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In this paper, we presented the results of analysis of experimental evidence for the decline of the human immune system functioning with age using mathematical model of immunosenescence. The most prominent changes in this system are related to the decline in the T-cellular immunity. These include the decline in the nai;ve T cells generation rate, shrinkage of the volume of the peripheral lymphoid tissue, decline of absolute and relative concentrations of nai;ve T cells in the blood, shortening of the average telomere length of T cells. These alterations in the immune system are responsible for sharp increase of morbidity and mortality caused by infectious agents at old ages. Analysis shows that concentrations of memory and nai;ve T cells in peripheral lymphoid tissue are the key variables in this process. Simulation experiments with our model show that the average life span of memory T cells must grow with age, and that decreasing of antigenic load led to considerable increase in organism's resistance in middle ages, but only to slight increase in old ages. Restriction in the rate of thymus involution resulted in an increase of organism's resistance to infections in old ages. However, this growth is accompanied by the decline of concentration of memory T cells and shortening of their life span. The proposed model describes the trade-off between concentrations of nai;ve and memory T cells and their potential to proliferate in human organism.

We consider the application of linear multistep methods (LMMs) for the numerical solution of initial value problem for stiff delay differential equations (DDEs) with several constant delays, which are used in mathematical modelling of immune response. For the approximation of delayed variables the Nordsieck's interpolation technique, providing an interpolation procedure consistent with the underlying linear multistep formula, is used. An analysis of the convergence for a variable-stepsize and structure of the asymptotic expansion of global error for a fixed-stepsize is presented. Some absolute stability characteristics of the method are examined. Implementation details of the code DIFSUB-DDE, being a modification of the Gear's DIFSUB, are given. Finally, an efficiency of the code developed for solution of stiff DDEs over a wide range of tolerances is illustrated on biomedical application model.