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Full-Text Articles in Physical Sciences and Mathematics

Using Physical And Social Sensors In Real-Time Data Streaming For Natural Hazard Monitoring And Response, Yelena Kropivnitskaya Aug 2016

Using Physical And Social Sensors In Real-Time Data Streaming For Natural Hazard Monitoring And Response, Yelena Kropivnitskaya

Electronic Thesis and Dissertation Repository

Technological breakthroughs in computing over the last few decades have resulted in important advances in natural hazards analysis. In particular, integration of a wide variety of information sources, including observations from spatially-referenced physical sensors and new social media sources, enables better estimates of real-time hazard. The main goal of this work is to utilize innovative streaming algorithms for improved real-time seismic hazard analysis by integrating different data sources and processing tools into cloud applications. In streaming algorithms, a sequence of items from physical and social sensors can be processed in as little as one pass with no need to store …


Studying Both Direct And Indirect Effects In Predator-Prey Interaction, Xiaoying Wang Aug 2016

Studying Both Direct And Indirect Effects In Predator-Prey Interaction, Xiaoying Wang

Electronic Thesis and Dissertation Repository

Studying and modelling the interaction between predators and prey have been one of the central topics in ecology and evolutionary biology. In this thesis, we study two different aspects of predator-prey interaction: direct effect and indirect effect.

Firstly, we study the direct predation between predators and prey in a patchy landscape.

Secondly, we study indirect effects between predators and prey.

Thirdly, we extend our previous model by incorporating a stage-structure into prey.

Finally, we further extend our previous model by incorporating spatial structures into modelling.


Bacteria-Phage Models With A Focus On Prophage As A Genetic Reservoir, Alina Nadeem Aug 2016

Bacteria-Phage Models With A Focus On Prophage As A Genetic Reservoir, Alina Nadeem

Electronic Thesis and Dissertation Repository

Temperate bacteriophages have the ability to incorporate their genetic material in the host's DNA, which may be utilized by later generations of phage to overcome the host's receptor-based defences. This effect of temperance can have major implications for the long-term survival of the phages as well as on bacteria-phage community evolution. To study the impact of prophage on microbial communities we have developed models simulating lytic and lysogenic infection and host and phage coevolution with a focus on prophage-phage recombination. Our results show that recombination can be crucial for the phage to survive host diversification, and a higher incidence of …


On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi Aug 2016

On Logarithmic Sobolev Inequality And A Scalar Curvature Formula For Noncommutative Tori, Sajad Sadeghi

Electronic Thesis and Dissertation Repository

In the first part of this thesis, a noncommutative analogue of Gross' logarithmic Sobolev inequality for the noncommutative 2-torus is investigated. More precisely, Weissler's result on the logarithmic Sobolev inequality for the unit circle is used to propose that the logarithmic Sobolev inequality for a positive element $a= \sum a_{m,n} U^{m} V^{n} $ of the noncommutative 2-torus should be of the form $$\tau(a^{2} \log a)\leqslant \underset{(m,n)\in \mathbb{Z}^{2}}{\sum} (\vert m\vert + \vert n\vert) \vert a_{m,n} \vert ^{2} + \tau (a^{2})\log ( \tau (a^2))^{1/2},$$ where $\tau$ is the normalized positive faithful trace of the noncommutative 2-torus. A possible approach to prove this …


Self-Assembled Copper Nanoparticle Superlattices On Graphene Thin Films, Tianhao Ouyang Aug 2016

Self-Assembled Copper Nanoparticle Superlattices On Graphene Thin Films, Tianhao Ouyang

Electronic Thesis and Dissertation Repository

Recently, Giovannetti et al. successfully demonstrated that some metals (such as Cu and Au) only have weak van der waals interaction with graphene and thus can only form weak bonding without severely shifting graphene’s band structure, which describes the energies range of the electrons in the material. Therefore, this opens up windows for graphene enhancement without greatly changing its properties. Furthermore, Zhou et al. later suggested the possibility of self-assembling periodic arrays of alkali atoms on graphene. In our group, graphene thin films fabricated in a cost effective way using solution-processed methods have been used extensively, including decorating …


Actuarial Modelling With Mixtures Of Markov Chains, Yuzhou Zhang Aug 2016

Actuarial Modelling With Mixtures Of Markov Chains, Yuzhou Zhang

Electronic Thesis and Dissertation Repository

Multi-state models are widely used in actuarial science because that they provide a convenient way of representing changes in people's statuses. Calculations are easy if one assumes that the model is a Markov chain. However, the memoryless property of a Markov chain is rarely appropriate.

