Stochastic Models of Compartmental Systems

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Abstract
This paper reviews a stochastic approach to compartmental modeling. The need for stochasticity in the model is motivated by two examples, one concerned with vanadium depuration in marine organisms and the other with the thermal resistance of the green sunfish, Lepomis cyanellus. A stochastic formulation is suggested which includes several sources of stochasticity due to some variability among particles in a given experiment and also to some variability between replicates of an experiment. Many different stochastic models of compartmental systems may be obtained from various combinations of these sources of stochasticity, and the mean and autocovariance functions of these models are derived for some one-compartment systems. The mean value functions are shown to be generally tractable forms which may be readily fitted to data; however, even in the case of time-invariant coefficients, the mean functions are often different from the sums of exponential functions given by the deterministic solution. The causal model cannot be identified on the basis of the mean value function alone; however the covariance structure is unique for each of the proposed causal models. The paper also illustrates some basic statistical analysis with these stochastic models. The estimation of parameters by weighted nonlinear least squares is illustrated by fitting some mean functions of these models to data from the previous examples. The procedure may generate RBAN estimators for both of the examples and an asymptotic goodness-of-fit statistic is formulated. The paper concludes with a short summary of promising areas for future research.

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