Yes at some point the new infections has to go down if for no other reason than there is no one left to infect. As you said it must be logistic, because after a while infected people simply bump into fewer and fewer uninfected people.

Thanks, I was wondering what the name of the curve was, because logically there is no way it ends up being totally exponential.

And to be more specific, you can roughly model it with a Gompertz function: https://en.wikipedia.org/wiki/Gompertz_function