I'm trying to model a process that has an horizontal asymptote when x goes to infinity, and that is (almost) monotonically increasing. It does not necessarily (but probably) decrease in the rate of increase.
Right now I found 3 types of equations (models):
Logistic: $$f(x)=\frac{c}{1+e^{\left(-b\cdot\left(x+a\right)\right)}}$$
Logarithmic: $$f(x)= c\cdot\ln\left(a\cdot x\ +b\right)\ +\ d$$
Rational: $$f(x)=\frac{a\cdot x}{x+b}+d$$
As of now the rational model has the best fit, but is very similar to the logarithmic model, and both of them overshoot values as x gets larger.
I'm using scipy.curve_optimize, but it doesn't always work as expected or converges. I want to explore a lot of equations. What I want to know is if these equations are flexible enough, and if not, what type of models are there that are more general but simple at the same time, for this case, and if there are ways to fit those models in a computationally efficient manner.