This thesis details research I have carried out in the field of quantum walks, which are the quantum analogue of classical random walks. Quantum walks have been shown to offer a significant speed-up compared to classical random walks for certain tasks and for this reason there has been considerable interest in their use in algorithmic settings, as well as in experimental demonstrations of such phenomena. One of the most interesting developments in quantum walk research is their application to spatial searches, where one searches for a particular site of some network or lattice structure. There has been much work done on the creation of discrete- and continuous-time quantum walk search algorithms on various lattice types. However, it has remained an issue that continuous-time searches on two-dimensional lattices have required the inclusion of additional memory in order to be effective, memory which takes the form of extra internal degrees of freedom for the walker. In this work, we describe how the need for extra degrees of freedom can be negated by utilising a graphene lattice, demonstrating that a continuous-time quantum search in the experimentally relevant regime of two-dimensions is possible. This is achieved through alternative methods of marking a particular site to previous searches, creating a quantum search protocol at the Dirac point in graphene. We demonstrate that this search mechanism can also be adapted to allow state transfer across the lattice. These two processes offer new methods for channelling information across lattices between specific sites and supports the possibility of graphene devices which operate at a single-atom level. Recent experiments on microwave analogues of graphene that adapt these ideas, which we will detail, demonstrate the feasibility of realising the quantum search and transfer mechanisms on graphene.

The following thesis is divided in two main topics. The first part studies variations of optimal prediction problems introduced in Shiryaev, Zhou and Xu (2008) and Du Toit and Peskir (2009) to a randomized terminal-time set up and different families of utility measures. The work presents optimal stopping rules that apply under different criteria, introduces a numerical technique to build approximations of stopping boundaries for fixed terminal time problems and suggest previously reported stopping rules extend to certain generalizations of measures. The second part of the thesis is concerned with analysing optimal wealth allocation techniques within a defaultable financial market similar to Bielecki and Jang (2007). It studies a portfolio optimization problem combining a continuous time jump market and a defaultable security; and presents numerical solutions through the conversion into a Markov Decision Process and characterization of its value function as a unique fixed point to a contracting operator. This work analyses allocation strategies under several families of utilities functions, and highlights significant portfolio selection differences with previously reported results.

There has been considerable recent interest in models for epidemics on networks describing social contacts. This thesis considers a stochastic SIR (Susceptible - Infective - Removed) model for the spread of an epidemic among a population of individuals, with a random network of social contacts, that is partitioned into households and in which individuals also make casual contacts, i.e. with people chosen uniformly at random from the population. The behaviour of the model as the population tends to infinity is investigated. A threshold parameter that governs whether or not the epidemic with an initial infective can become established is obtained, as is the probability that such an outbreak occurs and, if so, how large it will become. The behaviour of this model is then compared to that of a finite population using Monte Carlo simulations. The effect of the different transmission routes on the final outcome of an epidemic and the effect of introducing social contacts and clustering to the network on the performance of various vaccination strategies are also investigated.

Ca2+ as a universal messenger participates in a great variety of physiological functions and biological events such as cell maturation, chemotaxis or gene expression. These diverse functions are controlled through complex spatio-temporal calcium patterns. To date it is known that these patterns depend on stimuli type and concentration. However, the majority of these observations were from constant or step change stimulation protocols. Under these conditions two leading hypotheses for the stimulus encoding into cytosolic calcium responses were proposed, namely amplitude and frequency modulation. Under physiological conditions, however, cells often experience time dependent stimuli such as transient changes in neurotransmitter or oscillations in hormone concentrations. How cells transduce such dynamic stimuli into an appropriate response is an open question. We exposed HEK293 cells and astrocytes to dynamically varying time courses of carbachol and ATP, respectively, and investigated the corresponding cellular calcium activity. While single cells generally fail to follow the applied stimulation due to their intrinsic stochasticity and heterogeneity, faithful signal reconstruction is observed at the population level. We suggest eight possible population representation measures and using mutual information measure show that the area under the curve and total number of spikes are the most informative ones. Next we provide simple transfer functions that explain how dynamic stimulation is encoded into area under the curve and ensemble calcium spike rates. Cells in a physiological environment often experience diverse stimulation time courses which can be reproduced experimentally. Furthermore, cell populations may differ in the number of cells or exhibit various spatial distributions. In order to understand how these conditions affect population responses, we compute the single cell response to a given dynamic stimulus. Single cell variability and the small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances and outperform standard tools such as peri-stimulus time histograms and kernel smoothing. Having the single cell response model will allow us to compare responses of various sets of cells to the observed population response and consequently obtain insight into tissue-wide calcium oscillations for heterogeneous cell populations. Finally,in vivo astrocytes respond to a range of hormones and neurotransmitters. Furthermore these agonists can have different characteristics, for example glutamate is a fast excitatory transmitter, while ATP can be an inhibitory transmitter. Despite of this, how (or if at all) astrocytes differentiate between different agonists is still not clear. We hypothesize that astrocytes discriminates between different stimuli by exploiting the spatial-temporal complexity of calcium responses. We show how 2D A Trous wavelet decomposition combined with Bhattacharyya distance measure can be applied to test this hypothesis.

