# Branching Processes

**Teacher**: dr. Hrvoje Šikić, professor

**Associate**: dr. Snježana Lubura Strunjak, assistant professor

**Semester**: third

**ECTS**: 4

Required course

Students will learn the basic theory of Branching Processes, with the strong emphasis on applications in biomedicine. Students will be able to recognize various biomedical examples that can be modelled via branching processes. They will learn how to set up the model and how to analyse it using its generating function, moments, branching mechanism, asymptotic behaviour, extinction probability. They will also learn how to interpret the elements of mathematical analysis of their models in biomedical environment.

After the completion of this course, students are expected to be able to:

- Recognize, analyse, and construct regular branching processes and branching processes with immigration and emigration.
- Apply generating functions to determine basic properties of branching processes.
- Calculate extinction probability for various types of branching processes and interpret it within the context of different biomedical environment.
- Determine asymptotic properties of various branching processes.
- Set up complex biomedical models that may include multiple branching processes in coordinated action within a single model.

- Introduction. Conditional Probability. Wald’s Identity. Generating Function.
- Galton-Watson Process. Extinction Probability. Examples (Last Names Extinction; DNA Haplogroups). Compound Poisson Process.
- Asymptotic Behaviour. Exponential Growth: Stochastic vs. Deterministic Models. Strong Convergence.
- Branching Processes and Markov Property. Limit Theorems.
- Branching Processes with Immigration and Emigration. Branching Processes with Multiple Types.
- Case Studies:
- DNA and Chromosomes
- Cell Cycles
- Modelling of the Eye Lens Growth. Organ Growth Implications
- Basic Epidemiological Models