Joel C. Miller

# 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)

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

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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.

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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.

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