Sensitivity analysis for dynamic systems with time-lags

Many problems in bioscience for which observations are reported in the literature can be modelled by suitable functional differential equations incorporating time-lags (other terminology: delays) or memory effects, parameterized by scientifically meaningful constant parameters p or/and variable parameters (for example, control functions) u(t). It is often desirable to have information about the effect on the solution of the dynamic system of perturbing the initial data, control functions, time-lags and other parameters appearing in the model. The main purpose of this paper is to derive a general theory for sensitivity analysis of mathematical models that contain time-lags. In this paper, we use adjoint equations and direct methods to estimate the sensitivity functions when the parameters appearing in the model are not only constants but also variables of time. To illustrate the results, the methodology is applied numerically to an example of a delay differential model.