Skip to main content
SpringerLink
Log in

Introduction to Stochastic Models in Biology

  • Chapter
  • First Online:
Stochastic Biomathematical Models

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2058))

Abstract

This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential equations (ODEs). These models assume that the observed dynamics are driven exclusively by internal, deterministic mechanisms. However, real biological systems will always be exposed to influences that are not completely understood or not feasible to model explicitly. Ignoring these phenomena in the modeling may affect the analysis of the studied biological systems. Therefore there is an increasing need to extend the deterministic models to models that embrace more complex variations in the dynamics. A way of modeling these elements is by including stochastic influences or noise. A natural extension of a deterministic differential equations model is a system of stochastic differential equations (SDEs), where relevant parameters are modeled as suitable stochastic processes, or stochastic processes are added to the driving system equations. This approach assumes that the dynamics are partly driven by noise.

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aït-Sahalia, Y.: Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica 70(1), 223–262 (2002)

    Google Scholar 

  2. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations (I): convergence rate of the distribution function. Probab. Theor. Relat. Field 104(1), 43–60 (1996)

    Google Scholar 

  3. Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations (II): convergence rate of the density. Monte Carlo Meth. Appl. 2, 93–128 (1996)

    Google Scholar 

  4. Bibby, B.M., Sørensen, M.: Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1(1/2), 017–039 (1995)

    Google Scholar 

  5. Bibby, B.M., Sørensen, M.: On estimation for discretely observed diffusions: a review. Theor. Stoch. Process. 2(18), 49–56 (1996)

    Google Scholar 

  6. Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)

    Google Scholar 

  7. Dacunha-Castelle, D., Florens-Zmirou, D.: Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19(4), 263–284 (1986)

    Google Scholar 

  8. Davidian, M., Giltinan, D.M.: Nonlinear models for repeated measurements: An overview and update. J. Agr. Biol. Environ. Stat. 8, 387–419 (2003)

    Google Scholar 

  9. De la Cruz-Mesia, R., Marshall, G.: Non-linear random effects models with continuous time autoregressive errors: a Bayesian approach. Stat. Med. 25, 1471–1484 (2006)

    Google Scholar 

  10. Donnet, S., Samson, A.: Parametric inference for mixed models defined by stochastic differential equations. ESAIM Probab. Stat. 12, 196–218 (2008)

    Google Scholar 

  11. Donnet, S., Foulley, J.L., Samson, A.: Bayesian analysis of growth curves using mixed models defined by stochastic differential equations. Biometrics 66(3), 733–741 (2010)

    Google Scholar 

  12. Durham, G.B., Gallant, A.R.: Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econ. Stat. 20, 297–338 (2002)

    Google Scholar 

  13. Elerian, O., Chib, S., Shephard, N.: Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69(4), 959–993 (2001)

    Google Scholar 

  14. Eraker, B.: MCMC analysis of diffusion models with application to finance. J. Bus. Econ. Stat. 19(2), 177–191 (2001)

    Google Scholar 

  15. Favetto, B., Samson, A.: Parameter estimation for a bidimensional partially observed Ornstein-Uhlenbeck process with biological application. Scand. J. Stat. 37, 200–220 (2010)

    Google Scholar 

  16. Feller, W.: Diffusion processes in genetics. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 227–246. University of California Press, Berkeley (1951)

    Google Scholar 

  17. Fournier, L., Thiam, R., Cuénod, C.-A., Medioni, J., Trinquart, L., Balvay, D., Banu, E., Balcaceres, J., Frija, G., Oudard, S.: Dynamic contrast-enhanced CT (DCE-CT) as an early biomarker of response in metastatic renal cell carcinoma (mRCC) under anti-angiogenic treatment. J. Clin. Oncol. ASCO Annu. Meet. Proc. (Post-Meeting Edition) 25 (2007)

    Google Scholar 

  18. Hou, W., Garvan, C.W., Zhao, W., Behnke, M., Eyler, F., Wu, R.: A general model for detecting genetic determinants underlying longitudinal traits with unequally spaced measurements and nonstationary covariance structure. Biostatistics 6, 420–433 (2005)

