Good luck that most countries on the planet have already passed the inflection point, so finding the saturation point isn't that difficult anymore.
Interesting to see that just asking people to look into this will generate downvotes. It's just basic mathematics. And it really isn't that hard to verify...
In my area they made masks mandatory and cut down the protection of hospitals - now we have an infected nurse. I think politicians currently are using seriously wrong models for their actions.
I had a discussion with one of the "official" epidemiologists. It went like the discussion with you - just at the point where we are now he decided to go into all-caps swearing, said that he will never look into this, called his models perfect and then blocked me.
I'm not claiming that is what you were saying, I just wanted to call out this point in case other people read this that may come to this conclusion.
FWIW, the solution to f'/f = exp(-t) is f(t)=c*exp(-exp(t)), which actually looks like an epidemiological curve, but like any sigmoid it will be hard to determine the actual saturation point until after it has already passed the inflection point.