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. part 1: theory, Meccanica, 15, 9-20, (1980) · Zbl 0488.70015
[7] Degn, H., Discrete chaos is reversed random walk, Phys. rev, A26, 711-712, (1982)
[8] ()
[9] Dietz, K., The incidence of infectious diseases under the influence of seasonal fluctuations, (), 1-15 · Zbl 0333.92014
[10] Farmer, J.D.; Ott, E.; Yorke, J.A., The dimension of chaotic attractors, Physica, 7D, 153-180, (1983)
[11] Fine, P.E.M.; Clarkson, J.A., Measles in england and wales. I. an analysis of factors underlying seasonal patterns, Int. J. epidemiol, 11, 5-14, (1982)
[12] Ford, J., How random is a coin toss?, Physics today, 40-47, (1983), April
[13] Grassberger, P.; Proccacia, I., Measuring the strangeness of strange attractors, Physica, 9D, 189-208, (1983) · Zbl 0593.58024
[14] Kot, M.; Schaffer, W.M.; Truty, G.L.; Graser, D.J.; Olsen, L.F., Criteria for imposing order, Ecol. modell, (1988), in press
[15] Lindblad, P.; Degn, H., A compiler for digital computation in chemical kinetics and its application to oscillatory reaction schemes, Acta chem. scand, 21, 791-800, (1967)
[16] London, W.P.; Yorke, J.A., Recurrent outbreaks of measles, chicken pox and mumps. I. seasonal variations in contact rates, Amer. J. epidemiol, 98, 453-468, (1973)
[17] May, R.M., Simple mathematical models with very complicated dynamics, Nature (London), 261, 459-467, (1976) · Zbl 1369.37088
[18] Olsen, L.F., Poliomyelitis epidemics in Denmark over the period 1928-1958 were chaotic, Math. modell, (1988), in press
[19] Olsen, L.F., Low dimensional strange attractors in epidemics of childhood diseases in Copenhagen, Denmark, () · Zbl 0639.92012
[20] Olsen, L.F.; Degn, H., Chaos in biological systems, Q. rev. biophys, 18, 165-225, (1985)
[21] Rössler, O.E., Chaotic behaviour in simple reaction systems, Z. natürforsch, 31a, 259-264, (1976)
[22] Roux, J.-C.; Simoyi, H.; Swinney, H.L., Observations of a strange attractor, Physica, 8D, 257-266, (1983) · Zbl 0538.58024
[23] Ruelle, D., Sensitive dependence on initial conditions and turbulent behavior in dynamical systems, Ann. N. Y. acad. sci, 316, 408-416, (1979)
[24] Schaffer, W.M., Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology, IMA J. math. appl. med. biol, 2, 221-252, (1985) · Zbl 0609.92034
[25] Schaffer, W.M.; Kot, M., Nearly one-dimensional dynamics in an epidemic, J. theor. biol, 112, 403-427, (1985)
[26] Schaffer, W.M.; Kot, M., Do strange attractors govern ecological systems, Bioscience, 35, 342-350, (1985)
[27] Schaffer, W.M.; Olsen, L.F.; Truty, G.L.; Fulmer, S.L.; Graser, D.J., Periodic and chaotic dynamics in childhood infections, (), in press · Zbl 0665.92014
[28] Schaffer, W.M.; Truty, G.L., ()
[29] Takens, F., Detecting strange attractors in turbulence, (), 366-381
[30] Wolf, A., Quantifying chaos with Lyapunov exponents, (), 273-290
[31] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A., Determining Lyapunov exponents from a time series, Physica, 16D, 285-317, (1985) · Zbl 0585.58037
[32] Yorke, J.A.; London, W.P., Recurrent outbreaks of measles, chicken pox and mumps. II. systematic differences in contact rates and stochastic effects, Amer. J. epidemiol, 98, 469-482, (1973)
[33] Yorke, J.A.; Nathanson, N.; Pianigiani, G.; Martin, J., Seasonality and the requirements for perpetuation and eradication of viruses in populations, Amer. J. epidemiol, 109, 103-123, (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.