A Backward Technique for Demographic Noise in Biological Ordinary Differential Equation Models

Physical systems described by deterministic differential equations represent idealized situations since they ignore stochastic effects. In the context of biomathematical modeling, we distinguish between environmental or extrinsic noise and demographic or intrinsic noise, for which it is assumed that the variation over time is due to demographic variation of two or more interacting populations (birth, death, immigration, and emigration). The modeling and simulation of demographic noise as a stochastic process affecting units of populations involved in the model is well known in the literature, resulting in discrete stochastic systems or, when the population sizes are large, in continuous stochastic ordinary differential equations and, if noise is ignored, in continuous ordinary differential equation models. The inverse process, i.e., inferring the effects of demographic noise on a natural system described by a set of ordinary differential equations, is still an issue to be addressed. With this paper, we provide a technique to model and simulate demographic noise going backward from a deterministic continuous differential system to its underlying discrete stochastic process, based on the framework of chemical kinetics, since demographic noise is nothing but the biological or ecological counterpart of intrinsic noise in genetic regulation. Our method can, thus, be applied to ordinary differential systems describing any kind of phenomena when intrinsic noise is of interest.