On the problem of replacing an ordinary differential equation with constant time delay by the system of ordinary differential equations

In this paper we give and prove the comparison theorem between the solution of an ordinary differential equation with constant time delay and that of the corresponding so-called cascade system of ordinary differential equations (ODEs). This comparison problem arises, for example, when describing the plasma cell clone formation in an immune response by two different methods, viz. an equation with time delay, where the delay is done formation time, and the cascade system of ODEs which corresponds to a sequential cell division. We obtain the estimates that show how close the solutions of mathematical objects studied are and that the solutions coincide when the number of cascades is infinite. We give some test calculations, including the mathematical model of an infectious disease. Using the disease model, we prove and demonstrate the assertion that the stability conditions for the stationary solution of the equation with time delay and that of the cascade system of ODEs coincide when the number of cascades is infinite. In this paper we consider the comparison theorem between the solution of an ordinary differential equation with constant time delay and that of a so-called cascade system of ordinary differential equations (ODEs). The assertion given here may be called the inhomogeneous comparison theorem. Thus, we emphasize that the equations compared belong to different classes, viz. ODEs and ODEs with time delay unlike the familiar comparison theorems (see, e.g. [4]) which give the exponential estimates of the closeness of the solutions which belong to the same class of ODEs whose right-hand sides are 'similar* and the differ from each other insignificantly. There are some reasons or premises for considering the inhomogeneous comparison problem, two of them given below seem to be most important. First, it is a general mathematical premise. Needless to say that when solving complicated problems such as identification, optimization, parametrization ones [8] it is convenient to deal with the model of the original object studied in the form of ODEs rather than in form of ODEs with time delay. This is because in the process of calculations of these problems it is necessary to repeatedly use the original model studied, the integration step may be variable, the delay itself changes in the process of calculations, and so on. There are a lot of mathematical methods of solving ODEs and procedures of their numerical investigation, which is not the case with ODEs with time delay. An adaptation of these methods to delay systems requires much mental power, larger computer storage, the complication of the available complicated programs and, finally, increases the probability that an error will appear in the solution of the problem. Thus, the problem is whether it is possible to purely mathematically replace the delay model by the appropriate system of ODEs (without delay). Second, this mathematical problem admits a biological interpretation, at least, in problems of mathematical modelling in immunology [1,7]. For example, the plasma cell * Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 117951, GSP-1, Russia The work was supported by the Russian Foundation for the Basic Research (97-01-00080).