This thesis considers several mixtures of Markov chains to capture the heterogeneity of people's mortality rates, morbidity rates, recovery rates, and ageing speeds. This heterogeneity may be the result of unobservable factors that affect individuals' health. The focus of this thesis is on investigating the behaviours of intensities of the observable transitions in the mixture …


A Comparison Of Solution Methods For Mandelbrot-Like Polynomials, Eunice Y. S. Chan Aug 2016

A Comparison Of Solution Methods For Mandelbrot-Like Polynomials, Eunice Y. S. Chan

Electronic Thesis and Dissertation Repository

We compare two different root-finding methods, eigenvalue methods and homotopy methods, using three test problems: Mandelbrot polynomials, Fibonacci-Mandelbrot polynomials, and Narayana-Mandelbrot polynomials. For the eigenvalue methods, using both MATLAB and Maple, we computed the eigenvalues of a specialized recursively-constructed, supersparse, upper Hessenberg matrix, inspired by Piers Lawrence's original construction for the Mandelbrot polynomials, for all three families of polynomials. This led us to prove that this construction works in general. Therefore, this construction is genuinely a new kind of companion matrix. For the homotopy methods, we used a special-purpose homotopy, in which we used an equivalent differential equation to solve …


Anisotropic Waveform Tomography: Application To Crosshole Data For Transversely Isotropic Media, Shaun Michael Hadden Aug 2016

Anisotropic Waveform Tomography: Application To Crosshole Data For Transversely Isotropic Media, Shaun Michael Hadden

Electronic Thesis and Dissertation Repository

Anisotropic Traveltime Tomography and Full Waveform Inversion were applied first to synthetic and then to real data following the development of a transversely isotropic model for handling anisotropy. Best-fitting models of seismic velocity and Thomsen's anisotropy parameters were initially obtained from traveltime tomography, and then used as the starting models for Full Waveform Inversion. The use of a Laplace transform approach effectively damps late arriving S-wave artifacts that introduce errors into the modelling process. The results of the synthetic study highlights the tradeoffs in resolution between the two parameter classes, but verify anisotropic traveltime tomography as a valid method for …


Modeling The Mass Function Of Stellar Clusters Using The Modified Lognormal Power-Law Probability Distribution Function, Deepakshi Madaan Aug 2016

Modeling The Mass Function Of Stellar Clusters Using The Modified Lognormal Power-Law Probability Distribution Function, Deepakshi Madaan

Electronic Thesis and Dissertation Repository

We use the Modified Lognormal Power-law (MLP) probability distribution function to model the behaviour of the mass function (MF) of young and populous stellar populations in different environments. We begin by modeling the MF of NGC1711, a simple stellar population (SSP) in the Large Magellanic Cloud as a pilot case. We then use model selection criterion to differentiate between candidate models. Using the MLP we find that the stellar catalogue of NGC1711 follows a pure power-law behaviour below the completeness limit with the slope α = 2.75 for dN/dlnm ∝ m^(−α+1) in the mass range 0.89 M⊙ to 7.75 M⊙. …


Fractional Charge Methods For Correcting Approximate Kohn-Sham Potentials, Darya N. Komsa Aug 2016

Fractional Charge Methods For Correcting Approximate Kohn-Sham Potentials, Darya N. Komsa

Electronic Thesis and Dissertation Repository

The Kohn-Sham density functional theory relies on approximating the exchange-correlation energy functional or the corresponding potential. The behavior of the exchange-correlation potential as a function of position in a system can be used to detect and correct deficiencies of the parent functional. The too-fast decay of the potentials derived from common density functionals is a major problem, because it causes inaccurate Rydberg excitation energies and erroneous fractional charges in dissociating molecules. An efficient method to correct the shape of the exchange-correlation potential was proposed by Gaiduk et al. [A. P. Gaiduk, D. S. Firaha, and V. N. Staroverov, Phys. Rev. …


