Дисертації з теми "Mathematical models"

Chen, Donoho a Saunders (1998) studují problematiku hledání řídké reprezentace vektorů (signálů) s použitím speciálních přeurčených systémů vektorů vyplňujících prostor signálu. Takovéto systémy (někdy jsou také nazývány frejmy) jsou typicky vytvořeny buď rozšířením existující báze, nebo sloučením různých bazí. Narozdíl od vektorů, které tvoří konečně rozměrné prostory, může být problém formulován i obecněji v rámci nekonečně rozměrných separabilních Hilbertových prostorů (Veselý, 2002b; Christensen, 2003). Tento funkcionální přístup nám umožňuje nacházet v těchto prostorech přesnější reprezentace objektů, které, na rozdíl od vektorů, nejsou diskrétní. V této disertační práci se zabývám hledáním řídkých representací v přeurčených modelech časových řad náhodných veličin s konečnými druhými momenty. Numerická studie zachycuje výhody a omezení tohoto přístupu aplikovaného na zobecněné lineární modely a na vícerozměrné ARMA modely. Analýzou mnoha numerických simulací i modelů reálných procesů můžeme říci, že tyto metody spolehlivě identifikují parametry blízké nule, a tak nám umožňují redukovat původně špatně podmíněný přeparametrizovaný model. Tímto významně redukují počet odhadovaných parametrů. V konečném důsledku se tak nemusíme starat o řády modelů, jejichž zjišťování je většinou předběžným krokem standardních technik. Pro kratší časové řady (100 a méně vzorků) řídké odhady dávají lepší predikce v porovnání s těmi, které jsou založené na standardních metodách (např. maximální věrohodnosti v MATLABu - MATLAB System Identification Toolbox (IDENT)). Pro delší časové řady (500 a více) obě techniky dávají v podstatě stejně přesné predikce. Na druhou stranu řešení těchto problémů je náročnější, a to i časově, nicméně výpočetní doba je stále přijatelná.

In questa tesi vengono innanzitutto presentati due metodi matematici per lo studio di modelli biomedici e comportamentali. I modelli presentati sono tre: un modello per lo studio dell'evoluzione della malattia di Alzheimer, uno per lo studio dello sviluppo dei tumori e uno per la diffusione del Covid-19. Si riportano anche alcuni codici utilizzati per lo studio e lo sviluppo dei modelli trattati. Le conclusioni contengono alcuni possibili sviluppi degli argomenti trattati.

A cross-linking model for the activation of the A cell or immune accessory cell as a function of certain extracellular conditions is developed to determine the valency of the specific factor receptor on the A cell surface. It is found that such a determination can be made based on the FWHM of cross-linking curves which differ by a full order of magnitude between the bivalent receptor case and the monovalent receptor case. This determination can be made provided one can obtain accurate values for the equilibrium constants which characterize the system and provided that activation and IL-1 secretion is a linear function of cross-linking. It is also found that a determination of valence can be made if the equilibrium constants are such that substantial one receptor bridge formation takes place (one antibody molecule bound on both ends by the same receptor). This one-receptor bridge formation only takes place if the receptor is bivalent, and it presents itself in the cross-linking curve in a very distinctive manner. A second network model described as an ecological competition model of steady state lymphocyte populations is presented. This model, known as the symmetrical network theory is analysed numerically by integration of the differential equations and shown to provide a reasonable qualitative picture of the immune system's stable steady states, and offer a glimpse of state switching.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

Major efforts to sequence cancer genomes are now occurring throughout the world. Though the emerging data from these studies are illuminating, their reconciliation with epidemiologic and clinical observations poses a major challenge. Here we present mathematical models that begin to address this challenge. First we present a model of accumulation of driver and passenger mutations during tumor progression and derive a formula for the number of driver mutations as a function of the total number of mutations in a tumor. Fitting this formula to recent experimental data, we were able to calculate the selective advantage provided by a typical driver mutation. Second, we performed a quantitative analysis of pancreatic cancer metastasis genetic data. The results of this analysis define a broad time window for detection of pancreatic cancer before metastatic dissemination. Finally, we model the evolution of resistance to targeted cancer therapy. We apply our model to experimental data on the response to panitumumab, targeted therapy against colorectal cancer. Our modeling suggested that cells resistant to therapy were likely present in patients’ tumors prior to the start of therapy.
Mathematics

The phenomenon of kleptoparasitism - "food-stealing" - has frequently been observed, in a wide range of animal species. In this thesis, I extend the game-theoretic model of kleptoparasitism, proposed by Broom and Ruxton 1998, in a number of ways. Firstly, using their model, I investigate how quickly the equilibrium state of a kleptoparasitic population is reached. This work has been published (Luther and Broom 2004). I then investigate the case of a single homogenous population of kleptoparasites, finding which behaviours are Evolutionarily Stable Strategies. This is done with a variable probability that a challenger succeeds when attempting to steal food from a handler, and also allowing the possibility that the handler does not resist the attack. This work has been published (Broom et al 2004) I then consider populations of two groups, one stealing and the other only foraging, to find ESS's, particularly looking at situations where a mixed population can be an ESS, and other cases where pure populations are an ESS. I do this for indistinguishable groups, and then distinguishable groups. I show that a homogenous facultative population behaving in the Broom and Ruxton 1998 ESS has the same handling ratio as a mixed obligate population of kleptoparasites and foragers. Finally, I discuss some ornithological data on kleptoparasitism, and make a simple comparison with our models, to see if they are an accurate representation of the actual phenomenon

Cellular biophysics deals with, among other things, transport processes within cells. This thesis presents two studies where mathematical models have been used to explain how two of these processes occur.

Cellular membranes separate cells from their exterior environment and also divide a cell into several subcellular regions. Since the 1970s lateral diffusion in these membranes has been studied, one the most important experimental techniques in these studies is fluorescence recovery after photobleach (FRAP). A mathematical model developed in this thesis describes how dopamine 1 receptors (D1R) diffuse in a neuronal dendritic membrane. Analytical and numerical methods have been used to solve the partial differential equations that are expressed in the model. The choice of method depends mostly on the complexity of the geometry in the model.

Calcium ions (Ca²⁺) are known to be involved in several intracellular signaling mechanisms. One interesting concept within this field is a signaling microdomain where the inositol 1,4,5-triphosphate receptor (IP₃R) in the endoplasmic reticulum (ER) membrane physically interacts with plasma membrane proteins. This microdomain has been shown to cause the intracellular Ca²⁺ level to oscillate. The second model in this thesis describes a signaling network involving both ER membrane bound and plasma membrane Ca²⁺ channels and pumps, among them store-operated Ca²⁺ (SOC) channels. A MATLAB^® toolbox was developed to implement the signaling networks and simulate its properties. This model was also implemented using Virtual cell.

The results show a high resemblance between the mathematical model and FRAP data in the D1R study. The model shows a distinct difference in recovery characteristics of simulated FRAP experiments on whole dendrites and dendritic spines, due to differences in geometry. The model can also explain trapping of D1R in dendritic spines.

The results of the Ca²⁺ signaling model show that stimulation of IP3R can cause Ca²⁺ oscillations in the same frequency range as has been seen in experiments. The removing of SOC channels from the model can alter the characteristics as well as qualitative appearance of Ca²⁺ oscillations.

