Summary: The infected human organism releases antibodies that help to cope with deceases. Individual peculiarities of the immunity and the decease which are responsible for the formation of antibodies (for example, viruses or bacteria), resistance of an organism, etc., differ and so does the reaction of each organism with the same decease. Despite this fact, doctors as a rule offer a standard treatment plan which is not always optimal. Hence, it is important to define the individual peculiarities of immunity (the velocity of the immune response or the production of specific antibodies) and those of a decease (the velocity of propagation of viruses and bacteria and so on) for every patient separately by the blood and urine tests, etc.
In this article, we study the problem of determining the parameters of an infectious decease in the simplest mathematical model ‘antigen-antibody’ on the measurements of concentrations of antigens and antibodies at fixed times. Some functional describing the discrepancy between experimental and model data is examined. We obtain an explicit representation of the gradient of the objective functional with the use of solution to the corresponding adjoint problem. A comparative analysis of a numerical solution to an inverse problem obtained by the gradient method (the Landweber iteration) and the simplex method (the Nelder-Mead method) is exposed. It is demonstrated that the Nelder-Mead method defines a collection of local approximate values of the velocities of propagation of the immune response and the production of specific antibodies with a prescribed accuracy. The Landweber iteration calculates the minimizer of the objective functional which is closest to the initial approximation using sufficiently large number of iterations.