Nonnegative compartment dynamical system modelling with stochastic differential equations

Compartment models are widely used in various physical sciences and adequately describe many biological phenomena. Elements such as blood, gut, liver and lean tissue are characterized as homogeneous compartments, within which the drug resides for a time, later to transit to another compartment, perhaps recycling or eventually vanishing. We address the issue of compartment dynamical system modelling using multidimensional stochastic differential equations, rather than the classical approach based on the continuous-time Markov chain. Pure-jump processes are employed as perturbation to the deterministic compartmental dynamical system. Unlike with the Brownian motion, noise can be incorporated into both outflows and inter-compartmental flows without violating nonnegativity of the compartmental system, under mild technical conditions. The proposed formulation is easy to simulate, shares various essential properties with the corresponding deterministic ordinary differential equation, such as asymptotic behaviors in mean, steady states and average residence times, and can be made as close to the corresponding diffusion approximation as desired. Asymptotic mean-square stability of the steady state is proved to hold under some assumptions on the model structure. Numerical results are provided to illustrate the effectiveness of our formulation.