Cellulär biofysik behandlar bland annat transportprocesser i celler. I denna avhandling presenteras två studier där matematiska modeller har använts för att förklara hur två av dess processer uppkommer.

Cellmembran separerar celler från deras yttre miljö och delar även upp en cell i flera subcellulära regioner. Sedan 1970-talet har lateral diffusion i dessa membran studerats, en av de viktigaste experimentella metoderna i dessa studier är fluorescence recovery after photobleach (FRAP). En matematisk modell utvecklad i denna avhandling beskriver hur dopamin 1-receptorer (D1R) diffunderar i en neural dendrits membran. Analytiska och numeriska metoder har använts för att lösa de partiella differentialekvationer som uttrycks i modellen. Valet av metod beror främst på komplexiteten hos geometrin i modellen.

Kalciumjoner (Ca2+) är kända för att ingå i flera intracellulära signalmekanismer. Ett intressant koncept inom detta fält är en signalerande mikrodomän där inositol 1,4,5-trifosfatreceptorn (IP3R) i endoplasmatiska nätverksmembranet (ER-membranet) fysiskt interagerar med proteiner i plasmamembranet. Denna mikrodomän har visats vara orsak till oscillationer i den intracellulära Ca2+-nivån. Den andra modellen i denna avhandling beskriver ett signalerande nätverk där både Ca2+-kanaler och pumpar bundna i ER-membranet och i plasmamembranet, däribland store-operated Ca2+(SOC)-kanaler, ingår. Ett MATLAB®-verktyg utvecklades för att implementera signalnätverket och simulera dess egenskaper. Denna modell implementerades även i Virtual cell.

Resultaten visar en stark likhet mellan den matematiska modellen och FRAP-datat i D1R-studien. Modellen visar en distinkt skillnad i återhämtningsegenskaper hos simulerade FRAP-experiment på hela dendriter och dendritiska spines, beroende på skillnader i geometri. Modellen kan även förklara infångning av D1R i dendritiska spines.

Resultaten från Ca2+-signaleringmodellen visar att stimulering av IP3R kan orsaka Ca2+-oscillationer inom samma frekvensområde som tidigare setts i experiment. Att ta bort SOC-kanaler från modellen kan ändra karaktär hos, såväl som den kvalitativa uppkomsten av Ca2+-oscillationer.

The complex mechanisms responsible for mammalian wound healing raise many biological questions that are amenable to theoretical investigation. In the first part of this thesis, we consider the role of mitotic auto-regulation in adult epidermal wound healing. We develop a reaction-diffusion model for the healing process, with parameter values based on biological data. The model solutions compare well with experimental results on the normal healing of circular wounds, and we analyse the solutions in one spatial dimension as travelling waves. We then use the model to perform 'mathematical experiments' on the effects of adding mitosis-regulating chemicals and of varying the initial wound shape. Recent experiments suggest that in embryos, epidermal wound healing occurs not by lamellipodial crawling as in adults, but rather by contraction of a cable of filamentous actin at the wound edge. We focus on the formation of this cable as a response to wounding, and develop and analyse a mechanical model for the post-wounding equilibrium in the microfilament network. Our model reflects the well-documented phenomenon of stress-induced alignment of actin filaments, which has been neglected in previous mechanochemical models of tissue deformation. The model solutions reflect the key aspects of the experimentally observed response to wounding. In the final part of the thesis, we consider chemokinetic and chemotactic control of cell movement, which play an important role in many aspects of wound healing. We propose a new model which reflects the underlying receptor-based mechanisms, and apply it to endothelial cell movement in the Boyden chamber assay. We compare our model with a simpler scheme in which cells respond directly to gradients in extracellular chemical concentration, and for both models we use experimental data to make quantitative predictions on the values of the transport coefficients.

Chapter 1 is a brief overview of the Bayesian approach to blending mathematical models with data. For this introductory chapter, I do not claim any originality in the material itself, but only in the presentation, and in the choice of contents. Chapters 2, 3 and 4 are transcripts of published and submitted papers, with minimal cosmetic modifications. I now detail my contributions to each of these papers. Chapter 2 is a transcript of the published paper Long-time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems" [Sanz-Alonso and Stuart, 2015] written in collaboration with Andrew Stuart. The idea of building a unified framework for studying filtering of chaotic dissipative dynamical systems is from Andrew. My ideas include the truncation of the 3DVAR algorithm that allows for unbounded observation noise, using the squeezing property as the unifying arch across all models, and most of the links with control theory. I stated and proved all the results of the paper. I also wrote the first version of the paper, which was subsequently much improved with Andrew's input. Chapter 3 is a transcript of the published paper \Filter Accuracy for the Lorenz 96 Model: Fixed Versus Adaptive Observation Operators" [Law et al., 2016], written in collaboration with Kody Law, Abhishek Shukla, and Andrew Stuart. My contribution to this paper was in proving most of the theoretical results. I did not contribute to the numerical experiments. The idea of using adaptive observation operators is from Abhishek. Chapter 4 is a transcript of the submitted paper\Importance Sampling: Computational Complexity and Intrinsic Dimension" [Agapiou et al., 2015], written in collaboration with Sergios Agapiou, Omiros Papaspiliopoulos, and Andrew Stuart. The idea of relating the two notions of intrinsic dimension described in the paper is from Omiros. Sergios stated and proved Theorem 4.2.3. Andrew's input was fundamental in making the paper well structured, and in the overall writing style. The paper was written very collaboratively among the four of us, and some of the results were the fruit of many discussions involving different subsets of authors. Some of my inputs include: the idea of using metrics between probability measures to study the performance of importance sampling, establishing connections to tempering, the analysis of singular limits both for inverse problems and filtering, most of the filtering section and in particular the use of the theory of inverse problems to analyze different proposals in the filtering set-up, the proof of Theorem 4.2.1, and substantial input in the proof of all the results of the paper not mentioned before. This paper aims to bring cohesion and new insights into a topic with a vast literature, and I helped towards this goal by doing most of the literature review involved.

