Counting processes are an important class of stochastic processes with numerous interdisciplinary applications, including to evolving biological systems. Counting processes can be the basis for forward generative models, which can help develop a mechanistic understanding of a phenomenon, or for inferential methods.

In this thesis we develop a bottom-up microscopic model of a phenomenon in mitochondrial biology and ageing. Mitochondria are organelles that possess their own DNA, involved in crucial physiological functions. A typical eukaryotic cell hosts a population of mitochondrial DNA molecules, where mutations can expand and cause dysfunctions. With age, skeletal muscle suffers a reduction in strength and functionality; in mammals, this has been connected to the clonal expansion of mitochondrial deletion mutations. The mechanism driving this phenomenon remains poorly understood despite intense research.

We develop a stochastic population dynamics model corresponding to a novel evolutionary mechanism, termed stochastic survival of the densest, and we show that it can account for the expansion of mitochondrial mutations in skeletal muscle through a literature-parameterised model. We predict that a species can invade a system in a wave-like fashion, without having an explicit replicative advantage and even if preferentially eliminated. We establish that this mechanism is driven by the combined effect of stochasticity, differences in carrying capacity (or density) and spatial structure.

Part of the work for this thesis coincided with the COVID-19 pandemic. We look at another application of counting processes, modelling data collected within the SIREN study, part of the national response to the pandemic. We estimate parameters of counting processes modelling SARS‑CoV‑2 infection events, that correspond to the reduction in rate of infection associated with COVID-19 vaccination (vaccine effectiveness), with a previous infection, or both (hybrid protection). We highlight short-term vaccine effectiveness that subsequently wanes, and confirm the immune escape of the Omicron variant.