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Parameter Estimation and Model Selection

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Book cover Mathematical Immunology of Virus Infections

Abstract

In this chapter, we illustrate a data-driven methodology to formulation and calibration of mathematical models of immune responses. The maximum likelihood approach to parameter estimation, Tikhonov regularization method and information-theoretic criteria for model ranking and selection are presented for models formulated with ODEs, DDEs and PDEs. Experimental data on CFSE-based proliferation analysis of T cells and LCMV–CTL dynamics in a low dose experimental infection of mice are used.

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Notes

  1. 1.

    With little amendment we can consider the case where different components \(\{ {y}^{i}_j\}_{j=1}^{N_i}\) are associated with i-dependent times \(\{ {{\mathfrak t}}_j^i \}_{j=1}^{N_i}\).

  2. 2.

    Material of sects. 3.2.1–3.2.2 uses the results of the study Journal of Computational and Applied Mathematics, Vol. 184, C.T.H. Baker et al., Computational approaches to parameter estimation and model selection in immunology. Pages 50–76, Copyright \(\copyright \) 2005 with permission from Elsevier.

  3. 3.

    The data are regarded as fixed and assumed to have errors of a certain type.

  4. 4.

    The Akaike criterion is based upon Kullback–Leibler notion of information or distance between two probabilistic models (information loss) [99] approximated using the maximum likelihood estimation [98, 100].

  5. 5.

    Some modellers introduce as a variable the amount of virus-specific memory CTL, a subset of (ii) that is harder to quantify reliably.

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Authors and Affiliations

  1. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

    Gennady Bocharov

  2. Institut Camille Jordan, UMR 5208 CNRS, Centre National de la Recherche Scientifique (CNRS), Villeurbanne, France

    Vitaly Volpert

  3. RUDN University, Moscow, Russia

    Vitaly Volpert

  4. Institute of Immunobiology, Kantonsspital St. Gallen, St. Gallen, Switzerland

    Burkhard Ludewig

  5. Parc de Recerca Biomedica Barcelona, ICREA and Universitat Pompeu Fabra, Barcelona, Spain

    Andreas Meyerhans

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  1. Gennady Bocharov
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  4. Andreas Meyerhans
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Correspondence to Gennady Bocharov .

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Bocharov, G., Volpert, V., Ludewig, B., Meyerhans, A. (2018). Parameter Estimation and Model Selection. In: Mathematical Immunology of Virus Infections. Springer, Cham. https://doi.org/10.1007/978-3-319-72317-4_3

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