Journal of Computational Mathematics Vol.30, No.1, 2012, 59–79. http://www.global-sci.org/jcm doi:10.4208/jcm.1110-m11si12 CRITICAL ISSUES IN THE NUMERICAL TREATMENT OF THE PARAMETER ESTIMATION PROBLEMS IN IMMUNOLOGY * Tatyana Luzyanina Institute of Mathematical Problems in Biology, RAS, Pushchino, Moscow reg., Russia Email: tatyana luzyanina@impb.psn.ru Gennady Bocharov Institute of Numerical Mathematics, RAS, Moscow, Russia Email: bocharov@inm.ras.ru Abstract A robust and reliable parameter estimation is a critical issue for modeling in immunol- ogy. We developed a computational methodology for analysis of the best-fit parameter estimates and the information-theoretic assessment of the mathematical models formu- lated 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 vari- ational 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. Mathematics subject classification: 34K29, 92-08, 65K10. Key words: Mathematical modeling in immunology, Parameter estimation, Constrained optimization. 1. Introduction Mathematical immunology represents a rapidly growing field of applied mathematics. The key features of the immune system that make call for the application of mathematical modeling tools are: physical complexity, compartmental structure, non-linear response, threshold-type of regulation, memory or time-lag effects, inter-clonal competition and selection, redundancy [3]. Most mathematical models of immune responses are not obtained from first principles and therefore the model structure usually has no a priori proof of validity. The key elements of the data- and science-driven application of mathematical modeling to immunology are: (i) more than one model may correspond to a particular phenomenon; (ii) the computational techniques permit, given data of appropriate quality, to discriminate between rival mathematical models. Given a number of candidate models, one needs for each model to determine a set of actual parameters that is in a well-defined sense optimal and to order the resulting set of optimally parameterized models to indicate which is most appropriate, given the data [3]. * Received February 9, 2011 / Revised version received May 29, 2011 / Accepted June 5, 2011 / Published online January 9, 2012 /