Stochastic Differential Delay Equation, Moment Stability, and Application to Hematopoietic Stem Cell Regulation System

We study the moment stability of the trivial solution of a linear differential delay equation in the presence of additive and multiplicative white noise. The results established here are applied to examining the local stability of the hematopoietic stem cell (HSC) regulation system in the presence of noise. The stability of the first moment for the solutions of a linear differential delay equation under stochastic perturbation is identical to that of the unperturbed system. However, the stability of the second moment is altered by the perturbation. We obtain, using Laplace transform techniques, necessary and sufficient conditions for the second moment to be bounded. In applying the results to the HSC system, we find that the system stability is sensitive to perturbations in the stem cell differentiation and death rates, but insensitive to perturbations in the proliferation rate.