La chirurgie informatisée est une science nouvelle dont le but est de croiser la chirurgie avec les sciences de l’informatique afin d’aboutir à des améliorations significatives dans les deux domaines. Avec l’évolution des nouvelles techniques chirurgicales, une collaboration étroite entre chirurgiens et chercheurs est devenue à la fois inévitable et essentielle à l’optimisation des soins chirurgicaux. L’utilisation de modèles mathématiques est la pierre angulaire de ce nouveau domaine. Cette thèse démontre comment une approche systématique d’un problème clinique nous a amenés à répondre à des questions ouvertes dans le domaine chirurgical en utilisant des modèles mathématiques à grande échelle. De manière générale, notre approche inclut (i) une vision générale du problème, (ii) le ciblage du/des système(s) physiologique(s) à étudier pour y répondre, et (iii) un effort de modélisation mathématique, qui a toujours été poussé par la recherche d’un compromis entre complexité du système étudié et réalité physiologique. Nous avons consacré la première partie de cette thèse à l’optimisation des conditions limites à appliquer à un bio-réacteur utilisé pour démultiplier le tissu pulmonaire provenant d’un donneur. Un modèle géométrique de l’arbre trachéo-bronchique couplé à un modèle de dépôt de soluté nous a permis de déterminer l’ensemble des pressions à appliquer aux pompes servant le bio-réacteur afin d’obtenir une distribution optimale des nutriments à travers les cultures de tissus. Nous avons consacré la seconde partie de cette thèse au problème de resténose des greffes de veines utilisées pour contourner une occlusion artérielle. Nous avons reproduit l’apparition de resténose grâce à plusieurs modèles mathématiques qui permettent d’étudier les preuves cliniques et de tester des hypothèses cliniques avec un niveau croissant de complexité et de précision. Pour finir, nous avons développé un cadre de travail robuste pour tester les effets des thérapies géniques afin de limiter la resténose. Une découverte intéressante a été de constater qu’en contrôlant un groupe de gènes spécifique, la perméabilité à la lumière double après un mois de suivi. Grace aux résultats obtenus, nous avons démontré que la modélisation mathématique peut servir de puissant outil pour l’innovation chirurgicale
Computational surgery is a new science that aims to intersect surgery and computational sciences in order to bring significant improvements in both fields. With the evolution of new surgical techniques, a close collaboration between surgeons and computational scientists became unavoidable and also essential to optimize surgical care. A large usage of mathematical models is the cornerstone in this new field. The present thesis shows how a systematic approach to a clinical problem brought us to answer open questions in the field of surgery by using mathematical models on a large scale. In general, our approach includes (i) an overview of the problem, (ii) the individuation of which physiological system/s is/are to be studied to address the question, and (iii) a mathematical modeling effort, which has been always driven by the pursue of a compromise between system complexity and closeness to the physiological reality. In the first part, we focused on the optimization of the boundary conditions to be applied to a bioreactor used to re-populate lung tissue from donor. A geometrical model of tracheobronchial tree combined with a solute deposition model allowed us to retrieve the set of pressures to be applied to the pumps serving the bioreactor in order to reach an optimal distribution of nourishment across the lung scaffold. In the second part, we focused on the issue of post-surgical restenosis of vein grafts used to bypass arterial occlusions. We replicated the event of restenosis with several mathematical models that allow us to study the clinical evidences and to test hypothesis with an escalating level of complexity and accuracy. Finally, we developed a solid framework to test the effect of gene therapies aimed to limit the restenosis. Interestingly, we found that by controlling a specific group of genes, the lumen patency is double after a month of follow-up. With the results achieved, we proved how mathematical modeling can be used as a powerful tool for surgical innovation

Somitogenesis is an important pattern formation process in the developmental biology of vertebrates. The phenomenon has received wide attention from experimental, theoretical, and computational biologists. Numerous mathematical models of the process have been proposed, with the clock and wavefront mechanism rising to prominence over the last ten years. This work presents two multicellular mathematical models of somitogenesis. The first is a phenomenological phase oscillator model that reproduces both the clock and wavefront aspects of somitogenesis, but lacks a biological basis. The second is a biologically informed delay differential equation model of the clock-wave that is produced by coordinated oscillatory gene expression across many cells. Careful and efficient model construction, parameter estimation, and model validation identify important nonlinear mechanisms in the genetic control circuit of the somitogenesis clock. In particular, a graded control protein combined with differential decay of clock protein monomers and dimers is found to be a key mechanism for slowing oscillations and generating experimentally observed waves of gene expression. This represents a mode of combinatorial control that has not been previously examined in somitogenesis, and warrants further experimental and theoretical investigation.

Financial bubbles have been present in the history of financial markets from the early days up to the modern age. An asset is said to exhibit a bubble when its market value exceeds its fundamental valuation. Although this phenomenon has been thoroughly studied in the economic literature, a mathematical martingale theory of bubbles, based on an absence of arbitrage has only recently been developed. In this dissertation, we aim to further contribute to the developement of this theory. In the first part we construct a model that allows us to capture the birth of a financial bubble and to describe its behavior as an initial submartingale in the build-up phase, which then turns into a supermartingale in the collapse phase. To this purpose we construct a flow in the space of equivalent martingale measures and we study the shifting perception of the fundamental value of a given asset. In the second part of the dissertation, we study the formation of financial bubbles in the valuation of defaultable claims in a reduced-form setting. In our model a bubble is born due to investor heterogeneity. Furthermore, our study shows how changes in the dynamics of the defaultable claim's market price may lead to a different selection of the martingale measure used for pricing. In this way we are able to unify the classical martingale theory of bubbles with a constructive approach to the study of bubbles, based on the interactions between investors.
Finanz-Blasen sind seit der Entstehung der Finanzmärkte bis zur heutigen Zeit gegenwärtig. Es gilt, dass ein Vermögenswert eine Finanzblase aufweist, sobald dessen Marktwert die fundamentale Bewertung übersteigt. Obwohl dieses Phänomen in der Wirtschaftsliteratur ausgiebig behandelt wurde, ist eine mathematische Martingaltheorie von Blasen, die auf der Abwesenheit von Arbitragemöglichkeiten beruht, erst in letzter Zeit entwickelt worden. Das Ziel dieser Dissertation ist es einen Beitrag zur Weiterentwicklung dieser Theorie zu leisten. Im ersten Abschnitt konstruieren wir ein Model mit Hilfe dessen man die Entstehung einer Finanz-Blase erfassen und deren Verhalten anfänglich als Submartingal in der build-up phase beschrieben werden kann, welches dann in der collapse phase zu einem Supermartingal wird. Zu diesem Zweck entwickeln wir einen Zahlungsstrom im Raum der äquivalenten Martingalmaße und wir untersuchen die zu dem Vermögenswert passende Verschiebung des fundamentalen Werts. Der zweite Teil der Dissertation beschäftigt sich mit der Bildung von Finanz-Blasen bei der Bewertung von Forderungen, die mit Ausfallrisiken behaftet sind, in einer reduzierten Marktumgebung. In unserem Model ist die Entstehung einer Blase die Folge der Heterogenität der Investoren. Des Weiteren zeigen unsere Untersuchungen, inwieweit Veränderungen der Dynamik des Marktpreises einer risikobehafteten Forderung zu einer Veränderung des zur Bewertung verwendeten Martingalmaß es führen kann. Dadurch sind wir in der Lage die klassische Martingaltheorie von Finanz-Blasen mit einem konstruktivem Ansatz zur Untersuchung von Finanz-Blasen zu vereinigen, der auf den Interaktionen zwischen Marktteilnehmern basiert.

Diseases have the capacity to not only influence the dynamics of their hosts, but also interacting species like predators, prey and competitors. Likewise, interacting species can influence disease dynamics by altering the host's dynamics. The combination of these two effects is often called eco-epidemiology, the interaction of ecology and epidemiology. In this thesis, we explore this interplay of infectious diseases and predator--prey interactions, where the predator is a specialist. We start with an introductory chapter on modelling eco-epidemiology, with a particular focus on the myriad of different possible assumptions mathematical models in eco-epidemiology can have. In Chapter 2, we consider the effect predator--prey oscillations have on the endemic criteria for an infectious disease. In Chapter 3, we find a great variety of complex dynamics like tristability between endemic and disease-free states, quasi-periodic dynamics and chaos in a predator--prey model with an infectious disease in the predator. In Chapter 4, we consider the impact an infectious disease has on a group defending prey. Here, we find that the disease not only can coexist with a predator, it can actually help the predator survive where it could not in the absence of the disease, in stark contradiction to the principle of competitive exclusion which states that two exploiters should not coexist on a single resource. Lastly, in Chapter 5, we consider a spatial predator--prey model with a disease in the prey and focus on how preytaxis (the movement of predators along prey gradients) can alter various invasion scenarios. Through all these chapters, there is a common focus on the impact (endogenous) oscillations have in eco-epidemiology.

