Answer to this is complicated and boils down to 'it depends'.

Hypothetically, let's say a disease has been eradicated. Disease X was present in the population, a treatment and preventative measure was developed and this disease does not have reservoirs in other species. It could go extinct. Conversely, this disease may be closely related to another more benign disease. It is eradicated and the closely related but benign disease mutates to replace its cousin.

Overall the battle against a disease is never really over. The goal is containment and then eradication but eradication is likely never permanent or complete.

A,B.

Aside from the 'first' part. If Earth is devastated by an impactor, having a second base of humans is a good insurance policy.