Compartmental Modeling and Second-Moment Analysis of State Space Systems

Compartmental models involve nonnegative state variables that exchange mass, energy, or other quantities in accordance with conservation laws. Such models are widespread in biology and economics. In this paper a connection is made between arbitrary (not necessarily nonnegative) state space systems and compartmental models. Specifically, for an arbitrary state space model with additive white noise, the nonnegative-definite second-moment matrix is characterized by a Lyapunov differential equation. Kronecker and Hadamard (Schur) matrix algebra is then used to derive an equation that characterizes the dynamics of the diagonal elements of the second-moment matrix. Since these diagonal elements are nonnegative, they can be viewed, in certain cases, as the state variables of a compartmental model. This paper examines weak coupling conditions under which the steady-state values of the diagonal elements actually satisfy a steady-state compartmental model.