  We study topological indices of Fermionic time-reversal invariant topological
insulators in two dimensions, in the regime of strong Anderson localization. We
devise a method to interpolate between certain Fredholm operators arising in
the context of these systems. We use this technique to prove the bulk-edge
correspondence for mobility-gapped 2D topological insulators possessing a
(Fermionic) time-reversal symmetry (class AII) and provide an alternative route
to a theorem by Elgart-Graf-Schenker (2005) about the bulk-edge correspondence
for strongly-disordered integer quantum Hall systems. We furthermore provide a
proof of the stability of the $\mathbb{Z}_2$ index in the mobility gap regime.
These two-dimensional results serve as a model for the study of higher
dimensional $\mathbb{Z}_2$ indices.
