Some s-curves are never well-modeled by an exponential for any substantial part (of course the exponential will be approximately linear over a short enough rang

Some s-curves are never well-modeled by an exponential for any substantial part (of course the exponential will be approximately linear over a short enough range but then you might as well use a line), like for example arctan. Not all s-curves are logistic functions (although in the context of epidemiology, then I guess people usually just mean logistic by s-curve).