In this course I will give a brief introduction to stochastic individual-based models with delay. Conventional Markovian models involve reactions between different types of individuals, converting one type into another. The effects of these reactions are usually instantaneous; examples and applications include chemical reaction systems, models in game theory and evolution, biological processes and social dynamics. Markovian dynamics are associated with exponentially distributed waiting times between events.

 

In some applications it is more realistic to assume that the effects of reactions only occur at later times, with characteristic waiting time distributions. For example recovery from a disease does not follow an exponential distribution, instead the time-to-recovery is typically concentrated on a characteristic time window. Similar non-Markovian effects occur in transcription and translation processes in models of genetic networks.

 

The tentative contents of this set of lectures is as follows:

1. Markovian stochastic dynamics: brief introduction to master equation formalism, deterministic limit and rate equations, and stochastic differential equations in the diffusion approximation.

2. Motivation for delay dynamics: examples, history and overview.

3. Stability analysis for delay differential equations: theory and examples

4. Effects of noise in individual-based models with delay: examples, theory, simulation

5. Subdiffusion and stochastic transport processes with non-exponential waiting times

6. Summary and open questions