The floral transition is a developmental process through which some plants commit to flowering and stop producing leaves. This is controlled by changes in gene expression in the shoot apical meristem (SAM). Many of the genes involved are known, but their interactions are usually only studied one by one, or in small sets. While it might be necessary to properly ascertain the existence of regulatory interactions from a biological standpoint, it cannot really provide insight in the functioning of the floral-transition process as a whole. For this reason, a modelling approach has been used to integrate knowledge from multiple studies. Several approaches were applied, starting with ordinary differential equation (ODE) models. It revealed in two cases – one on rice and one on Arabidopsis thaliana – that the currently available data were not sufficient to build data-driven ODE models. The main issues were the low temporal resolution of the time series, the low spatial resolution of the sampling methods used on meristematic tissue, and the lack of gene expression measurements in studies of factors affecting the floral transition. These issues made the available gene expression time series of little use to infer the regulatory mechanisms involved. Therefore, another approach based on qualitative data was investigated. It relies on data extracted from published in situ hybridization (ISH) studies, and Boolean modelling. The ISH data clearly showed that shoot apical meristems (SAM) are not homogeneous and contain multiple spatial domains corresponding to coexisting steady-states of the same regulatory network. Using genetic programming, Boolean models with the right steady-states were successfully generated. Finally, the third modelling approach builds upon one of the generated Boolean models and implements its logic into a 3D tissue of SAM. As Boolean models cannot represent quantitative spatio-temporal phenomena such as passive transport, the model had to be translated into ODEs. This model successfully reproduced the patterning of SAM genes in a static tissue structure. The main biological conclusions of this thesis are that the spatial organization of gene expression in the SAM is a crucial part of the floral transition and of the development of inflorescences, and it is mediated by the transport of mobile proteins and hormones. On the modelling front, this work shows that quantitative ODE models, despite their popularity, cannot be applied to all situations. When the data are insufficient, simpler approaches like Boolean models and ODE models with qualitatively selected parameters can provide suitable alternatives and facilitate large-scale explorations of the space of possible models, due to their low computational cost.

Deoxyribonucleic acid (DNA) is a long polymer consisting of two chains of bases, in which the genetic information is stored. A base from one chain has a corresponding base on the other chain which together form a so-called base-pair. Molecular-dynamics simulations of a normal DNA duplex show that breathing events – the temporary opening of one or more base-pairs – typically occur on the microsecond time-scale. Using the molecular dynamics package AMBER, we analyse, for different twist angles in the range 30-40 degrees of twist, a 12 basepair DNA duplex solvated in a water box, which contains the ’rogue’ base difluorotoluene (F) in place of a thymine base (T). This replacement makes breathing occur on the nanosecond time-scale. The time spent simulating such large systems, as well as the variation of breathing length and frequency with helical twist, determined us to create a simplified model, which is capable to predict with accuracy the DNA behaviour. Starting from a nonlinear Klein-Gordon lattice model and adding noise and damping to our system, we obtain a new mesoscopic model of the DNA duplex, close to that observed in experiments and all-atom MD simulations. Defects are considered in the inter-chain interactions as well as in the along-chain interactions. The system parameters are fitted to AMBER data using the maximum likelihood method. This model enables us to discuss the role of the fluctuation-dissipation relations in the derivation of reduced (mesoscopic) models, the differences between the potential of mean force and the potential energies used in Klein-Gordon lattices and how breathing can be viewed as competition between the along-chain elastic energy, the inter-chain binding energy and the entropy term of the system’s free energy. Using traditional analysis methods, such as principal component analysis, data autocorrelation, normal modes and Fourier transform, we compare the AMBER and SDE simulations to emphasize the strength of the proposed model. In addition, the Fourier transform of the trajectory of the A-F base-pair suggests that DNA is a self-organised system and our SDE model is also capable of preserving this behaviour. However, we reach the conclusion that the critical DNA behaviour needs further investigations, since it might offer some information about bubble nucleation and growth and even about DNA transcription and replication.

