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Exponential Curve Fitting

Alright folks.

I feel a bit silly asking this, but maybe the problem is more difficult than I am giving it credit for.

Basically, I'm trying to replicate a function that I found in a PhD thesis for the purposes of research. However, the function is given only as a graph in the thesis, whereas I need the continuous equation for my application. So basically, it's a curve fitting exercise.

The curve is exponential, going through the points [0,0.1], [40,0.15], [50,0.25], and many more.

I think, to get this curve, it needs to be a really general form of the exponential equation, with at least 4 unknowns, possibly 5, to calibrate:

So, basically, how do I go about calibrating those parameters a, b, c, d, and e to get the curve I want?

The graph I'm trying to emulate is attached below. Any help greatly appreciated!