Applications of Probability Generating Functions to Biological Systems

Lecturer

A. Prof. Joel Miller – La Trobe University

Synopsis

Probability Generating Functions are commonly used to study birth-death processes. These appear across many biological systems, including infectious disease spread, within-host cancer or pathogen dynamics, evolutionary accumulation of beneficial/harmful mutations and the successful (or unsuccessful) establishment of an invasive species.

The subject will use Python. The purpose is to help illustrate concepts, not to test coding ability. Previous programming experience would be helpful but the material will be designed assuming no previous experience. Much of the code required will be provided through examples.

Course Overview

The material will be presented in a mixture of lectures and computer labs. Topics to be covered are described below. Some of the suggested Python topics may instead appear as options for the group project.

Week 1: Fundamentals and discrete-time branching processes:

  • Ordinary and exponential generating functions, radius of convergence, proving basic combinatorial identities (binomial, Catalan, Fibonacci).
  • Probability-generating functions (PGFs): interpretation of products (independent sums) and composition (mixed offspring), Sicherman
    dice as a worked example.
  • Important PGFs: Poisson, Binomial, Geometric, Negative Binomial.
  • Discrete-time branching processes (Galton–Watson), extinction probability, reproduction number,
  • Final cumulative size via coefficients of [G(z)]^j
  • Disease spread in small populations – Ball matrices.
  • Sizes near criticality.
  • Python: Jupyter-notebook primer, numpy, symbolic algebra, plotting.
  • Python: Simulating a Galton-Watson process. Compare simulations with analytic predictions. Inferring input parameters from simulated data.

Week 2:  Temporal analysis of Birth–death processes:

  • Discrete-time branching processes: relation to function composition.
  • Mean size E[N_n] = G'(1)^n, extinction probability by generation n, expected size conditional on non-extinction and implications for disease
    control policies.
  • Continuous-time birth–death processes and analogous results to discrete-time. The Malthusian parameter.
  • Forward and Backward Kolmogorov Equations.
  • PGF inversion techniques — saddle-point/Lugannani–Rice approximation and FFT coefficient extraction.
  • Python: Comparison of discrete-time simulation with predictions.
  • Python: Gillespie simulation for continuous-time model. Comparison with prediction.
  • Python: Implement saddle-point and FFT inversion, and assess inversion accuracy.

Week 3:  Multi-type branching and evolutionary genetics:

  • Multi-type branching processes, vector PGFs and the Jacobian threshold (Perron–Frobenius), Reproduction numbers
  • Application of multi-type branching processes to stem-cell differentiation and to joint cumulative size/current size distribution of single-type branching processes.
  • Muller’s ratchet.
  • Wright–Fisher recursion.
  • Python: estimate mutation rates from simulated ratchet data.
  • Python: Optimisation of offspring distribution (e.g., malaria merozoites per red blood cell)

Week 4: Network epidemics and synthesis:

  • Edge-based compartmental models (EBCMs) for SIR disease, degree-distribution PGFs G0, G1, reproduction numbers on networks.
  • Watts threshold model via PGFs, cascade conditions
  • Numerical toolbox wrap-up: automatic differentiation through PGF pipelines, Fisher information, comprehensive review and integration
    of course concepts.
  • Python: Simulate disease and Watts model with built-in methods of NetworkX and EoN, compare with PGF predictions.
  • Presentations

Each week will consist of three 2-hour classes and one 1-hour class. The longer 2hour classes will combine theoretical lectures and computer-based lab activities. The 1-hour class will primarily focus on computer-based labs, but it will also be an opportunity to review the lecture material. Some time will be provided in class for project work.

Prerequisites

This subject will draw on a wide range of topics, but does not require in-depth knowledge about any specific topic. For example, it is likely that students will have already seen all required material from Probability & Statistics, even if they have never taken a class focused on probability. See pre-enrollment quiz for details.

  • Calculus
  • Probability & Statistics

Assessment

  • 2x written assignments (due weeks 2 & 3) – 15% each
    • Depending on class size, we may have brief one-on-one discussions after submission
  • 1x group project applying methods from the subject to a biological problem with presentation and written report (due week 4) – 30%
  • 2 hour invigilated exam (plus 15 mins reading time) – 40%

Resources/pre-reading

  • Lecture notes for the subject will be available through links on the moodle website closer to the start date.
  • Other suggested reading materials (may be useful depending on group project choice):
    • A primer on the use of probability generating functions in infectious disease modeling. Joel C. Miller. I recommend the version on Arxiv https://arxiv.org/pdf/1803.05136 which has fewer typos than the official version.
    • Stem Cell Proliferation Catherine A Macken & Alan S. Perelson (particularly chapter 2. You can probably get a pdf of this online through your university).
    • The Accumulation of Deleterious Genese in a Population – Muller’s Ratchet. John Haigh.
    • PGFunk: https://cosmo-notes.github.io/pgfunk/chapters/index.html Guillaume St-Onge
    • generatingfunctionology. Herbert Wilf.
  • A working Python installation on your own machine or access to a university machine with Python. I expect to sort out details so that you can use
    python through any web browser using a remote server, but direct access to python is recommended.

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