Step by Step Perturbation of Discrete Models of Immunology

Abstract

A number of very different mathematical models are used to predict the response of the immune system to pathogenic microorganisms detected in the body and the corresponding course of viral disease. Usually,such models are based on the assumption that the body is a homogeneous envi-ronment in which all factors are evenly distributed.The article presents a generalized discrete model of Marchuk's infectious disease for the complex accounting of small diffusion «redistributions», con-centrated effects and the body's temperature response. The introduction of such additional terms into the basic model significantly complicates the orig-inal problem and aggravates the problem of constructing efficient algorithms for the numerical solution of such systems of differential equations with de-lays. It is noted that as a result of discretization of the original model problem using an implicit scheme, a nonlinear system of equations is obtained, the so-lution of which must be sought at each time step by iterations. Thus, the use of the corresponding classical Runge-Kutta schemes is very uneconomical from the point of view of calculations.The authors propose a step-by-step procedure for numerically asymp-totic approximation of the solution of the corresponding singularly per-turbed discrete problem with delay, which allows to combine the ad-vantages of implicit schemes and the cost-effectiveness of explicit schemes. The results of computer simulations are presented, which illus-trate the influence of diffuse «scattering»of antigens, delays and concen-trated sources of antigens on the nature of the infectious disease. It is em-phasized that the complex action of these factors can lead to a reduction of the initially supercritical concentration of antigens to a more acceptable level, which is important in forming a rational program of decision-making on the use of external «therapeutic»effects.