ИСТИНА

Immunology as a scientific discipline studies the response of an organism to antigenic invasion, the recognition of self and nonself, and all the biological, chemical and physical aspects of immune processes. Nonlinearity, threshold effects, feedback control loops, delays, compartmental organization, multiscale regulation inherent in the immune processes call for application of mathematics in modern immunology studies. Complex human diseases such as human immunodeficiency virus (HIV) infection require the development of multiscale models of the virus-host interaction [1]. An integrative model of HIV infection based on a combination of ODEs, PDEs and agentbased description of cells, viruses and cytokines is presented. It is applied to study the sensitivities of infection dynamics to biophysical parameters characterizing the transport and interaction processes, providing a basis for designing efficient therapies. Optimal treatment of virus infections requires application of multiple drugs which affects both the virus and the host organism physiology. Control approach for models of virus infections formulated with the help of delay differential equations on the base of optimal disturbances is described. The mathematical model of experimental infection of mice with lymphocytic choriomeningitis virus (LCMV), which is a gold standard in immunology, is considered. The state space of the model represents observable characteristics of the infection, i.e., the virus, precursor and effector T cells populations, and the cumulative viral load at time t: U(t) = [V (t),Ep(t),Ee(t),W(t)]^T . According to the model [2], their evolution is described by a system of nonlinear delay differential equations. A new method for constructing the multimodal impacts on the immune system in the chronic phase of a viral infection, based on the mathematical models with delayed argument, is formulated. The so called “optimal disturbances,” widely used in the aerodynamic stability theory for models without delays, are constructed for perturbing the steady states of the dynamical system in order to maximize the perturbationinduced response. An algorithm for computing the optimal disturbances is proposed. The concept of optimal disturbances is generalized to the systems with the delayed argument. [1] Bouchnita A., Bocharov G., Meyerhans A., Volpert V. Hybrid approach to model the spatial regulation of T cell responses, BMC Immunology, 18 (Suppl 1), No. 29(2017). [2] Bocharov G. A., Nechepurenko Yu. M., Khristichenko M.Yu., Grebennikov D. S. Optimal disturbances of steady states for an LCMV model, Russian Journal of Numerical Analysis and Mathematical Modelling, accepted (2017). 29