New Zealand Statistical Association 2024 Conference

Zehua Zang

University of Auckland

Branching processes with detection: Probabilistic analysis

This is joint work with Jesse Goodman, Simon Harris

The branching process is a stochastic model that describes the evolution of a population, where the offspring of each individual are produced independently of others and the past. This thesis explores a less explored aspect of branching processes by integrating a detection mechanism wherein each individual in a population has a probability of being detected. This augmentation provides insights for applications in fields such as infectious disease research.

We study four models in this thesis: discrete and continuous-time Galton-Watson processes and discrete and continuous-time multi-type branching processes. For these four models, we examine the distribution of the first detection time, establish limit theorems, and analyse the asymptotic behaviour of the detected processes. Our analysis also addresses questions, such as in the case of continuous-time branching processes, we derive an explicit generating function expression for the cluster size at the first detection and the application of a coupling technique in multi-type branching processes.

This study extends the theoretical framework of branching processes and emphasises their practical applications in real-world scenarios.