Cobalt Mediated Oxidative Cyclizations: The Diastereoselective Synthesis Of Trans-Tetrahydrofuran Rings, Cory D.N. Palmer Aug 2016

Cobalt Mediated Oxidative Cyclizations: The Diastereoselective Synthesis Of Trans-Tetrahydrofuran Rings, Cory D.N. Palmer

Electronic Thesis and Dissertation Repository

The Mukaiyama oxidative cyclization has been a major area of study within the Pagenkopf group and within the field of THF containing natural products. This document contains a brief discussion of the major divisions of THF synthesis; SN2, cycloadditions, and transition mediated strategies. The development of the Mukaiyama cyclization is described from the earliest applications of cobalt acac type ligand in Mukaiyama’s work to the initial discovery of the cyclization reactivity. Initial finding from the Mukaiyama group is shown, accompanied by its applications to the synthesis of mucosin, aplysiallene, and bullatacin. Previous attempts at improved catalyst design are …


Mechanism Of Ethanol Partial Oxidation Over Titania Supported Vanadia Catalysts: Geometric And Electronic Structure Consequences On Reaction Kinetics, Dongmin Yun Aug 2016

Mechanism Of Ethanol Partial Oxidation Over Titania Supported Vanadia Catalysts: Geometric And Electronic Structure Consequences On Reaction Kinetics, Dongmin Yun

Electronic Thesis and Dissertation Repository

Supported vanadium oxide catalysts (VOx) are used for a wide range of industrial processes. A significant amount of research has been carried on this specific system with the aim to identify the critical parameters involved in catalytic activity. The results on this relative large body of research indicate that the catalyst support plays a critical role in determining the catalytic activity of vanadia and the use of titanium dioxide as support renders a catalyst with superior activity. Studies on supported vanadia on titanium oxide (VOx/TiO2) have provided an understanding on support interactions that result …


K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, Ivan Kobyzev Aug 2016

K-Theory Of Root Stacks And Its Application To Equivariant K-Theory, Ivan Kobyzev

Electronic Thesis and Dissertation Repository

We give a definition of a root stack and describe its most basic properties. Then we recall the necessary background (Abhyankar’s lemma, Chevalley-Shephard-Todd theorem, Luna’s etale slice theorem) and prove that under some conditions a quotient stack is a root stack. Then we compute G-theory and K-theory of a root stack. These results are used to formulate the theorem on equivariant algebraic K-theory of schemes.


Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega Aug 2016

Moduli Space And Deformations Of Special Lagrangian Submanifolds With Edge Singularities, Josue Rosario-Ortega

Electronic Thesis and Dissertation Repository

Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this thesis we use elliptic theory for edge- degenerate differential operators on singular manifolds to study general deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.


New Chemistry Of Donor-Acceptor Cycloalkanes And Studies Towards The Synthesis Of Streptorubin B, Naresh Vemula Aug 2016

New Chemistry Of Donor-Acceptor Cycloalkanes And Studies Towards The Synthesis Of Streptorubin B, Naresh Vemula

Electronic Thesis and Dissertation Repository

This dissertation presents two separate chapters within the broad area of synthetic organic chemistry. The first chapter describes the annelation chemistry of donor-acceptor (DA) cyclopropanes and cyclobutanes for the synthesis of heterocycles. The Yb(OTf)3-catalyzed [4+2] cycloaddition between DA cyclobutanes and nitrosoarenes facilitated the synthesis of tetrahydro-1,2-oxazines in good to excellent yields as single diastereomers. Additionally, an unexpected deoxygenation occurred with electron-rich nitrosoarenes under MgI2-catalysis that afforded pyrrolidine products. The GaCl3-catalyzed [4+2] cycloaddition of DA cyclobutanes and cis-diazenes provided hexahydropyridazine derivatives in good to excellent yields as single diastereomers. Furthermore, a procedure to make …


Aging Comets And Their Meteor Showers, Quanzhi Ye Jul 2016

Aging Comets And Their Meteor Showers, Quanzhi Ye

Electronic Thesis and Dissertation Repository

Comets are thought to be responsible for the terrestrial accretion of water and organic materials. Comets evolve very quickly, and will generally deplete their volatiles in a few hundred revolutions. This process, or the aging of comets, is one of the most critical yet poorly understood problems in planetary astronomy. The goal of this thesis is to better understand this problem by examining different parts of the cometary aging spectrum of Jupiter-family comets (JFCs), a group of comets that dominates the cometary influx in the near-Earth space, using both telescopic and meteor observations. We examine two representative JFCs and the …


