Probability generating functions for computational epidemiology

Probability generating functions are extremely useful to characterize complex combinatorial objects which arise in various fields, such as network science or the study of epidemiological processes.

Here we a list a few references that might be of interest in combination with this tutorial:

• A primer on the use of probability generating functions in infectious disease modeling [Miller, 2018]

• generatingfunctionology [Wilf, 2005]. This is a book on the theory of probability generating functions including various applications in discrete mathematics.

Many theoretical results can be obtained using probability generating functions—critical points of phase transition, asymptotic behavior of functions, and the moments of distributions of interest to name a few.

Here, our main goal is to provide a hands-on tutorial on computational methods based on probability generating functions. Indeed, numerical approximations are often more flexible when trying to to tackle real world scenarios. Our main applications will be in epidemiology, on how to characterize contagion on networks or in complex population by this set of tools.

Table of content

PGF 101

• Definition and properties
• Numerical inversion
• Multivariate PGF

Applications

• Branching process
• Contagion on networks
• Percolation on hypergraphs

Exercises

• Part I
• Part II: branching process
• Part III: statistical inference

References

Mil18

Joel C. Miller. A primer on the use of probability generating functions in infectious disease modeling. Infectious Disease Modelling, 3:192–248, 2018. doi:10.1016/j.idm.2018.08.001.

Wil05

Herbert S. Wilf. generatingfunctionology. CRC press, 2005.