Probability Generating Functions

A Probability Generating Function (PGF) is a function of the form

p_0 x^0 + p_1 x^1 + p_2 x^2 + . . . (I hope to sort out math text better in the future)

where we can think of p_n as the probability of n events happening.

These have many applications in infectious disease, where usually we interpret the event as being a transmission.

A few publications

Here are a few examples of where I have used PGFs in my research.

PGF Methodology

I wrote a 60+ page paper describing how to apply PGFs to infectious disease modelling. The paper is really a mini-book, complete with exercises and appendics. It was initially intended to be a short 4-page internal note at the Institute for Disease Modeling (where I was at the time) explaining how to use PGFs to predict epidemic probability. But it kept expanding.

Edge-based Compartmental Models

The EBCM (edge-based compartmental model) approach that I developed based on a paper by Erik Volz uses PGFs to reduce the number of equations needed to model a heterogeneous population to just a handful. Without PGFs,, the number of equations we would need would be proportional to the number of degrees observed in the population.

On the 'publications' page, references 23, 24, 26, 32, and 33 develop this. I recommend starting with reference 26.

Understanding Super-Spreading

Early in an epidemic, there is a big difference between an epidemic where every individual causes 2 infections or where 1% would cause 200 infections and 99% would cause none, even though in both cases the average is the same. An epidemic is guaranteed from one infection in the first case, and unlikely in the second.

PGFs allow us to study the possible different outcomes. We applied this to understanding the early spread of COVID-19