Compartment Models with Memory

SIAM Review, Volume 65, Issue 3, Page 774-805, August 2023.
The beauty and simplicity of compartment modeling makes it a useful approach for simulating dynamics in an amazingly wide range of applications, which are growing rapidly especially in global carbon cycling, hydrological network flows, and epidemiology and population dynamics. These contexts, however, often involve compartment-to-compartment flows that are incongruent with the conventional assumption of complete mixing that results in exponential residence times in linear models. Here we detail a general method for assigning any desired residence time distribution to a given intercompartmental flow, extending compartment modeling capability to transport operations, power-law residence times, diffusions, etc., without resorting to composite compartments, fractional calculus, or partial differential equations (PDEs) for diffusive transport. This is achieved by writing the mass exchange rate coefficients as functions of age-in-compartment as done in one of the first compartment models in 1917, at the cost of converting the usual ordinary differential equations to a system of first-order PDEs. The PDEs are readily converted to a system of integral equations for which a numerical method is devised. Example calculations demonstrate incorporation of advective lags, advective-dispersive transport, power-law residence time distributions, or diffusive domains in compartment models.