Non-linear compartmental models

Extract

The linear compartmental model arises when ‘particles’ move independently between (or out of) a system of compartments in a stochastically similar way. With a given ‘initial’ particle count, the subsequent compartmental particle counts follow multinomial probability distributions (Faddy (1976)) for Markov or semi-Markov movement processes. One immediate consequence of this is that the variance of the compartmental particle count is always less than the mean, with the result that the coefficient of variation is very small for large mean counts. In applications (e.g, Faddy, Jones and Edwards (1976)) this underestimation of the variation can be a shortcoming of a compartmental analysis. Clustering, introduced by Matis and Wehrly (1981), where ‘clusters’ of particles may move together, is a way in which increased variability may be attained. Increased variability in general will result from relaxing the main assumption that gives rise to a linear model: independent behaviour of the particles. Such nonlinear compartmental models can be generally difficult to handle; in this paper some particular cases are discussed and illustrated with reference to one- and twocompartment systems.