In this thesis several problems concerning the stochastic modelling of emerging infections are considered. Mathematical modelling is often the only available method of predicting the extent of an emerging disease and assessing proposed control measures, as there may be little or no available data on previous outbreaks. Only stochastic models capture the inherent randomness in disease transmission observed in real-life outbreaks, which can strongly influence the outcome of an emerging epidemic because case numbers will initially be small compared with the population size. Chapter 2 considers a model for diseases in which some of the cases exhibit no symptoms and are therefore difficult to observe. Examples of such diseases include influenza, mumps and polio. This chapter investigates the problem of determining whether or not the epidemic has died out if a period containing no symptomatic individuals is observed. When modelling interventions, it is realistic to include a delay between observing the presence of infection and the implementation of control measures. Chapter 3 quantifies the effect that the length of such a delay has on an epidemic amongst a population divided into households. As well as a constant delay, an exponentially distributed delay is also considered. Chapter 4 develops a model for the spread of an emerging strain of influenza in humans. By considering the probability that an outbreak will be contained within a region in which an intervention strategy is active, it becomes possible to quantify and therefore compare the effectiveness of intervention strategies.

This thesis considers mathematical models of the interaction between social and territorial behaviour in animals, mainly by probabilistic methods. Chapter 1 introduces the Resource Dispersion Hypothesis, which suggests that territorial behaviour plus dispersed food resources can explain the existence of social groups, and describes an existing model of the process, due to Carr and Macdonald. In Chapter 2 the model of Carr and Macdonald is analysed, and in Chapter 3 an improved model is suggested and its main properties derived, primarily using renewal theory. Chapters 4 and 5 consider various spatial models for territory formation, and the effect, of spatial factors on social behaviour, using analytic and simulation-based methods. Chapter 6 considers the evolution of social behaviour using both discrete-time deterministic models and branching processes to investigate the viability of different strategies of social behaviour in the presence of dispersed resources.

This thesis mainly focuses on the statistical modelling of a selection of games, namely, the minority game, the urn model and the Hawk-Dove game. Chapters 1 and 2 give a brief introduction and survey of the field. In Chapter 3, the key characteristics of the minority game are reproduced. In addition, the minority game is extended to include wealth distribution and leverage effect. By assuming that each player has initial wealth which rises and falls according to profit and loss, with the potential of borrowing and bankruptcy, we find that modelled wealth distribution may be power law distributed and leverage increases the instability of the system. In Chapter 4, to explore the effects of memory, we construct a model where agents with memories of different lengths compete for finite resources. Using analytical and numerical approaches, our research demonstrates that an instability exists at a critical memory length; and players with different memory lengths are able to compete with each other and achieve a state of co-existence. The analytical solution is found to be connected to the well-known urn model. Additionally, our findings reveal that the temperature is related to the agent's memory. Due to its general nature, this memory model could potentially be relevant for a variety of other game models. In Chapter 5, our main finding is extended to the Hawk-Dove game, by introducing the memory parameter to each agent playing the game. An assumption is made that agents try to maximise their profits by learning from past experiences, stored in their finite memories. We show that the analytical results obtained from these two games are in agreement with the results from our simulations. It is concluded that the instability occurs when agents' memory lengths reach the critical value. Finally, Chapter 6 provides some concluding remarks and outlines some potential future work.

We develop several new algorithms using molecular simulation to investigate the nucleation barrier of a single, freely-jointed polymer chain. In the first part of the thesis, we use a free particle model to develop a new biasing technique, which uses an automated feedback mechanism to overcome the poor sampling of crystal states in a thermodynamic system. Our feedback technique does not require any prior knowledge of the nucleation barrier and enables good representative sampling of all available states of interest. In the second part of the thesis, we simulate the nucleation barrier of the single, freely-jointed, square-well chain. We use our feedback technique and parallel tempering with a nonstandard temperature distribution to overcome poor sampling of crystal states and configuration mixing issues respectively. We also provide some comparative analysis of different choices of configurational order parameters for the single chain. Finally, we apply stretching to the chain to approximate flow-induced crystallisation and investigate the effect of different degrees of stretch on the nucleation barrier. We verify the quality of our simulation with careful monitoring of several criteria, including the acceptance ratios of configuration swaps between simulations with adjacent temperatures, evolution of the energy traces as a result of configuration swaps between tempering levels, and ensuring effective de-correlation of configurations through reptation moves. Our simulations provide strong reproducible results for the base, the peak and beyond the peak of the barrier for the quiescent and stretched single chain. We observe a remarkably strong effect of modest stretching on the nucleation barrier for a single chain, which can potentially lead to dramatic effects on the nucleation rate. Our simulation code has been made publicly available, with details provided in an appendix.