Inverse problems for delay differential equations using the Collage Theorem

We present a theoretical framework to solve inverse problems for systems of delayed ordinary differential equations (delay ODEs) that allows us to estimate the values of unknown parameters, such as coefficients and time delays, using available time series data. This work builds on similar results for non-delayed ODEs, inspired by the Collage Theorem. We discuss technical details related to the implementation of the method, including the use of non-convex optimization to recover unknown delay values. The performance of the method is demonstrated using simulated and noised datasets to recover parameters in models applied to human health. These include an additive delay model for the population dynamics of malaria transmission, and a distributed delay model for the homeostasis of glucose and insulin in the bloodstream.