Membrane filtration is a simple concept for water purification: water containing particulate contaminants is forced through a semi-permeable membrane that rejects the particulates leaving clean water to flow out. Nevertheless, there are many complex features of membrane filtration, the most important of which is the accumulation of the particulates at the membrane surface. This leads ultimately to fouling of the membrane and a reduction in the efficiency of the process. Concentration polarization is the precursor of fouling, that is, a high concentration of contaminants develops in front of the membrane without the contaminants attaching to each other or the membrane surface. However, several types of acute membrane fouling develop from the layer formed in concentration polarization, including internal fouling, pore blocking and caking. Addressing these and related problems has been at the forefront of membrane research since the process' inception. In this thesis we develop mathematical models of aspects of crossflow and directflow filtration operating at constant flux. We begin by addressing questions related to the initial stages of concentration polarization in crossflow systems. In particular, we study the influence of particulates on the viscosity of the filtrate, and show how the filtration efficiency may be improved by tailoring the wall permeability to reduce the effects of osmosis. We then address the development of membrane fouling and caking in directflow systems: the transmembrane pressure difference, the possibility of elastic deformations during filtration, and the influence of these on the development of fouling and caking are all considered. We show that even small elastic effects can worsen fouling and suggest how the process can be operated to avoid this. We then discuss further opportunities for mathematical modelling in this area.

1. Higher mathematics: manual / V. P. Denisiuk, V. M. Bobkov, L. I. Grishina and others. –K. : NAU, 2006. – Part 1. – 268 p. 2. Larson R. College Algebra / R. Larson, R. Hosteller. – Houghton Mifflin, 1997. – 545 p. 3. Mizrahi A. Calculus and Analytic Geometry / A. Mizrahi, M. Sullivan. – California: Wadsmonth Publishing Company, 1987. – 1083 p.
Exponential functions are useful in modeling many physical phenomena, such as populations, interest rates, radioactive decay, and the amount of medicine in the bloodstream. An exponential model is of the form A = A_0×bˆ(t/c) , where we have A_0 is the initial amount of whatever is being modelled, t is elapsed time.
Експоненціальні функції корисні для моделювання багатьох фізичних явищ, таких як популяції, процентні ставки, радіоактивний розпад та кількість ліків у крові. Експоненціальна модель має вигляд A = A_0×bˆ(t/c), де A_0 - це початкова кількість того, що моделюється, t - час, що минув.

The goal of this paper is to explain the mathematics involved in modeling biochemical oscillations. We first discuss several important biochemical concepts fundamental to the construction of descriptive mathematical models. We review the basic theory of differential equations and stability analysis as it relates to two-variable models exhibiting oscillatory behavior. The importance of the Hopf Bifurcation will be discussed in detail for the central role it plays in limit cycle behavior and instability. Once we have exposed the necessary mathematical framework, we consider several specific models of biochemical oscillators in three or more variables. This will include a detailed analysis of Goodwin's equations and their modification first studied by Painter. Additionally, we consider the consequences of introducing both distributed and discrete time delay into Goodwin's model. We will show that the presence of distributed time lag modifies Goodwin's model in no significant way. The final section of the paper will discuss discrete time lag in the context of a minimal model of the circadian rhythm. In the main, this paper will address mathematical, as opposed to biochemical, issues. Nevertheless, the significance of the mathematics to the biochemistry will be considered throughout.
Master of Science

Mathematical models provide a framework to confirm or reject hypotheses via the generation of quantitative predictions and offer rich insights into the processes that drive complex biological phenomena. In this thesis, we develop mathematical models that integrate experimental data and use these models to explore cellular dysfunction at different scales. The core of this thesis focuses on the selection of high-avidity T cells in cancer vaccines. High-avidity T cells, unlike low-avidity T cells, are adept at killing cancer cells and are essential for durable anti-tumour immunity. Using an ordinary differential equation (ODE) model, we show that we can optimise dosages to elicit high-avidity T cells. We find that increased numbers of immune cells known as immature dendritic cells, can also promote high-avidity T cells. We then reduce the complexity of our model and perform a thorough sensitivity analysis. We then study how immune cells regulate PD-L1 in the tumour niche. PD-L1 is an immunosuppressive molecule that tumours upregulate. Intriguingly, PD-L1 expression does not always correlate with tumour progression. To understand why we develop an ODE model that we calibrate to in vitro data. Using this model, we show that PD-L1 expression equilibrates in response to changes in immune activity via a feedforward circuit. This finding explains why some patients may respond to therapies targeting PD-L1 despite being PD-L1 negative. The last part of this thesis tests whether the spatial arrangement of cardiac mitochondria affects bioenergetics, as speculated by scholars. To test this, we develop an agent-based model of mitochondrial structure linked to a validated model of energy production and show that cardiac bioenergetics are robust to changes in fission and fusion over a physiological range. This thesis contains several foundational models. We expect the findings from this thesis to be a starting point for further interdisciplinary modelling and experimental work.

The purpose of this thesis is to develop various advanced linear algebra techniques that apply to image processing. With the increasing use of computers and digital photography, being able to manipulate digital images efficiently and with greater freedom is extremely important. By applying the tools of linear algebra, we hope to improve the ability to process such images. We are especially interested in developing techniques that allow computers to manipulate images with the least amount of human guidance. In Chapter 2 and Chapter 3, we develop the basic definitions and linear algebra concepts that lay the foundation for later chapters. Then, in Chapter 4, we demonstrate techniques that allow a computer to rotate an image to the correct orientation automatically, and similarly, for the computer to correct a certain class of color distortion automatically. In both cases, we use certain properties of the eigenvalues and eigenvectors of covariance matrices. We then model color clashing and color variation in Chapter 5 using a powerful tool from linear algebra known as the Perron-Frobenius theorem. Finally, we explore ways to determine whether an image is a blur of another image using invariant functions. The inspiration behind these functions are recent applications of Lie Groups and Lie algebra to image processing.

This thesis studies mathematical models of a population of cells invading the surrounding environment or another living population. A classical single-species model is reformulated using a moving boundary to track the position of the moving front of the invading population. The moving boundary is also used to separate two populations. Other models studied are coupled partial differential equations to describe the interaction of a population with another one. Different types of interaction are represented: the degradation of healthy skin by cancer and the growth of bone tissue on substrate.

