  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.
