A primer on the use of probability generating functions in infectious disease modeling

TL;DR

This paper introduces probability generating functions (PGFs) as a powerful tool for modeling early outbreak dynamics of infectious diseases in large populations, providing practical methods and a Python package for researchers.

Contribution

It offers a comprehensive tutorial on applying PGFs to infectious disease modeling, including derivations, applications, and a Python package, with minimal new theoretical results.

Findings

PGFs predict epidemic probability and outbreak size distributions.

PGFs simplify SIR models with fewer ODEs in large populations.

The paper provides practical tools and exercises for applying PGFs.

Abstract

We explore the application of probability generating functions (PGFs) to invasive processes, focusing on infectious disease introduced into large populations. Our goal is to acquaint the reader with applications of PGFs, moreso than to derive new results. PGFs help predict a number of properties about early outbreak behavior while the population is still effectively infinite, including the probability of an epidemic, the size distribution after some number of generations, and the cumulative size distribution of non-epidemic outbreaks. We show how PGFs can be used in both discrete-time and continuous-time settings, and discuss how to use these results to infer disease parameters from observed outbreaks. In the large population limit for susceptible-infected-recovered (SIR) epidemics PGFs lead to survival-function based models that are equivalent the the usual mass-action SIR models but with fewer ODEs. We use these to explore properties such as the final size of epidemics or even the dynamics once stochastic effects are negligible. We target this tutorial to biologists and public health researchers who want to learn how to apply PGFs to invasive diseases, but it could also be used in an introductory mathematics course on PGFs. We include many exercises to help demonstrate concepts and to give practice applying the results. We summarize our main results in a few tables. Additionally we provide a small python package which performs many of the relevant calculations.