The research activity carried out during the PhD course was focused on the development of mathematical models of some cognitive processes and their validation by means of data present in literature, with a double aim: i) to achieve a better interpretation and explanation of the great amount of data obtained on these processes from different methodologies (electrophysiological recordings on animals, neuropsychological, psychophysical and neuroimaging studies in humans), ii) to exploit model predictions and results to guide future research and experiments. In particular, the research activity has been focused on two different projects: 1) the first one concerns the development of neural oscillators networks, in order to investigate the mechanisms of synchronization of the neural oscillatory activity during cognitive processes, such as object recognition, memory, language, attention; 2) the second one concerns the mathematical modelling of multisensory integration processes (e.g. visual-acoustic), which occur in several cortical and subcortical regions (in particular in a subcortical structure named Superior Colliculus (SC)), and which are fundamental for orienting motor and attentive responses to external world stimuli. This activity has been realized in collaboration with the Center for Studies and Researches in Cognitive Neuroscience of the University of Bologna (in Cesena) and the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA). PART 1. Objects representation in a number of cognitive functions, like perception and recognition, foresees distribute processes in different cortical areas. One of the main neurophysiological question concerns how the correlation between these disparate areas is realized, in order to succeed in grouping together the characteristics of the same object (binding problem) and in maintaining segregated the properties belonging to different objects simultaneously present (segmentation problem). Different theories have been proposed to address these questions (Barlow, 1972). One of the most influential theory is the so called “assembly coding”, postulated by Singer (2003), according to which 1) an object is well described by a few fundamental properties, processing in different and distributed cortical areas; 2) the recognition of the object would be realized by means of the simultaneously activation of the cortical areas representing its different features; 3) groups of properties belonging to different objects would be kept separated in the time domain. In Chapter 1.1 and in Chapter 1.2 we present two neural network models for object recognition, based on the “assembly coding” hypothesis. These models are networks of Wilson-Cowan oscillators which exploit: i) two high-level “Gestalt Rules” (the similarity and previous knowledge rules), to realize the functional link between elements of different cortical areas representing properties of the same object (binding problem); 2) the synchronization of the neural oscillatory activity in the γ-band (30-100Hz), to segregate in time the representations of different objects simultaneously present (segmentation problem). These models are able to recognize and reconstruct multiple simultaneous external objects, even in difficult case (some wrong or lacking features, shared features, superimposed noise). In Chapter 1.3 the previous models are extended to realize a semantic memory, in which sensory-motor representations of objects are linked with words. To this aim, the network, previously developed, devoted to the representation of objects as a collection of sensory-motor features, is reciprocally linked with a second network devoted to the representation of words (lexical network) Synapses linking the two networks are trained via a time-dependent Hebbian rule, during a training period in which individual objects are presented together with the corresponding words. Simulation results demonstrate that, during the retrieval phase, the network can deal with the simultaneous presence of objects (from sensory-motor inputs) and words (from linguistic inputs), can correctly associate objects with words and segment objects even in the presence of incomplete information. Moreover, the network can realize some semantic links among words representing objects with some shared features. These results support the idea that semantic memory can be described as an integrated process, whose content is retrieved by the co-activation of different multimodal regions. In perspective, extended versions of this model may be used to test conceptual theories, and to provide a quantitative assessment of existing data (for instance concerning patients with neural deficits). PART 2. The ability of the brain to integrate information from different sensory channels is fundamental to perception of the external world (Stein et al, 1993). It is well documented that a number of extraprimary areas have neurons capable of such a task; one of the best known of these is the superior colliculus (SC). This midbrain structure receives auditory, visual and somatosensory inputs from different subcortical and cortical areas, and is involved in the control of orientation to external events (Wallace et al, 1993). SC neurons respond to each of these sensory inputs separately, but is also capable of integrating them (Stein et al, 1993) so that the response to the combined multisensory stimuli is greater than that to the individual component stimuli (enhancement). This enhancement is proportionately greater if the modality-specific paired stimuli are weaker (the principle of inverse effectiveness). Several studies have shown that the capability of SC neurons to engage in multisensory integration requires inputs from cortex; primarily the anterior ectosylvian sulcus (AES), but also the rostral lateral suprasylvian sulcus (rLS). If these cortical inputs are deactivated the response of SC neurons to cross-modal stimulation is no different from that evoked by the most effective of its individual component stimuli (Jiang et al 2001). This phenomenon can be better understood through mathematical models. The use of mathematical models and neural networks can place the mass of data that has been accumulated about this phenomenon and its underlying circuitry into a coherent theoretical structure. In Chapter 2.1 a simple neural network model of this structure is presented; this model is able to reproduce a large number of SC behaviours like multisensory enhancement, multisensory and unisensory depression, inverse effectiveness. In Chapter 2.2 this model was improved by incorporating more neurophysiological knowledge about the neural circuitry underlying SC multisensory integration, in order to suggest possible physiological mechanisms through which it is effected. This endeavour was realized in collaboration with Professor B.E. Stein and Doctor B. Rowland during the 6 months-period spent at the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA), within the Marco Polo Project. The model includes four distinct unisensory areas that are devoted to a topological representation of external stimuli. Two of them represent subregions of the AES (i.e., FAES, an auditory area, and AEV, a visual area) and send descending inputs to the ipsilateral SC; the other two represent subcortical areas (one auditory and one visual) projecting ascending inputs to the same SC. Different competitive mechanisms, realized by means of population of interneurons, are used in the model to reproduce the different behaviour of SC neurons in conditions of cortical activation and deactivation. The model, with a single set of parameters, is able to mimic the behaviour of SC multisensory neurons in response to very different stimulus conditions (multisensory enhancement, inverse effectiveness, within- and cross-modal suppression of spatially disparate stimuli), with cortex functional and cortex deactivated, and with a particular type of membrane receptors (NMDA receptors) active or inhibited. All these results agree with the data reported in Jiang et al. (2001) and in Binns and Salt (1996). The model suggests that non-linearities in neural responses and synaptic (excitatory and inhibitory) connections can explain the fundamental aspects of multisensory integration, and provides a biologically plausible hypothesis about the underlying circuitry.

