# American Institute of Mathematical Sciences

March  2013, 18(2): 417-435. doi: 10.3934/dcdsb.2013.18.417

## Reduction and identification of dynamic models. Simple example: Generic receptor model

 1 Department of Mathematics and Physics, Lappeenranta University of Technology, P.O.Box 20, FIN-53851 Lappeenranta, Finland 2 32 Campus Drive, Department of Mathematical Sciences, University of Montana, Missoula, MT 59812 3 Finnish Meteorological Institute, P.O. Box 503, FI-00101 Helsinki, Finland

Received  March 2011 Revised  January 2012 Published  November 2012

Identification of biological models is often complicated by the fact that the available experimental data from field measurements is noisy or incomplete. Moreover, models may be complex, and contain a large number of correlated parameters. As a result, the parameters are poorly identified by the data, and the reliability of the model predictions is questionable. We consider a general scheme for reduction and identification of dynamic models using two modern approaches, Markov chain Monte Carlo sampling methods together with asymptotic model reduction techniques. The ideas are illustrated using a simple example related to bio-medical applications: a model of a generic receptor. In this paper we want to point out what the researchers working in biological, medical, etc., fields should look for in order to identify such problematic situations in modelling, and how to overcome these problems.
Citation: Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417
##### References:
 [1] Y. Bard, "Nonlinear Parameter Estimation," Academic Press, New York - London, 1974. [2] A. Gelman and J. Carlin, "Bayesian Data Analysis," $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, FL, 2004. [3] H. Haario, E. Saksman and J. Tamminen, An adaptive metropolis algorithm, Bernoulli, 7 (2001), 223-242. doi: 10.2307/3318737. [4] H. Haario, M. Laine, A. Mira and E. Saksman, DRAM: Efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354. doi: 10.1007/s11222-006-9438-0. [5] H. Haario, L. Kalachev and M. Laine, Reduced Models for Algae Growth, Bull. Math. Biol., 71 (2009), 1626-1648. doi: 10.1007/s11538-009-9417-7. [6] J. Kevorkian and J. Cole, "Singular Perturbation Methods in Applied Mathematics," Springer-Verlag, Berlin-Heidelberg-New York, 1981. [7] J. Murray, "Mathematical Biology," $2^{nd}$ edition, Springer-Verlag, Berlin-Heidelberg-New York, 1993. doi: 10.1007/b98869. [8] R. O'Malley, "Singular Perturbations Methods for Ordinary Differential Equations," Springer-Verlag, Berlin-Heidelberg-New York, 1991. doi: 10.1007/978-1-4612-0977-5. [9] A. Vasil'eva, V. Butuzov and L. Kalachev, "The Boundary Function Method for Singular Perturbation Problems," SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970784.

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##### References:
 [1] Y. Bard, "Nonlinear Parameter Estimation," Academic Press, New York - London, 1974. [2] A. Gelman and J. Carlin, "Bayesian Data Analysis," $2^{nd}$ edition, Chapman & Hall/CRC, Boca Raton, FL, 2004. [3] H. Haario, E. Saksman and J. Tamminen, An adaptive metropolis algorithm, Bernoulli, 7 (2001), 223-242. doi: 10.2307/3318737. [4] H. Haario, M. Laine, A. Mira and E. Saksman, DRAM: Efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354. doi: 10.1007/s11222-006-9438-0. [5] H. Haario, L. Kalachev and M. Laine, Reduced Models for Algae Growth, Bull. Math. Biol., 71 (2009), 1626-1648. doi: 10.1007/s11538-009-9417-7. [6] J. Kevorkian and J. Cole, "Singular Perturbation Methods in Applied Mathematics," Springer-Verlag, Berlin-Heidelberg-New York, 1981. [7] J. Murray, "Mathematical Biology," $2^{nd}$ edition, Springer-Verlag, Berlin-Heidelberg-New York, 1993. doi: 10.1007/b98869. [8] R. O'Malley, "Singular Perturbations Methods for Ordinary Differential Equations," Springer-Verlag, Berlin-Heidelberg-New York, 1991. doi: 10.1007/978-1-4612-0977-5. [9] A. Vasil'eva, V. Butuzov and L. Kalachev, "The Boundary Function Method for Singular Perturbation Problems," SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970784.
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