Stochastic modelling of biological processes

In this monograph, we have provided a general framework of how noise can arise and be incorporated in deterministic continuous biomathematical models and in deterministic discrete chemical reaction systems. As with continuous models, we consider ordinary differential equation models and delay differential equation models, either with discrete or distributed delays, arising in epidemiology, immunology and ecology. We also distinguish between external (or environmental) noise and internal (or demographic) noise. We study the effects of noise on the dynamical behaviour of the systems and, in particular, on the stability of the positive equilibria, both theoretically and numerically. For discrete biochemical systems such as Genetic Regulatory Networks, we consider the effects of intrinsic noise due to the uncertainty of what reaction will take place and when a reaction will take place. We provide extensive simulations of the solutions of a model of bacteriophage infection and one of transcription regulation in three possible modelling regimes (discrete stochastic, continuous stochastic and continuous deterministic) via Stochastic Simulation Algorithms, stochastic delay and deterministic delay differential equations.