Probability generating functions for epidemiological time series
Abstract
Probability generating functions (PGFs) are a fundamental tool in mathematical epidemiology to characterize the stochastic evolution of epidemics via branching process theory. However, they are rarely used as operational tools in real-time during an outbreak, the main barrier being a lack of a systematic, computational methodology to describe epidemiological time series, such as cases or hospital admissions over time. Here, we introduce a general and computationally efficient framework for outbreak analytics based on PGFs. We first develop a method to encode joint distributions of incidences and their noisy measurements by realistic surveillance systems. Then, by transforming PGFs to cumulant generating functions (CGFs), we enable efficient evaluation of these joint distributions. We demonstrate the practicality of our approach for Bayesian inference and short-term forecasts, offering a significant computational advantage over simulation-based methods. Altogether, our framework operationalizes generating functions and paves the way for their broader adoption in computational epidemiology.
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BibTeX
@misc{boudreau2026probability,
title = {{Probability generating functions for epidemiological time series}},
author = {Mariah C. Boudreau and Guillaume St-Onge},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17138},
eprint = {2610.17138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17138},
}
RIS
TY - UNPB
TI - Probability generating functions for epidemiological time series
AU - Mariah C. Boudreau
AU - Guillaume St-Onge
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17138
UR - https://arxiv.org/abs/2610.17138
AB - Probability generating functions (PGFs) are a fundamental tool in mathematical epidemiology to characterize the stochastic evolution of epidemics via branching process theory. However, they are rarely used as operational tools in real-time during an outbreak, the main barrier being a lack of a systematic, computational methodology to describe epidemiological time series, such as cases or hospital admissions over time. Here, we introduce a general and computationally efficient framework for outbreak analytics based on PGFs. We first develop a method to encode joint distributions of incidences and their noisy measurements by realistic surveillance systems. Then, by transforming PGFs to cumulant generating functions (CGFs), we enable efficient evaluation of these joint distributions. We demonstrate the practicality of our approach for Bayesian inference and short-term forecasts, offering a significant computational advantage over simulation-based methods. Altogether, our framework operationalizes generating functions and paves the way for their broader adoption in computational epidemiology.
ER -