Identifiability of linear compartmental models: the singular locus

Abstract: This work addresses the problem of identifiability, that is, the question of whether parameters can be recovered from data, for linear compartmental models. Using standard differential algebra techniques, the question of whether a given model is generically locally identifiable is equivalent to asking whether the Jacobian matrix of a certain coefficient map, arising from input-output equations, is generically full rank. A natural next step is to study the set of parameter values where the Jacobian matrix drops in rank, which we refer to as the locus of non-identifiable parameter values, or, for short, the singular locus. In this work, we give a formula for coefficient maps in terms of acyclic subgraphs of the model's underlying directed graph and, then, study the case when the singular locus is defined by a single equation, the singular-locus equation. We prove that the singular-locus equation can be used to determine when submodels are generically locally identifiable. We also determine the singular-locus equation for two families of linear compartmental models, cycle and mammillary (star) models with input and output in a single compartment. We also state a conjecture for the corresponding equation for a third family: catenary (path) models. Finally, we introduce the identifiability degree, which is the number of parameter values that map to a generic input-output data vector. This degree was previously computed for mammillary and catenary models, and here we determine this degree for cycle models.