J. Comp. Math., 30 (2012), pp. 59-79.

Critical Issues in the Numerical Treatment of the Parameter Estimation Problems in Immunology

Tatyana Luzyanina 1, Gennady Bocharov 2

1 Institute of Mathematical Problems in Biology, RAS, Pushchino, Moscow reg., Russia
2 Institute of Numerical Mathematics, RAS, Moscow, Russia

Received 2011-2-09 Accepted 2011-6-05
Available online 2012-1-09


A robust and reliable parameter estimation is a critical issue for modeling in immunology. We developed a computational methodology for analysis of the best-fit parameter estimates and the information-theoretic assessment of the mathematical models formulated with ODEs. The core element of the methodology is a robust evaluation of the first and second derivatives of the model solution with respect to the model parameter values. The critical issue of the reliable estimation of the derivatives was addressed in the context of inverse problems arising in mathematical immunology. To evaluate the first and second derivatives of the ODE solution with respect to parameters, we implemented the variational equations-, automatic differentiation and complex-step derivative approximation methods. A comprehensive analysis of these approaches to the derivative approximations is presented to understand their advantages and limitations.

Key words: Mathematical modeling in immunology, Parameter estimation, Constrained optimization.

AMS subject classifications: 34K29, 92-08, 65K10.

Email: tatyana_luzyanina@impb.psn.ru, bocharov@inm.ras.ru

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