Studies on Branching Processes: Time-Varying Crump-Mode-Jagers models with applications to epidemiology and a law of large number for structured bran

We first study a time-varying extension of a generalised branching process commonly called Crump-Mode-Jagers (CMJ) process in the literature. With a focus on the derivation of integral equations related to processes counted by random characteristics, we connect this time-varying model CMJ process to the development of an infectious disease, and in this context, show that the classical incidence renewal equation commonly applied in epidemiology is thus grounded on strong mathematical foundations.
Then we treat of a structured branching population where individuals are assumed to behave independently and are characterised by a trait that follows Markovian dynamics. In the supercritical case, under a new set of conditions with respect to the non-uniform convergence of the first moment semi group and an 'L log L' condition, we provide a general Kesten-Stigum theorem and unify part of the literature. We prove L1 and almost sure convergence of a super-critical structured branching process towards its martingale limit. Our new set of assumptions, in particular, has the advantage to be more easily verified than the conditions of the classical result of Asmussen and Hering (1976) which required uniform convergence.

The primary CDT impact will be training 75 PhD graduates as the next generation of leaders in statistics and statistical machine learning. These graduates will lead in industry, government, health care, and academic research. They will bridge the gap between academia and industry, resulting in significant knowledge transfer to both established and start-up companies. Because this cohort will also learn to mentor other researchers, the CDT will ultimately address a UK-wide skills gap. The students will also be crucial in keeping the UK at the forefront of methodological research in statistics and machine learning.
After graduating, students will act as multipliers, educating others in advanced methodology throughout their career. There are a range of further impacts:
- The CDT has a large number of high calibre external partners in government, health care, industry and science. These partnerships will catalyse immediate knowledge transfer, bringing cutting edge methodology to a large number of areas. Knowledge transfer will also be achieved through internships/placements of our students with users of statistics and machine learning.
- Our Women in Mathematics and Statistics summer programme is aimed at students who could go on to apply for a PhD. This programme will inspire the next generation of statisticians and also provide excellent leadership training for the CDT students.
- The students will develop new methodology and theory in the domains of statistics and statistical machine learning. It will be relevant research, addressing the key questions behind real world problems. The research will be published in the best possible statistics journals and machine learning conferences and will be made available online. To maximize reproducibility and replicability, source code and replication files will be made available as open source software or, when relevant to an industrial collaboration, held as a patent or software copyright.