Identifiability of linear compartmental models: the effect of moving inputs, outputs, and leaks

Abstract

A mathematical model is identifiable if its parameters can be recovered from data. Here we investigate, for linear compartmental models, whether (local, generic) identifiability is preserved when parts of the model – specifically, inputs, outputs, leaks, and edges – are moved, added, or deleted. Our results are as follows. First, for certain catenary, cycle, and mammillary models, moving or deleting the leak preserves identifiability. Next, for cycle models with up to one leak, moving inputs or outputs preserves identifiability. Thus, every cycle model with up to one leak (and at least one input and at least one output) is identifiable. Next, we give conditions under which adding leaks renders a cycle model unidentifiable. Finally, for certain cycle models with no leaks, adding specific edges again preserves identifiability. Our proofs, which are algebraic and combinatorial in nature, rely on results on elementary symmetric polynomials and the theory of input-output equations for linear compartmental models.

Keywords:

• Identifiability
• linear compartmental models
• input-output equation
• cycle model

2010 Mathematics Subject Classifications:

• 92C42
• 97M60
• 37N25

Acknowledgments

This research was initiated by SG in the REU in the Department of Mathematics at Texas A&M University, which was funded by the NSF (DMS-1757872). NO and AS were partially supported by the NSF (DMS-1752672). We thank Gleb Pogudin for helpful discussions and suggestions which improved this work. We are grateful to Nicolette Meshkat, who conveyed to us the proof of Theorem 3.13, which we had stated as a conjecture in an earlier preprint. We thank a referee for useful suggestions, especially for alternate proofs of Lemma 2.10, Proposition 3.6, and Theorem 3.13.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Ovchinnikov et al. considered identifiability over $\mathbb{C}$, whereas we consider identifiability over $\mathbb{R}$, but these are equivalent in our setting (cf. the discussion in [21, § 3]).

2 This $n\ge 3$ assumption comes from the fact that the n = 1 and n = 2 cases reduce to catenary models.

3 By convention, $e_0^{\left[\ell \right]}:=1$.

4 By convention, $h_0^j:=1$

Additional information

Funding

This research was initiated by SG in the REU in the Department of Mathematics at Texas A&M University, which was funded by the National Science Foundation – NSF (DMS-1757872). NO and AS were partially supported by the National Science Foundation – NSF (DMS-1752672).