The research activity carried out during the PhD course was focused on the development of mathematical models of some cognitive processes and their validation by means of data present in literature, with a double aim: i) to achieve a better interpretation and explanation of the great amount of data obtained on these processes from different methodologies (electrophysiological recordings on animals, neuropsychological, psychophysical and neuroimaging studies in humans), ii) to exploit model predictions and results to guide future research and experiments. In particular, the research activity has been focused on two different projects: 1) the first one concerns the development of neural oscillators networks, in order to investigate the mechanisms of synchronization of the neural oscillatory activity during cognitive processes, such as object recognition, memory, language, attention; 2) the second one concerns the mathematical modelling of multisensory integration processes (e.g. visual-acoustic), which occur in several cortical and subcortical regions (in particular in a subcortical structure named Superior Colliculus (SC)), and which are fundamental for orienting motor and attentive responses to external world stimuli. This activity has been realized in collaboration with the Center for Studies and Researches in Cognitive Neuroscience of the University of Bologna (in Cesena) and the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA). PART 1. Objects representation in a number of cognitive functions, like perception and recognition, foresees distribute processes in different cortical areas. One of the main neurophysiological question concerns how the correlation between these disparate areas is realized, in order to succeed in grouping together the characteristics of the same object (binding problem) and in maintaining segregated the properties belonging to different objects simultaneously present (segmentation problem). Different theories have been proposed to address these questions (Barlow, 1972). One of the most influential theory is the so called “assembly coding”, postulated by Singer (2003), according to which 1) an object is well described by a few fundamental properties, processing in different and distributed cortical areas; 2) the recognition of the object would be realized by means of the simultaneously activation of the cortical areas representing its different features; 3) groups of properties belonging to different objects would be kept separated in the time domain. In Chapter 1.1 and in Chapter 1.2 we present two neural network models for object recognition, based on the “assembly coding” hypothesis. These models are networks of Wilson-Cowan oscillators which exploit: i) two high-level “Gestalt Rules” (the similarity and previous knowledge rules), to realize the functional link between elements of different cortical areas representing properties of the same object (binding problem); 2) the synchronization of the neural oscillatory activity in the γ-band (30-100Hz), to segregate in time the representations of different objects simultaneously present (segmentation problem). These models are able to recognize and reconstruct multiple simultaneous external objects, even in difficult case (some wrong or lacking features, shared features, superimposed noise). In Chapter 1.3 the previous models are extended to realize a semantic memory, in which sensory-motor representations of objects are linked with words. To this aim, the network, previously developed, devoted to the representation of objects as a collection of sensory-motor features, is reciprocally linked with a second network devoted to the representation of words (lexical network) Synapses linking the two networks are trained via a time-dependent Hebbian rule, during a training period in which individual objects are presented together with the corresponding words. Simulation results demonstrate that, during the retrieval phase, the network can deal with the simultaneous presence of objects (from sensory-motor inputs) and words (from linguistic inputs), can correctly associate objects with words and segment objects even in the presence of incomplete information. Moreover, the network can realize some semantic links among words representing objects with some shared features. These results support the idea that semantic memory can be described as an integrated process, whose content is retrieved by the co-activation of different multimodal regions. In perspective, extended versions of this model may be used to test conceptual theories, and to provide a quantitative assessment of existing data (for instance concerning patients with neural deficits). PART 2. The ability of the brain to integrate information from different sensory channels is fundamental to perception of the external world (Stein et al, 1993). It is well documented that a number of extraprimary areas have neurons capable of such a task; one of the best known of these is the superior colliculus (SC). This midbrain structure receives auditory, visual and somatosensory inputs from different subcortical and cortical areas, and is involved in the control of orientation to external events (Wallace et al, 1993). SC neurons respond to each of these sensory inputs separately, but is also capable of integrating them (Stein et al, 1993) so that the response to the combined multisensory stimuli is greater than that to the individual component stimuli (enhancement). This enhancement is proportionately greater if the modality-specific paired stimuli are weaker (the principle of inverse effectiveness). Several studies have shown that the capability of SC neurons to engage in multisensory integration requires inputs from cortex; primarily the anterior ectosylvian sulcus (AES), but also the rostral lateral suprasylvian sulcus (rLS). If these cortical inputs are deactivated the response of SC neurons to cross-modal stimulation is no different from that evoked by the most effective of its individual component stimuli (Jiang et al 2001). This phenomenon can be better understood through mathematical models. The use of mathematical models and neural networks can place the mass of data that has been accumulated about this phenomenon and its underlying circuitry into a coherent theoretical structure. In Chapter 2.1 a simple neural network model of this structure is presented; this model is able to reproduce a large number of SC behaviours like multisensory enhancement, multisensory and unisensory depression, inverse effectiveness. In Chapter 2.2 this model was improved by incorporating more neurophysiological knowledge about the neural circuitry underlying SC multisensory integration, in order to suggest possible physiological mechanisms through which it is effected. This endeavour was realized in collaboration with Professor B.E. Stein and Doctor B. Rowland during the 6 months-period spent at the Department of Neurobiology and Anatomy of the Wake Forest University School of Medicine (NC, USA), within the Marco Polo Project. The model includes four distinct unisensory areas that are devoted to a topological representation of external stimuli. Two of them represent subregions of the AES (i.e., FAES, an auditory area, and AEV, a visual area) and send descending inputs to the ipsilateral SC; the other two represent subcortical areas (one auditory and one visual) projecting ascending inputs to the same SC. Different competitive mechanisms, realized by means of population of interneurons, are used in the model to reproduce the different behaviour of SC neurons in conditions of cortical activation and deactivation. The model, with a single set of parameters, is able to mimic the behaviour of SC multisensory neurons in response to very different stimulus conditions (multisensory enhancement, inverse effectiveness, within- and cross-modal suppression of spatially disparate stimuli), with cortex functional and cortex deactivated, and with a particular type of membrane receptors (NMDA receptors) active or inhibited. All these results agree with the data reported in Jiang et al. (2001) and in Binns and Salt (1996). The model suggests that non-linearities in neural responses and synaptic (excitatory and inhibitory) connections can explain the fundamental aspects of multisensory integration, and provides a biologically plausible hypothesis about the underlying circuitry.

Thesis (Ph. D.)--Illinois State University, 2001.
Title from title page screen, viewed March 30, 2006. Dissertation Committee: Graham A. Jones, James Cottrill (co-chairs), Linnea Sennott. Includes bibliographical references (leaves 107-111) and abstract. Also available in print.

Mathematics
Ph.D.
Microbial biofilms are defined as clusters of microbial cells living in self-produced extracellular polymeric substances (EPS), which always attached to various kinds of surfaces. In this thesis, we studied several mathematical models of biofilm in the human body and marble environment. Some related background of biofilm growth and some basic existing numerical models were introduced in the first chapter. In the first project, we introduced how biofilm affects the local oxygen concentration near the neutrophil cells in the human body with three one-dimensional reaction-diffusion models from different geometries. In nature, microbial biofilm development can be observed on almost all kinds of stone monuments and can also be associated with the problem of monument conservation. In the second part of my research, we built the deliquescence models for biofilm growth environment in the first model and added biomass into consideration in the second one. Also, we analyzed the stability of the equilibria. In the third part, we applied the weather data collected from the weather station on the roof of the Jefferson Memorial to the deliquescence model with biofilm. Furthermore, compared the simulation result for biofilm growth in cold and warm weathers. In the last part of this thesis, we analyzed the biofilm activity with support vector regression. The machine learning model we obtained can be used to find the growth trends of biofilm for any pair of temperature and relative humidity data.
Temple University--Theses

Cancer is a deadly, complex disease with 14 million new cases diagnosed every year and the endeavour to develop a cure is a global multidisciplinary effort. The complexity of cancer and the resulting vast volume of data derived from its research necessitates a robust and cutting-edge system of mathematical and statistical modelling. This thesis proposes novel mathematical models of quantification and modelling applied to heterogeneous preclinical cancers, focusing on the translation of animal studies into patients with particular emphasis on tumour stroma. The first section of this thesis (quantification) will present different techniques of extracting and quantifying data from bioanalytical assays. The overall aim will be to present and discuss potential methods of obtaining data regarding tumour volume, stromal morphology, stromal heterogeneity, and oxygen distribution. Firstly, a 3D scanning technique will be discusses. This technique aims to assess tumour volume in mice more precisely than the current favoured method (callipers) and record any cutaneous symptoms as well, with the potential to revolutionise tumour growth analysis. Secondly, a series of image processing methods will be presented which, when applied to tumour histopathology, demonstrate that tumour stromal morphology and its microenvironment play a key role in tumour physiology. Lastly, it will be demonstrated through the integration of in-vitro data from various sources that oxygen and nutrient distribution in tumours is very irregular, creating metabolic niches with distinct physiologies within a single tumour. Tumour volume, oxygen, and stroma are the three aspects central to the successful modelling of tumour drug responses over time. The second section of this thesis (modelling) will feature a mathematical oxygen-driven model - utilising 38 cell lines and 5 patient-derived animal models - that aims to demonstrate the relationship between homogeneous oxygen distribution and preclinical tumour growth. Finally, all concepts discussed will be merged into a computational tumour-stroma model. This cellular automaton (stochastic) model will demonstrate that tumour stroma plays a key role in tumour growth and has both positive (at a molecular level) and negative (at both a molecular and tissue level) effects on cancers. This thesis contains a useful set of algorithms to help visualise, quantify, and understand tissue phenomena in cancer physiology, as well as providing a series of platforms to predict tumour outcome in the preclinical setting with clinical relevance.

