Skip to main content
SpringerLink
Log in

Exotica

  • Chapter
  • First Online:
Epidemics

Part of the book series: Use R! ((USE R))

  • 4393 Accesses

Abstract

Chapter 9 discussed how a linear approximation to the perennially nonlinear dynamics of infectious disease can provide important insights on invasion, stability, and resonant periodicity. As remarked by Nisbet and Gurney (1982) more generally, linear approximation can often provide remarkably useful insights for nonlinear ecological systems as long as they are not too nonlinear.

This chapter uses the following R-packages: deSolve, pomp, and nlts.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Interestingly Ruelle (1993) paraphrases Henri Poincaré as defining chance as sensitive dependence on unknown initial conditions as far back as 1908.

  2. 2.

    Modeling chicken pox, a herpes virus that can reactivate in older individuals in the form of zoster, Ferguson et al. (1996) showed that the SEIR model cannot sustain multiannual (or chaotic) childhood dynamics in the presence of “immigration” of the virus from an adult carrier group.

  3. 3.

    The method was originally proposed as a nonparametric method to estimate the “order” of a time series (Cheng and Tong 1992).

References

  • Bailey, B. A., Ellner, S., & Nychka, D. W. (1997). Chaos with confidence: Asymptotics and applications of local lyapunov exponents. Nonlinear dynamics and time series: Building a bridge between the natural and statistical sciences (pp. 115–133). Providence, RI: American Mathematical Society.

    Google Scholar 

  • Bjørnstad, O. N., & Grenfell, B. T. (2001). Noisy clockwork: Time series analysis of population fluctuations in animals. Science, 293(5530), 638–643.

    Article  Google Scholar 

  • Cheng, B., & Tong, H. (1992). On consistent nonparametric order determination and chaos. Journal of the Royal Statistical Society. Series B (Methodological), 54, 427–449.

    MathSciNet  MATH  Google Scholar 

  • Cushing, J., Dennis, B., Desharnais, R., & Costantino, R. (1998). Moving toward an unstable equilibrium: Saddle nodes in population systems. Journal of Animal Ecology, 67, 298–306.

    Article  Google Scholar 

  • Dalziel, B. D., Bjørnstad, O. N., van Panhuis, W. G., Burke, D. S., Metcalf, C. J. E., & Grenfell, B. T. (2016). Persistent chaos of measles epidemics in the prevaccination United States caused by a small change in seasonal transmission patterns. PLoS Computational Biology, 12(2), e1004655.

    Article  Google Scholar 

  • Dennis, B., Desharnais, R. A., Cushing, J., Henson, S. M., & Costantino, R. (2003). Can noise induce chaos? Oikos, 102(2), 329–339.

    Article  Google Scholar 

  • Eckmann, J.-P., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57(3), 617.

    Article  MathSciNet  Google Scholar 

  • Ellner, S., & Turchin, P. (2005). When can noise induce chaos and why does it matter: A critique. Oikos, 111(3), 620–631.

    Article  Google Scholar 

  • Fan, J., Yao, Q., & Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika, 83(1), 189–206.

    Article  MathSciNet  Google Scholar 

  • Ferguson, N. M., Anderson, R. M., & Garnett, G. P. (1996). Mass vaccination to control chickenpox: The influence of zoster. Proceedings of the National Academy of Sciences, 93(14), 7231–7235.

    Article  Google Scholar 

  • Gammaitoni, L., Hänggi, P., Jung, P., & Marchesoni, F. (1998). Stochastic resonance. Reviews of Modern Physics, 70(1), 223.

    Article  Google Scholar 

  • Grenfell, B., Bjørnstad, O., & Kappey, J. (2001), Travelling waves and spatial hierarchies in measles epidemics. Nature, 414(6865), 716–723.

    Article  Google Scholar 

  • Grenfell, B. T., Bjørnstad, O. N., & Finkenstadt, B. F. (2002). Dynamics of measles epidemics: Scaling noise, determinism, and predictability with the tsir model. Ecological Monographs, 72(2), 185–202.

    Article  Google Scholar 

  • Lavine, J. S., King, A. A., Andreasen, V., & Bjørnstad, O. N. (2013). Immune boosting explains regime-shifts in prevaccine-era pertussis dynamics. PLoS One, 8(8), e72086.

    Article  Google Scholar 

  • Lavine, J. S., King, A. A., & Bjørnstad, O. N. (2011). Natural immune boosting in pertussis dynamics and the potential for long-term vaccine failure. Proceedings of the National Academy of Sciences, 108(17), 7259–7264.

    Article  Google Scholar 

  • Loader, C. (2006). Local regression and likelihood. Berlin: Springer Science & Business Media.

    MATH  Google Scholar 

  • Nisbet, R. M., & Gurney, W. (1982). Modelling fluctuating populations. Chichester: John Wiley and Sons Limited.

    MATH  Google Scholar 

  • Rand, D., & Wilson, H. (1991). Chaotic stochasticity: A ubiquitous source of unpredictability in epidemics. Proceedings of the Royal Society of London B: Biological Sciences, 246(1316), 179–184.

    Article  Google Scholar 

  • Rohani, P., Keeling, M. J., & Grenfell, B. T. (2002). The interplay between determinism and stochasticity in childhood diseases. The American Naturalist, 159(5), 469–481.

    Article  Google Scholar 

  • Ruelle, D. (1993). Chance and chaos. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Sugihara, G., Grenfell, B., & May, R. M. (1990). Distinguishing error from chaos in ecological time series. Philosophical Transactions of the Royal Society of London Series B, 330, 235–250.

    Article  Google Scholar 

  • Wiesenfeld, K., & Moss, F. (1995). Stochastic resonance and the benefits of noise: From ice ages to crayfish and squids. Nature, 373(6509), 33–36.

    Article  Google Scholar 

  • Yao, Q., & Tong, H. (1994). On prediction and chaos in stochastic systems. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 348(1688), 357–369.

    Article  Google Scholar 

  • Yao, Q., & Tong, H. (1998). A bootstrap detection for operational determinism. Physica D: Nonlinear Phenomena, 115(1–2), 49–55.

    Article  Google Scholar 

  • Ye, H., Beamish, R. J., Glaser, S. M., Grant, S. C., Hsieh, C.-H., Richards, L. J., et al. (2015). Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. Proceedings of the National Academy of Sciences, 112(13), E1569–E1576.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Infectious Disease Dynamics, Pensylvania State University, University Park, PA, USA

    Ottar N. Bjørnstad

Authors
  1. Ottar N. Bjørnstad
    View author publications

    You can also search for this author in PubMed Google Scholar

Appendix: Making a Pomp-Simulator

Appendix: Making a Pomp-Simulator

Doing the computations involved in Sects. 10.5 and 10.7 are computationally expensive. The pomp-package includes a Csnippet-function that will compile C code on the fly to speed up calculations. The following provides the C code used in the simulations of the stochastic SEIR model.

We first define the Csnippet for the deterministic skeleton of the unobserved process:

Then the Csnippet for the stochastic simulator

pomp wants Csnippets for the observational process also (even if we only use the object for simulation).

We need initial conditions

Finally we can build the pomp object. The dat-data object defines the times for the stochastic simulation. We are not working with data, so the reports column is just a dummy.

The pomp-package has numerous functions to simulate deterministic and stochastic trajectories from pomp-objects.

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bjørnstad, O.N. (2018). Exotica. In: Epidemics. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-97487-3_10

Download citation

Publish with us

Policies and ethics

Search

Navigation