Stochastic Modelling of Biological Processes: Noise and Delay in Biomathematics and Genetic Regulatory Networks

The monograph provides a general framework of how noise and delay can arise and be incorporated in continuous biomathematical models and in discrete chemical reaction systems. The continuous models are taken from epidemiology, immunology and ecology. The delays are either discrete or distributed. The distinction between external (or environmental) and internal (or demographic) noise is pointed out. The effects of external noise on the stability of the positive equilibria are studied both theoretically and numerically. On the other hand, for discrete biochemical systems such as Genetic Regulatory Networks, the effects of intrinsic noise can be crucial in obtaining true realistic models. As applications, a model of transcription regulation and one of bacteriophage infection are studied in three possible modelling regimes (discrete stochastic, continuous stochastic and continuous deterministic) via stochastic simulation algorithms, stochastic delay and deterministic delay differential equations.