    Google Scholar 

  19. Iacus, S.M.: Simulation and Inference for Stochastic Differential Equations. With R examples. Springer, New York (2008)

    Google Scholar 

  20. Jaffrézic, F., Meza, C., Lavielle, M., Foulley, J.L.: Genetic analysis of growth curves using the SAEM algorithm. Genet. Sel. Evol. 38, 583–600 (2006)

    Google Scholar 

  21. Karlin, S., Taylor, H.M.: A Second Course in Stochastic Processes. Academic, New York (1981)

    Google Scholar 

  22. Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, New York (1999)

    Google Scholar 

  23. Kutoyants, T.: Parameter Estimation for Stochastic Processes. Helderman Verlag, Berlin (1984)

    Google Scholar 

  24. Øksendal, B.: Stochastic Differential Equations. An Introduction with Applications, 6th edn. Universitext. Springer, Berlin (2003)

    Google Scholar 

  25. Pedersen, A.: A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat. 22(1), 55–71 (1995)

    Google Scholar 

  26. Pedersen, A.R.: Statistical analysis of gaussian diffusion processes based on incomplete discrete observations. Research Report, Department of Theoretical Statistics, University of Aarhus, 297 (1994)

    Google Scholar 

  27. Prakasa Rao, B.: Statistical Inference for Diffusion Type Processes. Arnold, London (1999)

    Google Scholar 

  28. Robert, C.P.: Bayesian computational methods. In: Handbook of Computational Statistics, pp. 719–765. Springer, Berlin (2004)

    Google Scholar 

  29. Roberts, G.O., Stramer, O.: On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88(3), 603–621 (2001)

    Google Scholar 

  30. Rosen, M.A., Schnall, M.D.: Dynamic contrast-enhanced magnetic resonance imaging for assessing tumor vascularity and vascular effects of targeted therapies in renal cell carcinoma. Clin. Cancer Res. 13(2), 770–6 (2007)

    Google Scholar 

  31. Sørensen, M.: Parametric inference for discretely sampled stochastic differential equations. In: Andersen, T.G., Davis, R.A., Kreiss, J.P., Mikosch, T. (eds.) Handbook of Financial Time Series, pp. 531–553. Springer, Heidelberg (2009)

    Google Scholar 

  32. Sørensen, M.: Estimating functions for diffusion-type processes. In: Kessler, M., Lindner, A., Sørensen, M. (eds.) Statistical Methods for Stochastic Differential Equations. Chapmann & Hall/CRC Monographs on Statistics & Applied Probability, London (2012)

    Google Scholar 

  33. Spyrides, M.H., Struchiner, C.J., Barbosa, M.T., Kac, G.: Effect of predominant breastfeeding duration on infant growth: a prospective study using nonlinear mixed effect models. J. Pediatr. 84, 237–243 (2008)

    Google Scholar 

  34. Taylor, H.M., Karlin, S.: An Introduction to Stochastic Modeling, 3rd edn. Academic, San Diego, CA (1998)

    Google Scholar 

  35. Zimmerman, D., Núnez-Antón, V.: Parametric modelling of growth curve data: an overview. Test 10, 1–73 (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Susanne Ditlevsen

  2. CNRS UMR8145, Laboratoire MAP5, Université Paris Descartes, 45 rue des Saints Pères, 75006, Paris, France

    Adeline Samson

Authors
  1. Susanne Ditlevsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adeline Samson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Susanne Ditlevsen .

Editor information

Editors and Affiliations

  1. College of Sciences, Department of Mathematics, King Saud University, King Khalid Road, Diriyah Campus, Riyadh, 11451, Saudi Arabia

    Mostafa Bachar

  2. Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, Graz, 8010, Austria

    Jerry Batzel

  3. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, Copenhagen, 2100, Denmark

    Susanne Ditlevsen

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ditlevsen, S., Samson, A. (2013). Introduction to Stochastic Models in Biology. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation