METHOD OF DISCRETE SINGULARITIES FOR ACCOUNTING FOR CONVECTION WHEN MODELING DYNAMICS OF INFECTIOUS DISEASE UNDER DIFFUSION PERTURBATIONS AND CONCENTRATED INFLUENCES

Abstract

A wide range of various mathematical models have been developed to predict the dynamics of an infectious disease. As a rule, such models do not take into account spatial effects, which are associated with both the uneven distribution of active factors and their convective transfer by the intercellular fluid. The paper proposes a variant of taking into account convection when modeling the process of an infectious disease under conditions of diffusion perturbations and concentrated influences. Based on the reduction of original singularly perturbed model problem with delay to a sequence of problems without delay, an effective step-by-step procedure for numerically asymptotically approximating the solution as a perturbation of the solution of the corresponding degenerate problems is synthesized. To find the velocity field, it is proposed to model the movement of fluid in the intercellular medium as a potential flow in the source-drain system. It is emphasized that this approach can be used for a wide range of configurations of model areas with sufficient variability of boundary conditions. The results of computer modeling are presented, which illustrate the influence of diffusion scattering and convection on the development of a viral disease under the conditions of injections of immunological drugs. It is shown that as a result of diffusion scattering and convective transfer of viral elements, their concentration in the epicenter of infection decreases over time, which leads to a corresponding decrease in the "severity" of the disease. It is also shown that with an uneven field of movement speed of the intercellular fluid, there will be zones with a less intense influx of both own and donor antibodies. As a result, the amount of antibodies available in these zones may be insufficient to neutralize antigens, which may lead to the emergence of new epicenters of infection here. The importance of taking such effects into account, in particular, when forming effective treatment programs, is indicated.