3. Parameter Identification in ODE Models

In most systems biological models, a series of parameters enters that need to be discussed. In the preceding Sect. 2.​5.​3, we presented an example, where simulations for a large set of parameters have been performed and analyzed. This is the situation, when no measurements are available. The present chapter deals with the case, when measurements are available and can be used, in principle, to identify at least part of the parameters. Such parameter identification problems in ODE models typically arise as nonlinear least squares problems, see Sect. 3.1. They are solved by Gauss-Newton methods, which require the numerical solution of linear least squares problems within each iteration. For pedagogical reasons, the order of these three topics is reversed in our presentation. Therefore, in Sect. 3.2, linear least squares problems are discussed first including the important issue of automatic detection of rank deficiencies in matrix factorization. Clearly, not all data sets are equally well suited to fit all unknown parameters of a given model. Next, in Sect. 3.3, the class of “adequate” nonlinear least squares problems is defined, both theoretically and computationally, for which the local Gauss-Newton method converges. Globalization via a damping strategy is presented. The case of possible non-convergence is treated in detail to find out which part originates from an insufficient model and which one from “bad” initial guesses for the Gauss-Newton iteration. In Sect. 3.4, all pieces of the text presented so far are glued together to apply to the ODE models, which are the general topic of the book. Finally, in Sect. 3.5, three examples with increasing complexity are presented. First, the notorious predator-prey problem is revisited, which turns out to be quite standard. Next, in order to connect the advocated computational ideas with modelling intuition, a simple illustrative example is worked out in algorithmic detail. Last, a more complex parameter identification problem related to a model of the human menstrual cycle is discussed in detail.