If you're willing to abstract a bit from populations of animals to populations of bacteria, there is the minimum selective concentration (MSC), which is the smallest amount of antibiotic you can add to the growth medium and detect antibiotic resistant bacteria competing out non-antibiotic-resistant bacteria:

https://revive.gardp.org/resource/minimal-selective-concentr...

> Has anybody modeled what percent of a population has to die from something for the protective gene to become widespread?

The question is incoherent. The gene spreads if the organisms carrying it average more children. It unspreads if they average less. All of them could be dying of the same thing, and it wouldn't matter.

The rate of spread is given by the https://en.wikipedia.org/wiki/Selection_coefficient , but cause of death isn't relevant.