Branching Processes with Detection: Probabilistic Analysis
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-studied aspect of branching processes by integrating a detection mechanism, wherein each individual in a population has a probability of being detected. This augmentation offers new insights for applications in fields such as infectious disease research. We study four models in this thesis: the discrete and continuous-time Galton--Watson processes, and the discrete and continuous-time multi-type branching processes. For each of these 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 specific questions, for example, in the case of continuous-time branching processes, we obtain an explicit generating function expression for the cluster size at the time of first detection, and we apply a coupling technique in the context of multi-type branching processes. This study extends the theoretical framework of branching processes and highlights their practical relevance to real-world scenarios.
Branching Process, Detection, Galton Watson Process, Probability Theory
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