DELAY MODELING OF MATHEMATICAL MODELS OF BIOLOGY AND IMMUNOLOGY

Abstract

Systems of differential-difference equations are mathematical models of many applied problems of biology, ecology, medicine, economics. The variety of mathematical models of real dynamic processes is due to the fact that their evolution does not occur instantaneously, but with some delays that have different biological interpretations. The introduction of delay allows you to build adequate mathematical models and describe new effects and phenomena in physics, ecology, immunology and other sciences. The exact solution of differential-difference equations can be found only in the simplest cases, so algorithms for finding approximate solutions of such equations are important. In this paper, a family of difference schemes is constructed for the approximate finding of solutions to initial problems with delay. Special cases are generalized Euler difference schemes. The conditions for the convergence of the generalized explicit Euler difference scheme are established. To automate the numerical simulation of systems with delays, an application program has been developed, which is used to approximate the solutions of SIR models with two delays.