On the Stability of Continuous-Time Systems With Stochastic Delay: Applications to Gene Regulatory Circuits

In this paper the dynamics and stability of a linear system with stochastic delay are investigated. We assume that the delay may take finitely many different values and its dynamics are modeled by a continuous-time Markov chain. Semi-discretization is used to derive the dynamics of the second moment which leads to necessary and sufficient stability conditions for the trivial solution. We apply these results to investigate the stability of the steady state of an auto-regulatory gene-protein network. We demonstrate that stochastic delay may stabilize the system when the corresponding deterministic system with average delay is unstable.