Optimization methods for solving inverse immunology and epidemiology problems

Inverse problems for systems of nonlinear ordinary differential equations are studied. In these problems, the unknown coefficients and initial data must be found given additional information about the solution to the corresponding direct problems; this information is obtained by measurements made at some specified points in time. Examples of inverse immunology and epidemiology problems arising in the analysis of infectious diseases progression, in the study of HIV dynamics, and spread of tuberculosis in highly endemic regions taking treatment into account are discussed. In the case when the solution to the inverse problem is not unique, three approaches to the study of identifiability of mathematical models are considered. A numerical solution algorithm based on the minimization of a quadratic objective functional is proposed. At the first stage, neighborhoods of the global minimizers are found, and gradient methods are used at the second stage. The gradient of the objective functional is calculated by solving the corresponding adjoint problem. Numerical results are discussed.