A hyperbolic curve doesn't have an underlying meaning modeling a process beyond being a curve which goes vertical at a chosen point. It's a bad curve to fit to a process. Exponentials make sense to model a compounding or self-improving process.

You have not read far enough.

But this is a phase change process.

Also, the temptation to shitpost in this thread ...

I read TFA. They found a best fit to a hyperbola. Great. One more data point will break the fit. Because it's not modeling a process, it's assigning an arbitrary zero point. Bad model.