In this thesis, I investigate several aspects of life cycle evolution using mathematical models. (1) We expect natural selection to favour organisms that reproduce as often and as quickly as possible. However, many species delay development unless particular environments or rare disturbance events occur. I use models to ask when delayed development (e.g., seed dormancy) in long-lived species can be favoured by selection. I find that long-lived plants experience `immaturity risk': the risk of death due to a population-scale disturbance, such as a fire, before reproducing. This risk can be sufficient to favour germination in the disturbance years only. I show how demographic models can be constructed in order to estimate the contribution of this mechanism (and two other mechanisms) to the evolution of dormancy in a particular environment. (2) All sexually reproducing eukaryotes alternate between haploid and diploid phases. However, selection may not occur in both phases to the same extent. I use models to investigate the evolution of the time spent in haploid versus diploid phases. The presence of a homologous gene copy in diploids has important population genetic effects because it can mask the other gene copy from selection. A key innovation of my investigation is to allow haploids and homozygous diploids to have different fitnesses (intrinsic fitness differences). This reveals a novel hypothesis for the evolution of haploid-diploid strategies (where selection occurs in both phases), where the genetic effects of ploidy are balanced against intrinsic fitness differences. (3) Many sex chromosome systems are characterized by a lack of recombination between sex chromosome types. The predominant explanation for this phenomenon involves differences in selection between diploid sexes. I develop a model for the evolution of recombination between the sex chromosomes in which there is a period of selection among haploid genotypes in pollen or sperm. I find that a period of haploid selection can also drive the evolution of suppressed recombination between sex chromosomes, which should become enriched for genes selected in the haploid phase. This model predicts that the tempo of sex chromosome evolution can depend on the degree of competition among haploids.
Science, Faculty of
Botany, Department of
Graduate

Thesis (Ph. D.) -- University of Maryland, College Park, 2007.
Thesis research directed by: Business and Management: Marketing. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

We here analyse a number of class-D amplifier topologies. Class-D amplifiers operate by converting an audio input signal into a high-frequency square wave output, whose lower-frequency components can accurately reproduce the input. Their high power efficiency and potential for low distortion makes them suitable for use in a wide variety of electronic devices. By calculating the outputs from a classical class-D design implementing different sampling schemes we demonstrate that a more recent method, called the Fourier transform/Poisson resummation method, has many advantages over the double Fourier series method, which is the traditional technique employed for this analysis. We thereby show that when natural sampling is used the input signal is reproduced exactly in the low-frequency part of the output, with no distortion. Although this is a known result, our calculations present the method and notation that we later develop. The classical class-D design is prone to noise, and therefore negative feedback is often included in the circuit. Subsequently we incorporate the Fourier transform/Poisson resummation method into a formalised and succinct analysis of a first-order negative feedback amplifier. Using perturbation expansions we derive the audio-frequency part of the output, demonstrating that negative feedback introduces undesirable distortion. Here we reveal the next order terms in the output compared with previous work, giving further insight into the nonlinear distortion. We then further extend the analysis to examine two more complex negative feedback topologies, namely a second-order and a derivative negative feedback design. Modelling each of these amplifiers presents an increased challenge due to the differences in their respective circuit designs, and in addition, for the derivative negative feedback amplifier we must consider scaling regimes based on the relative magnitudes of the frequencies involved. For both designs we establish novel expressions for the output, including the most significant distortion terms.

DNA replication and associated processes take place in all living organisms with the same constitutions. The knowledge of the duplication process, chromatin building and repair mechanisms has increased explosively over the last years, but the complex interplay of different proteins and their mechanisms are not conceived properly. During DNA replication, the DNA has to be unpacked, duplicated and finally repacked into chromatin. These steps require different proteins, e.g. new histone proteins on demand to secure an error-free and undelayed DNA replication. This thesis includes different mathematical models for DNA replication, repair and chromatin formation, which are based on experimental results. Three models of chromatin formation provide a simplified description of histone gene expression and protein synthesis during G1/S/G2 phase and include the contribution of different regulatory elements. Furthermore, all models present two different mechanisms of regulation to test possible scenarios of newly synthesised histones and free DNA binding sites. The basic model presents a single histone gene, which codes for a single histone protein. The stem-loop binding protein (SLBP) acts as a master regulator, which is only present during S phase. Different analyses of early S-phase, over- and underexpressed replication and the down-regulation of SLBP proof the model under extreme conditions. This basic model serves as a template for further scenarios with several genes and different histone families. For this, a second model is realised to simulate imbalances in the histone mRNA synthesis and translation. Additionally, a third model tests a gene knock-out and mRNA silencing. The initial histone model is able to qualitatively reproduce experimental observations and shows basic regulatory principles. The adaptation with several genes and different histone families presents qualitatively different system responses for the discussed regulatory mechanisms and illustrates the ability to compensate the effect of mRNA silencing.

Carding is an essential pre-spinning process whereby masses of dirty tufted fibres are cleaned, disentangled and refined into a smooth coherent web. Research and development in this `low-technology' industry have hitherto depended on empirical evidence. In collaboration with the School of Textile Industries at the University of Leeds, a mathematical theory has been developed that describes the passage of fibres through the carding machine. The fibre dynamics in the carding machine are posed, modelled and simulated by three distinct physical problems: the journey of a single fibre, the extraction of fibres from a tuft or tufts and many interconnecting, entangled fibres. A description of the life of a single fibre is given as it is transported through the carding machine. Many fibres are sparsely distributed across machine surfaces, therefore interactions with other neighbouring fibres, either hydrodynamically or by frictional contact points, can be neglected. The aerodynamic forces overwhelm the fibre's ability to retain its crimp or natural curvature, and so the fibre is treated as an inextensible string. Two machine topologies are studied in detail, thin annular regions with hooked surfaces and the nip region between two rotating drums. The theoretical simulations suggest that fibres do not transfer between carding surfaces in annular machine geometries. In contrast to current carding theories, which are speculative, a novel explanation is developed for fibre transfer between the rotating drums. The mathematical simulations describe two distinct mechanisms: strong transferral forces between the taker-in and cylinder and a weaker mechanism between cylinder and doffer. Most fibres enter the carding machine connected to and entangled with other fibres. Fibres are teased from their neighbours and in the case where their neighbours form a tuft, which is a cohesive and resistive fibre structure, a model has been developed to understand how a tuft is opened and broken down during the carding process. Hook-fibre-tuft competitions are modelled in detail: a single fibre extracted from a tuft by a hook and diverging hook-entrained tufts with many interconnecting fibres. Consequently, for each scenario once fibres have been completely or partially extracted, estimates can be made as to the degree to which a tuft has been opened-up. Finally, a continuum approach is used to simulate many interconnected, entangled fibre-tuft populations, focusing in particular on their deformations. A novel approach describes this medium by density, velocity, directionality, alignment and entanglement. The materials responds to stress as an isotropic or transversely isotropic medium dependent on the degree of alignment. Additionally, the material's response to stress is a function of the degree of entanglement which we describe by using braid theory. Analytical solutions are found for elongational and shearing flows, and these compare very well with experiments for certain parameter regimes.

