Applying Stiff Integrators for Ordinary Differential Equations and Delay Differential Equations to Problems with Distributed Delays

There exist excellent codes for an efficient numerical treatment of stiff and differential-algebraic problems. Let us mention Radau5, which is based on the three-stage Radau IIA collocation method, and its extension to problems with discrete delays, Radar5. The aim of the present work is to present a technique that permits a direct application of these codes to problems having a right-hand side with an additional distributed delay term (which is a special case of an integro-differential equation). Models with distributed delays are of increasing importance in pharmacodynamics and pharmacokinetics for the study of the interaction between drugs and the body. The main idea is to approximate the distribution kernel of the integral term by a sum of exponential functions or by a quasi-polynomial expansion and then to transform the distributed (integral) delay term into a set of ordinary differential equations (ODEs). This set is typically stiff, and, for some distribution kernels (e.g., Pareto distribution), it contains discrete delay terms with constant delay. The original equations augmented by this set of ODEs can have a very large dimension, and a careful treatment of the solution of the arising linear systems is necessary. The use of the codes Radau5 and Radar5 is illustrated with three examples (two test equations and one problem taken from pharmacodynamics). The driver programs for these examples are publicly available from the home pages of the authors.