  Rate equation models are extensively used to describe the many-body states of
laser driven atomic gases. We show that the properties of the rate equation
model used to describe nonlinear optical effects arising in interacting Rydberg
gases can be understood by considering the excitation of individual
super-atoms. From this we deduce a simple semi-analytic model that accurately
describes the Rydberg density and optical susceptibility for different
dimensionalities. We identify the previously reported universal dependence of
the susceptibility on the Rydberg excited fraction as an intrinsic property of
the rate equation model that is rooted in one-body properties. Benchmarking
against exact master equation calculations, we identify regimes in which the
semi-analytic model is particularly reliable. The performance of the model
improves in the presence of dephasing which destroys higher order atomic
coherences.
