Cholera Dynamics and El Niño-Southern Oscillation

For example, see J. L. Bryden, Epidemic Cholera in the Bengal Presidency (Office of the Superintendent of Government Printing, Calcutta, India, 1871).

Colwell R. R., Science 274, 2025 (1996).

Epstein P. R., Ford T. E., Colwell R. R., Lancet 342, 1216 (1993).

Lobitz B., et al., Proc. Natl. Acad. Sci. U.S.A. 97, 1438 (2000).

Checkley W., et al., Lancet 355, 442 (2000).

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Broomhead D. S., King G. P., Physica D 20, 217 (1986).

Vautard R., Ghil M., Physica D 35, 395 (1989).

Examples of such responses to seasonal forcing in nonlinear models for disease dynamics can be found in W. M. Schaffer et al., in The Ubiquity of Chaos, S. Krasner, Ed. (American Association for the Advancement of Science, Washington, DC, 1990), pp. 138–166;

Schwartz I. B., Smith H. L., J. Math. Biol. 18, 233 (1983);

; and

Schwartz I. B., J. Math. Biol. 30, 473 (1992).

Ellner S., Turchin P., Am. Nat. 145, 343 (1995).

D. W. Nychka, S. Ellner, A. R. Gallant, D. McCaffrey, J. R. Stat. Soc. B 54 399 (1992).

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M. Casdagli, in Nonlinear Modeling and Forecasting, M. Casdagli and S. Eubank, Eds. (Addison-Wesley, New York, 1992).

To fit f we used the feedforward neural network (FNN) model

$f$

$+$

(2)

where G is a sigmoid function such as G(y) = e^y/(1 + e^y). Given k and the set of independent variables (x₁, x₂, … x_d), the model parameters (β_i, γ_ij, μ_i) were estimated by ordinary least squares. Models with different values of k or a different set of x's were compared with a GCV criterion function

$V_{\text{c}}$

(3)

where p is the number of fitted parameters, and n is the sample size. V₁ is the standard GCV criterion [see G. Wahba, Spline Models for Observational Data (Society for Industrial and Applied Mathematics, Philadelphia, 1990)]. We used c = 2 based on (11). This slight over-penalization of model complexity creates a small bias toward simpler models but greatly reduces the chances of spuriously selecting an overly complex model. The FNN models were fitted with FUNFITS, a suite of S/Fortran functions that run in S-Plus (see www.cgd.ucar.edu/stats/Funfits/).

We evaluate the significance of the improvement in fit between a “full” model that incorporates a predictor variable and a “reduced” model that omits the variable. The bootstrap test procedure consists of generating a large number of artificial time series with the reduced model and fitting each of these time series with both the full and reduced models. The artificial time series are generated from the reduced model by adding a vector of randomized residuals to the vector of predictions from the reduced model. In the few cases where the resulting values are negative, we replace them by a lower threshold of 0.1 (equal to the minimum value observed in the data). The improvement in fit between the full and reduced models on the original data is compared to the improvements in fit on the artificial time series, in which any apparent improvement is an artifact of the larger number of parameters and variables in the full model. Let Δ_ir² denote the difference in r² between the full and reduced models for the ith time series (with i = 0 being the original data and i = 1, 2, . . . n being the artificial data). Let p be the fraction of Δ_ir², i > 0 values that are larger than Δ₀r². The reduced model is then rejected in favor of the full model at significance level α if p < α.

See, for example,

Finkenstädt B., Keeling M., Grenfell B. T., Proc. R. Soc. London Ser. B 265, 753 (1998).

Klein S. A., Soden B. J., Lau N., J. Clim. 12, 917 (1999).

Monthly measurements of upper-tropospheric humidity for the period 1979–92 [

Schmetz J., Vanderberg L., Geijo C., Holmlund K., Adv. Space Res. 16, 69 (1995);

Soden B. J., Bretherton F. P., J. Geophys. Res. 101, 9333 (1996)].

Monthly measurements of cloud cover for the period 1983–91 are from the International Satellite Cloud Climatology Project [

Rossow W. B., Schiffer R. A., Bull. Am. Meteorol. Soc. 72, 2 (1991)].

Monthly measurements of top-of-atmosphere absorbed solar radiation for the period 1985–89 are from the Earth Radiation Budget Experiment [

Harrison E. F., et al., J. Geophys. Res. 95, 18687 (1990)].

A supplementary figure is available at Science Online at www.sciencemag.org/feature/data/1051490.shl.

Franco A. A., et al., Am. J. Epidemiol. 146, 1067 (1997).

Ghil M., Mo K., J. Atmos. Sci. 48, 752 (1991).

___, J. Atmos. Sci. 48, 780 (1991).

We thank K. Siddique and G. Fuchs for assistance with the cholera data; R. B. Sack, J. Trantj, and the Office of Global Programs at the National Oceanic and Atmospheric Administration for stimulating this work; B. Soden for the cloud cover and radiation data; and M. A. Rodriguez-Arias for computing assistance. M.P. was supported by a James S. McDonnell Foundation Centennial Fellowship and by The Knut and Alice Wallenbergs Foundation; S.P.E. was supported by a grant from the Mellon Foundation to S.P.E. and N.G. Hairston Jr.; X.R. received partial support from the Commissionat per Universitats i Recerca.