When modelling biological systems one often faces transport phenomena of which the details are irrelevant to the model, e.g.\ viral kinematics inside a host, the prepubescent development of an individual, etc. Two abstractions are possible, either one introduces an intermediate compartment and models transport as typical flow-rates between these compartments, or one introduces time-delay. Although the former is more common, the latter can yield richer models. In particular, time-delay can introduce intricate periodic and chaotic behaviours even in the scalar case, as illustrated by the celebrated Mackey--Glass equations. These models admit more complicated behaviours because the dependence on past values makes them inherently infinite-dimensional.

The numerical study of a system`s attractors and their basins is well-established for finite dimensional systems. In this work, we use ideas from approximation theory to develop analogous techniques for time-delayed models. Whilst similar to the techniques used in the delay-free case, the treatment of infinite-dimensionality yields novel challenges. In particular, we now have to find an interpretable projection to visualize the found basins. Throughout the presentation, we will illustrate these techniques using an SIR model with the inclusion of waning and boosting of immunity.