Mathematics thesis and dissertation collection
http://hdl.handle.net/1842/2284
2019-09-23T15:24:20ZMutation frequencies in a birth-death branching process
http://hdl.handle.net/1842/36163
Mutation frequencies in a birth-death branching process
Cheek, David Michael
A growing population of cells accumulates genetic mutations. We study stochastic
models of this process. Cells divide and die as a branching process, and a
cell's genetic information is a sequence of nucleotides which mutates randomly at
division. Motivated by biologically realistic parameters, we consider that few cells
grow to many cells and mutation rates are small, proving approximations in this
limit. In particular we are interested in mutation frequencies and their dependency
structure along the genetic sequence; the relevance of the evolutionary tree
and selection are discussed. Amongst other results, we recover a power-law distribution
for mutation frequencies which is consistent with previously published
cancer genetic data.
2019-08-20T00:00:00ZCancer recurrence times and early detection from branching process models
http://hdl.handle.net/1842/36158
Cancer recurrence times and early detection from branching process models
Avanzini, Stefano
Cancer is among the leading causes of death worldwide. While primary tumors
are often treated effectively, they can spawn secondary cancers called metastases
which dramatically decrease chances of survival. In order to develop successful
therapies, it is thus crucial to estimate the time until metastases appearance
and improve our ability to detect primary tumors before metastases are generated.
The estimation of the time to cancer recurrence depends on the dynamics
of tumor growth and metastases seeding. For early detection, promising results
have recently been obtained with liquid biopsies, id est the analysis of specific
biomarker levels in blood samples. This thesis investigates these problems by
studying mathematical models of cancer evolution and liquid biopsies based on
the theory of branching processes.
Firstly, we consider first passage times to a given size in branching birth-death
processes. We derive their probability distribution and first moments conditioned
on non-extinction, comparing the results obtained for supercritical, critical and
subcritical processes. Such results for hitting times are presented both in exact
form and in their asymptotic limit for large sizes. In this limit we show that their
probability distribution asymptotically converges to extreme value types.
Second, we present a semi-stochastic model of cancer recurrence. The primary
tumor is described by a deterministically growing population of cells initiating
metastases at a rate proportional to its size. Each metastasis is then modelled
by a branching birth-death process with the same net growth rate. In this framework
we discuss several features of the time to cancer relapse, defined as the first
time that any metastasis reaches a given detectable size. We apply this model
to different cancer types and compare its predictions with data collected from
clinical literature.
Third, we present a multi-type branching process model of biomarker shedding.
We focus on the case of circulating tumor DNA fragments shed in the
bloodstream by both cancerous and healthy cells. We model the population of
tumor cells as a supercritical branching birth-death process and take the healthy
cells population to be constant in size. As DNA fragments cannot reproduce or
divide, their amount is described by a pure death process with immigration. By
applying this model, we provide quantitative estimates for the number of circulating
tumor DNA fragments detectable in a blood sample, conditioned on the
primary tumor size. Comparing our estimates with clinical observations we then
discuss the potential of liquid biopsies for early cancer detection.
2019-11-28T00:00:00ZStochastic dispersive PDEs with additive space-time white noise
http://hdl.handle.net/1842/36113
Stochastic dispersive PDEs with additive space-time white noise
Tolomeo, Leonardo
In this thesis, we will discuss the Cauchy problem for some nonlinear dispersive PDEs with
additive space-time white noise forcing. We will focus on two different models: the stochastic
nonlinear beam equation (SNLB) with power nonlinearity posed on the three dimensional torus,
and the stochastic nonlinear wave equation with cubic nonlinearity in two dimensions, posed
both on the torus and on the Euclidean space (SNLW).
For (SNLB), we will present a joint work with R. Mosincat, O. Pocovnicu and Y. Wang,
which settles local well-posedness for every nonlinearity of the type |u|p-1u, and global welllposedness
for p < 11=3. In the case p = 3, we also consider a damped version of the equation,
for which we can show invariance of the Gibbs measure. Moreover, we describe the long time-behaviour
of the flow, by showing unique ergodicity of the Gibbs measure, and convergence to
equilibrium for smooth initial data.
In the case of (SNLW) with cubic nonlinearity, we consider a renormalised version of the
equation, which was introduced by M. Gubinelli, H. Koch and T. Oh. In their work, they
established local well-posedness on the two-dimensional torus. We show global existence for
these solutions (joint with M. Gubinelli, H. Koch and T. Oh, and local and global wellposedeness
for the same equation posed on the two-dimensional Euclidean space.
2019-11-28T00:00:00ZQuantitative propagation of chaos of McKean-Vlasov equations via the master equation
http://hdl.handle.net/1842/36096
Quantitative propagation of chaos of McKean-Vlasov equations via the master equation
Tse, Alvin Tsz Ho
McKean-Vlasov stochastic differential equations (MVSDEs) are ubiquitous in kinetic theory
and in controlled games with a large number of players. They have been intensively studied
since McKean, as they pave a way to probabilistic representations for many important nonlinear/
nonlocal PDEs. Classically, their simulation involves using standard particle systems,
which replace the evolving law in MVSDEs by the evolving empirical measure of the particles.
However, this type of simulation is costly in terms of computational complexity, due to the
interaction between the particles.
Apart from classical techniques in stochastic analysis, the approach in this thesis relies
heavily on the calculus on Wasserstein space, presented by P. Lions in his course at Collège de
France. An important object in our study, is a PDE written on the product space of the space
of time horizon and the Wasserstein space, which is a generalisation of the classical Feynman-
Kac PDE. This PDE, namely the master equation, provides a new insight into the study of
mean-field limits of particles and consequently allows us to solve many problems on MVSDEs
that are very difficult/impossible to solve by classical techniques.
The layout of the thesis is as follows. We start by a recap on classical results of MVSDEs
(Chapter 2), followed by a full exposition of Wasserstein calculus on the results that we need
(Chapter 3). Chapters 4 and 5 propose approximating systems to MVSDEs (as alternatives to
the classical particle system) via Romberg extrapolation and Antithetic Multi-level Monte-Carlo
estimation respectively, which are less costly in terms of computational complexity. Finally, in
Chapter 6, we explore the converse: given a standard particle system, we hope to find an
alternative mean-field limit that gives a better approximation to the standard particle system.
2019-11-28T00:00:00Z