Filamentous fungi are important in an enormous variety of ways to our life, with examples ranging from bioremediation, through the food and drinks industry to human health. These organisms can form huge networks stretching metres and even kilometres. However, their mode of growth is by the extension of individual hyphal tips only a few microns in diameter. Tip growth is mediated by the incorporation of new wall building materials at the soft apex. Just how this process is controlled (in fungi and in cell elongation in other organisms) has been the subject of intense study over many years and has attracted considerable attention from mathematical modellers. In this thesis, we consider mathematical models of fungal tip growth that can be classified as either geometrical or biomechanical. In every model we examine, a 2-D axisymmetric semihemisphere-like curve represents half the medial section of fungal tip geometry. A geometrical model for the role of the Spitzenkorper in the tip growth was proposed by Bartnicki-Garcia et al (1989), where a number of problems with the mathematical derivation were pointed out by Koch (2001). A suggestion is given as an attempt to revise the derivation by introducing a relationship between arc length of a growing tip, deposition of wall-building materials and tip curvature. We also consider two types of geometrical models as proposed by Goriely et al (2005). The first type considers a relationship between the longitudinal curvature and the function used to model deposition of wall-building materials. For these types of models, a generalized formulae for the tip shape is introduced, which allows localization of deposition of wall-building materials to be examined. The second type considers a relationship between longitudinal and latitudinal curvatures and the function used to model deposition of wall-building materials. For these types of models, a new formulation of the function used to model deposition of wall-building materials is introduced. Finally, a biomechanical model as proposed by Goriely et al (2010). Varying arc length of the stretchable region on the tip suggests differences in geometry of tip shape and the effective pressure profile. The hypothesis of orthogonal growth is done by focusing only on the apex of a "germ tube". Following that, it suggests that material points on the tip appear to move in a direction perpendicular to the tip either when surface friction is increased or decreased.

In this thesis we develop mathematical models to understand the process of ecological invasion when the invading species also carries a disease that is harmful to the native species. In particular we focus on a key case study system of the invasion of grey squirrels and replacement of red squirrels in the UK, in which the shared disease, squirrelpox, has been suggested as a key driver of the rapid expansion of grey squirrels. Our initial study focused on examining the viability of red squirrels in the stronghold forests of Kidland and Uswayford in Northumberland. These are commercially managed forests that Forestry Commission England manage to improve red squirrel population viability. Through close collaboration with the Forestry Commission, we developed a mathematical model that could test squirrel population viability for a range of felling and replanting strategies. Our findings have been used to direct the forest design plans that will be implemented in these forests. Our second study used spatial, stochastic modelling techniques to model the replacement of red squirrels and subsequent control of grey squirrels on the Isle of Anglesey. Our findings indicated that the replacement of red squirrels by grey squirrels on the island was largely driven by competitive interactions. However, on a local level squirrelpox epidemics could occur and lead to mortality in red squirrel populations. Our model was also fitted to data on the control and eradication of grey squirrels and reintroduction of red squirrels that took place on the Isle of Anglesey between 1998-2013. Our fitted model was then used to examine the best conservation strategies to protect the red squirrels on Anglesey. Our final study compared key findings on the process of disease-mediated invasion in deterministic and stochastic model frameworks. The deterministic frameworks predict that a wave of disease can spread through a native population in advance of a wave of replacement of the invading species. A stochastic representation of this system indicated that this wave of disease in advance of the wave of replacement may not occur if the disease is too virulent to the native species. However, if the disease is supported by the invading species, it will still mediate the invasion at the interface between the native and invading species where local epidemic disease outbreaks can occur. In general this thesis shows that mathematical models are powerful tools for the conservation management of native species under threat from invasion.

The classical Black-Scholes analysis determines a unique, continuous, trading strategy which allows one to hedge a financial option perfectly and leads to a unique price for the option. It assumes, however, that there are no transaction costs involved in implementing this strategy, and the stock market is absolutely liquid. In this work some new results are obtained to accommodate costs of hedging, which occur in practice, and market imperfections into the option pricing framework. In Part One transaction charges are dealt with by means of the mean-variance technique, originally developed by Markowitz. This approach is based on the minimisation of the variance of the outcome at expiry subject to spending at most a given initial endowment. Since "perfect" replication is no longer possible in this case, there will always be an unavoidable element of risk associated with writing an option. Therefore, the option price is now not unique. A mean-variance approach makes option pricing relatively easy and meaningful to an investor, who is supposed to choose a point on the mean-deviation locus. In the limit of zero transaction costs, the problem naturally reduces to the Black-Scholes valuation method, unlike alternative approaches based on the utility-maximisation. The stochastic optimisation problem obtained is dealt with by means of the stochastic version of Pontryagin's maximum principle. This technique is believed to be applied to this kind of problem for the first time. In general the resulting free-boundary problem has to be solved numerically, but for a small level of proportional transaction costs an asymptotic solution is possible. Regions of short term and long term dynamics are identified and the intermediate behaviour is obtained by matching these regions. The perturbation analysis of the utility-maximisation approach is also revised in this work, and amendments are obtained. In addition, the maximum principle is applied to the Portfolio Selection problem of Markowitz. The dynamical rebalancing technique developed in this work proves more efficient than the classical static approach, and allows investors to obtain portfolios with lower levels of risk. The model presented in Part Two is an attempt to quantify the concept of liquidity and establish relations between various measures of market performance. Informational inefficiency is argued to be the main reason for the unavailability of an asset at its equilibrium price. A mathematical model to describe the asset price behaviour together with arbitrage considerations enable us to estimate the component of the bid-ask spread arising from the outstanding information. The impact of the market liquidity on hedging an option with another option as well as the underlying asset itself is also examined. Although in the last case uncertainty cannot be completely eliminated from the hedged portfolio, a unique risk-minimising strategy is found.

In this thesis, we derive, extend and generalise various aspects of impact theory and splash dynamics. Our methods throughout will involve isolating small parameters in our models, which we can utilise using the language of matched asymptotics. In Chapter 1 we briefly motivate the field of impact theory and outline the structure of the thesis. In Chapter 2, we give a detailed review of classical small-deadrise water entry, Wagner theory, in both two and three dimensions, highlighting the key results that we will use in our extensions of the theory. We study oblique water entry in Chapter 3, in which we use a novel transformation to relate an oblique impact with its normal-impact counterpart. This allows us to derive a wide range of solutions to both two- and three-dimensional oblique impacts, as well as discuss the limitations and breakdown of Wagner theory. We return to vertical water-entry in Chapter 4, but introduce the air layer trapped between the impacting body and the liquid it is entering. We extend the classical theory to include this air layer and in the limit in which the density ratio between the air and liquid is sufficiently small, we derive the first-order correction to the Wagner solution due to the presence of the surrounding air. The model is presented in both two dimensions and axisymmetric geometries. In Chapter 5 we move away from Wagner theory and systematically derive a series of splash jet models in order to find possible mechanisms for phenomena seen in droplet impact and droplet spreading experiments. Our canonical model is a thin jet of liquid shot over a substrate with a thin air layer trapped between the jet and the substrate. We consider a variety of parameter regimes and investigate the stability of the jet in each regime. We then use this model as part of a growing-jet problem, in which we attempt to include effects due to the jet tip. In the final chapter we summarise the main results of the thesis and outline directions for future work.