  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.
