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Predicting Unobserved Exposures from Seasonal Epidemic Data

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Abstract

We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious individuals, our method enables a data-based model prediction of the number of unobserved exposed individuals over very long times.

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Notes

  1. We note the fact that while the inclusion of noise terms on other components makes the analysis more difficult, it will not affect the predictions as long as we stay away from bifurcation points.

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Acknowledgements

The authors gratefully acknowledge support from the Office of Naval Research, and the National Institutes of Health. E.F. is supported by Award Number N0017310-2-C007 from the Naval Research Laboratory (NRL). I.B.S. was supported by the NRL Base Research Program N0001412WX30002, and by Award Number R01GM090204 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences or the National Institutes of Health.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ, 07043, USA

    Eric Forgoston

  2. Nonlinear Systems Dynamics Section, Plasma Physics Division, Code 6792, US Naval Research Laboratory, Washington, DC, 20375, USA

    Ira B. Schwartz

Authors
  1. Eric Forgoston
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Correspondence to Eric Forgoston.

Appendices

Appendix A: Deterministic Model Reduction

The F 1, F 2, and F 3 expressions found in Eqs. (13a)–(13c) are given as follows:

(24a)
(24b)
(24c)

Appendix B: Stochastic Model Reduction

The stochastic, transformed equations are given as follows:

(25a)
(25b)
(25c)

As discussed in the article, we can use the deterministic center manifold result to reduce the stochastic model. Substituting the deterministic center manifold equation given by Eq. (21) into the full system of stochastic, transformed equations gives the following reduced stochastic model that describes the dynamics on the center manifold:

(26a)
(26b)

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Forgoston, E., Schwartz, I.B. Predicting Unobserved Exposures from Seasonal Epidemic Data. Bull Math Biol 75, 1450–1471 (2013). https://doi.org/10.1007/s11538-013-9855-0

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