  Compartmental ordinary differential equation (ODE) models are used
extensively in mathematical biology. When transit between compartments occurs
at a constant rate, the well-known linear chain trick can be used to show that
the ODE model is equivalent to an Erlang distributed delay differential
equation (DDE). Here, we demonstrate that compartmental models with non-linear
transit rates and possibly delayed arguments are also equivalent to a scalar
distributed delay differential equation. To illustrate the utility of these
equivalences, we calculate the equilibria of the scalar DDE, and compute the
characteristic function-- without calculating a determinant. We derive the
equivalent scalar DDE for two examples of models in mathematical biology and
use the DDE formulation to identify physiological processes that were otherwise
hidden by the compartmental structure of the ODE model.
