Introduction to Stochastic Models in Biology

This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential equations (ODEs). These models assume that the observed dynamics are driven exclusively by internal, deterministic mechanisms. However, real biological systems will always be exposed to influences that are not completely understood or not feasible to model explicitly. Ignoring these phenomena in the modeling may affect the analysis of the studied biological systems. Therefore there is an increasing need to extend the deterministic models to models that embrace more complex variations in the dynamics. A way of modeling these elements is by including stochastic influences or noise. A natural extension of a deterministic differential equations model is a system of stochastic differential equations (SDEs), where relevant parameters are modeled as suitable stochastic processes, or stochastic processes are added to the driving system equations. This approach assumes that the dynamics are partly driven by noise. All biological dynamical systems evolve under stochastic forces, if we define stochasticity as the parts of the dynamics that we either cannot predict or understand or that we choose not to include in the explicit modeling. To be realistic, models of biological systems should include random influences, since they are concerned with subsystems of the real world that cannot be sufficiently isolated from effects external to the model. The physiological justification to include erratic behaviors