Узагальнення математичної моделі противірусної імунної відповіді Марчука-Петрова з урахуванням впливу малих просторово розподілених дифузійних збурень

Abstract

To study the patterns of development of the body's immune defenses against viruses and pathogenic bacteria, a very diverse range of mathematical models has been developed. The well-known antiviral immune response model by Marchuk and Petrov, which describes the mechanisms of immune protection of cellular and humoral types, is based on the assumption that the environment of the «organism» is homogeneous and all process components are instantly mixed.The article summarizes the mathematical model by Marchuk and Petrov in order to take into account small spatially distributed diffusion effects on the development of viral disease. The corresponding singularly perturbed model problem with delays is reduced to a sequence of problems without delay, for which the corresponding asymptotic developments of solutions are obtained. The numerical experiments results characterizing the influence of spatially distributed diffusion factors of viral disease on the development of the immune response are presented. The model decrease of the antigens number maximum level in the infection epicenter due to their diffusion «erosion» in the viral disease process is illustrated. It is emphasized that even if the initial antigens amount in some part of the infected area exceeds a certain critical value (immunological barrier), diffusion «redistribution» for a certain period of time reduces above critical values of antigen concentrations to a level below critical, and their further neutralization can be provided by the immune protection level available in an organism before infection. That is, within this model for some time there is confidence that the development of an viral disease acute form not only not occurs, but takes place to an asymptotically stable steady state, which characterizes the state of a healthy organism.