The best option for badly rotten Travertine
A good grade Travertine material should withstand many many years wear and tear however all to often i see some poor grade material that has found its way on the floor in a market that is increasingly cost driven .
In addition a poor cleaning and maintenance regime can also decrease the longevity of the material , keeping loose grit at bay and reducing chemical damage or pet urine, often many household cleaners can be hi alkaline cleaners or on the Acid side of the PH scale over a period of time this have a negative impact.
regular maintenance should also include filling any small pop outs, minor holes that will eventually open up there are fillers available for this and grout if grout fill Travertine. Its worth mentioning that any seals applied should also be fully breathable.
However some tiles will likely need replacing so ts always worth keeping a few tiles back from the origional installation as matching tile batches from new supplies will likely turn out in colour and pattern being different.