An extensive study for solving the initial value problem for systems of delay differential equations (DDE-IVPs) of general parametric type is presented in this paper. In particular, the authors contribute to the determination and calculation of the derivatives of the solution, assuming a discontinuity propagation.

“A theorem on the differentiability of solutions of DDE-IVPs with respect to parameters,” including the case with discontinuity in the initial time, is proved and discussed. This result gives the sufficient conditions for more precise local differentiability of solutions identifying a piecewise continuity of the derivative with possible “jumps at the propagated discontinuity times.”

Special attention is paid to numerical methods for solving the problem. A method based on the concept of internal numerical differentiation (IND) is developed for “computing the derivatives of the solution of DDE-IVPs with respect to parameters.” This approach leads to a reduction of the computational costs, allowing for effective calculation and error control of discontinuities in the derivative. A computer implementation, collocation solver for DDEs (COLSOL-DDE), is presented, which is a newly developed Fortran95 program for solving the problem and computation of derivatives using the proposed IND method combined with implicit continuous Runge-Kutta methods of collocation type. The proposed approach is applied for the numerical solution of two examples: a test problem with a known exact solution and a DDE model of a genetic regulatory network. The numerical results obtained for different parameters and the numerical analysis performed and compared with other numerical approaches demonstrate the “reliability and efficiency of the developed IND-based method.”

The work is an example of a thorough mathematical investigation and solution of the given problem in its theoretical and numerical aspects, including its computer realization with detailed numerical analysis. This increases its applicability for solving real, practical problems.