Branching out into Structural Identifiability Analysis with Maple: Interactive Exploration of Uncontrolled Linear Time-Invariant Structures

Suppose we wish to predict the behaviour of a physical system. We may choose to represent the system by model structure $S$ (a set of related mathematical models defined by parametric relationships between system variables), and a parameter set $\Theta$. Each parameter vector in $\Theta$ is associated with a completely specified model in $S$. We use $S$ with system observations in estimating the "true" (unknown) parameter vector. Inconveniently, multiple parameter vectors may cause $S$ to approximate the data equally well. If we cannot distinguish between such alternatives, and these lead to dissimilar predictions, we cannot confidently use $S$ in decision making. This result may render efforts in data collection and modelling fruitless. This outcome occurs when $S$ lacks the property of structural global identifiability (SGI). Fortunately, we can test various classes of structures for SGI prior to data collection. A non-SGI result may guide changes to our structure or experimental design towards obtaining a better outcome. We aim to assist the testing of structures for SGI through bespoke Maple 2020 procedures. We consider continuous-time, uncontrolled, linear time-invariant state-space structures. Here, the time evolution of the state-variable vector ${\bf x}$ is modelled by a system of constant-coefficient, ordinary differential equations. We utilise the "transfer function" approach, which is also applicable to the "compartmental" subclass (mass is conserved). Our use of Maple's "Explore" enables an interactive consideration of a parent structure and its variants, obtained as the user changes which components of ${\bf x}$ are observed, or have non-zero initial conditions. Such changes may influence the information content of the idealised output available for the SGI test, and hence, its result. Our approach may inform the interactive analysis of structures from other classes.