Applications of Probability Generating Functions to Biological Systems

Lecturer

A. Prof. Joel Miller – La Trobe University

Synopsis

Many birth-death processes can be modelled using probability generating functions.  These have been applied to models of species/strain diversification, infection spread, invasive species, and tumour growth.

This subject will be based on my paper A primer on the use of probability generating functions in infectious disease modeling – ScienceDirect. J Infectious Disease Modelling, Vol 3, 2018, pages 192-248.

Course Overview

Week 1: Introduction

  • Introduction to birth death processes.
  • Properties of Probability Generating Functions, and isomorphisms with birth-death processes. Sicherman Dice

Week 2:  Basic modelling approaches

  • Establishment probability and early growth rates: Discrete Time and Continuous Time.
  • Kolmogorov Equations (primarily the Forward Kolmogorov Equations)

Week 3:  Applications

  • Infectious Disease Models: SIS (primarily early growth and establishment probability)
  • Infectious Disease Models: SIR (same as SIS plus models of long-term dynamics and outbreak sizes)

Week 4: 

  • Tumour/Virus strains: diversification and accumulation of advantageous/deleterious mutations.
  • Review

Prerequisites

  • Linear Algebra (good understanding of eigenvalues and eigenvectors)
  • Differential Equations (Ability to relate differential equations to rates of increase and decrease of quantities, no need to have detailed knowledge of solution techniques)
  • Probability (A single semester subject would suffice)

Assessment

TBA

Attendance requirements

TBA

Resources/pre-reading

TBA

Not sure if you should sign up for this course?

Check back for pre-enrolment QUIZ details so you can self-evaluate and get a measure of the key foundational knowledge required.

Dr Joel Miller, La Trobe University