Yup: For time t, upper asymptote b, arbitrary constant k, value at time t y(t),
d/dt y(t) = k y(t) (b - y(t))
Solve that and get a sigmoid, lazy S, curve. Yup, the solution, easy closed form, first calculus, is in terms of exponentials.
Could use that to model, say, the growth of immunity to Covid-19. Once it was used to project the growth of FedEx -- pleased the BoD and saved the company.
Addition: Ah, will save you a little first calculus. In TeX,
$$ y(t) = { y(0) b e^{bkt} \over y(0) \big ( e^{bkt} - 1 \big ) + b} $$