Application of the Global Optimization Methods for Solving the Parameter Estimation Problem in Mathematical Immunology

Mathematical modeling is widely used in modern immunology. The availability of biologically meaningful and detailed mathematical models permits studying the complex interactions between the components of a biological system and predicting the outcome of the therapeutic interventions. However, the incomplete theoretical understanding of the immune mechanism leads to the uncertainty of model structure and the need of model identification. This process is iterative and each step requires data-based model calibration. When the model is highly detailed, the considerable part of model parameters can not be measured experimentally or found in literature, so one has to solve the parameter estimation problem. Using the maximum likelihood framework, the parameter estimation leads to minimization problem for least square functional, when the observational errors are normally distributed. In this work we presented different computational approaches to the treatment of global optimization problem, arising in parameter estimation. We consider two high-dimensional mathematical models of HIV (human immunodeficiency virus)-infection dynamics as examples. The ODE (ordinary differential equations) and DDE (delay differential equations) versions of models were studied. For these models we solved the parameter estimation problem using a number of numerical global optimization techniques, including the optimization method, based on the tensor-train decomposition (TT). The comparative analysis of obtained results showed that the TT-based optimization technique is in the leading group of the methods ranked according to their performance in the parameter estimation for ODE and DDE versions of both models.