Development And Optimization Of Solid Polymer Electrolyte For Lithium Ion Batteries, Ladan Eskandarian Jul 2016

Development And Optimization Of Solid Polymer Electrolyte For Lithium Ion Batteries, Ladan Eskandarian

Electronic Thesis and Dissertation Repository

This thesis focuses on the development of new poly(ethylene oxide) (PEO)-based solid polymer electrolytes (SPEs) in order to enhance their ionic conductivity at ambient temperature and fabricate the prototypes of novel Li ion batteries using these SPEs. Different types of SPEs have been developed: (i) blends of high molecular weight PEO and low molecular weight poly(vinyl acetate) (PVAc); (ii) composites of high molecular weight PEO and titanium dioxide (TiO2)nanoparticles; and (iii) blend-based composite electrolytes consisting of PEO and PVAc with dispersed TiO2. The SPEs were characterized by scanning electron microscopy (SEM), thermal gravimetric analysis (TGA), and …


The Survival Probability Of Beneficial De Novo Mutations In Budding Viruses, With An Emphasis On Influenza A Viral Dynamics, Jennifer Ns Reid Jul 2016

The Survival Probability Of Beneficial De Novo Mutations In Budding Viruses, With An Emphasis On Influenza A Viral Dynamics, Jennifer Ns Reid

Electronic Thesis and Dissertation Repository

A deterministic model is developed of the within-host dynamics of a budding virus, and coupled with a detailed life-history model using a branching process approach to follow the fate of de novo beneficial mutations affecting five life-history traits: clearance, attachment, eclipse, budding, and cell death. Although the model can be generalized for any given budding virus, our work was done with a major emphasis on the early stages of infection with influenza A virus in human populations. The branching process was then interleaved with a stochastic process describing the disease transmission of this virus. These techniques allowed us to predict …


Digging Deeper With Diffuse Correlation Spectroscopy, Kyle J. Verdecchia Jul 2016

Digging Deeper With Diffuse Correlation Spectroscopy, Kyle J. Verdecchia

Electronic Thesis and Dissertation Repository

Patients with neurological diseases are vulnerable to cerebral ischemia, which can lead to brain injury. In the intensive care unit (ICU), neuromonitoring techniques that can detect flow reductions would enable timely administration of therapies aimed at restoring adequate cerebral perfusion, thereby avoiding damage to the brain. However, suitable bedside neuromonitoring methods sensitive to changes of blood flow and/or oxygen metabolism have yet to be established.

Near-infrared spectroscopy (NIRS) is a promising technique capable of non-invasively monitoring flow and oxygenation. Specifically, diffuse correlation spectroscopy (DCS) and time-resolved (TR) NIRS can be used to monitor blood flow and tissue oxygenation, respectively, and …


Omcvd Gold Nanoparticles Covalently Attached To Polystyrene For Biosensing Applications, Sivayini Kandeepan Jul 2016

Omcvd Gold Nanoparticles Covalently Attached To Polystyrene For Biosensing Applications, Sivayini Kandeepan

Electronic Thesis and Dissertation Repository

Remarkable developments and successes were witnessed in the fabrication and implementation of optical sensors based on localized surface plasmon resonance (LSPR) for the investigation of chemical and biological material quantities and to detect lethal diseases such as cancer in early stages. Gold nanoparticles (AuNPs) are ideal candidate for sensing purposes due to their chemical stability, ease of surface functionalization and strong LSPR in the visible range. Although there are several designs of sensors published, most of them are still limited to small scale research laboratory use partly due to their high cost of fabrication and waste management, in particular critical …


Uniform Approximation On Riemann Surfaces, Fatemeh Sharifi Jul 2016

Uniform Approximation On Riemann Surfaces, Fatemeh Sharifi

Electronic Thesis and Dissertation Repository

This thesis consists of three contributions to the theory of complex approximation on

Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is usually not possible to approximate f uniformly by functions holomorphic on all of R. Firstly, we show, however, that for every open Riemann surface R and every closed subset E of R; there is closed subset F of E, which approximates E extremely well, such that every function holomorphic on F can be approximated much …


Gauss-Bonnet-Chern Type Theorem For The Noncommutative Four-Sphere, Mitsuru Wilson Jul 2016

Gauss-Bonnet-Chern Type Theorem For The Noncommutative Four-Sphere, Mitsuru Wilson

Electronic Thesis and Dissertation Repository

We introduce a pseudo-Riemannian calculus of modules over noncommutative al- gebras in order to investigate to what extent the differential geometry of classical Riemannian manifolds can be extended to a noncommutative setting. In this frame- work, it is possible to prove an analogue of the Levi-Civita theorem. It states that there exists at most one connection, which satisfies torsion-free condition and metric compatibility condition, on a given smooth manifold with fixed metric. More signif- icantly, the corresponding curvature operator has the same symmetry properties as the classical curvature tensors. We consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and the …


Hyperpolarized 3he Magnetic Resonance Imaging Phenotypes Of Chronic Obstructive Pulmonary Disease, Damien Pike Jul 2016

Hyperpolarized 3he Magnetic Resonance Imaging Phenotypes Of Chronic Obstructive Pulmonary Disease, Damien Pike

Electronic Thesis and Dissertation Repository

Chronic obstructive pulmonary disease (COPD) is the third leading cause of death in the world. Identifying clinically relevant COPD phenotypes has the potential to reduce the global burden of COPD by helping to alleviate symptoms, slow disease progression and prevent exacerbation by stratifying patient cohorts and forming targeted treatment plans. In this regard, quantitative pulmonary imaging with hyperpolarized 3He magnetic resonance imaging (MRI) and thoracic computed tomography (CT) have emerged as ways to identify and measure biomarkers of lung structure and function. 3He MRI may be used as a tool to probe both functional and structural properties of the lung …


Anthropogenic Particles In Natural Sediment Sinks: Microplastics Accumulation In Tributary, Beach And Lake Bottom Sediments Of Lake Ontario, North America, Anika M. Ballent Jul 2016

Anthropogenic Particles In Natural Sediment Sinks: Microplastics Accumulation In Tributary, Beach And Lake Bottom Sediments Of Lake Ontario, North America, Anika M. Ballent

Electronic Thesis and Dissertation Repository

Surface waters of the Great Lakes are known to be contaminated with microplastics, however, microplastics in the sediments of the region are poorly documented. This study provides a baseline of micro- and macro-plastics contamination in nearshore, tributary and beach sediments of Lake Ontario and the upper St. Lawrence River. Microplastics were quantified and characterized by morphology and composition using visual identification and Raman spectroscopy. Microplastics are most concentrated in nearshore sediments in the vicinity of urban and industrial regions. Concentrations in Humber Bay and Toronto Harbour consistently measured > 500 particles per kg dry sediment, and maximum concentrations of ~28,000 particles …


Optimizing Respiratory Gated Intensity Modulated Radiation Therapy Planning And Delivery Of Early-Stage Non-Small Cell Lung Cancer, Ilma Xhaferllari Jul 2016

Optimizing Respiratory Gated Intensity Modulated Radiation Therapy Planning And Delivery Of Early-Stage Non-Small Cell Lung Cancer, Ilma Xhaferllari

Electronic Thesis and Dissertation Repository

Stereotactic ablative body radiotherapy (SABR) is the standard of care for inoperable early-stage non-small cell lung cancer (NSCLC) patients. However, thoracic tumours are susceptible to respiratory motion and, if unaccounted for, can potentially lead to dosimetric uncertainties. Respiratory gating is one method that limits treatment delivery to portions of the respiratory cycle, but when combined with intensity-modulated radiotherapy (IMRT), requires rigorous verification. The goal of this thesis is to optimize respiratory gated IMRT treatment planning and develop image-guided strategies to verify the dose delivery for future early-stage NSCLC patients.

Retrospective treatment plans were generated for various IMRT delivery techniques, including …


Climbing Up Cloud Nine: Performance Enhancement Techniques For Cloud Computing Environments, Mohamed Abusharkh Jul 2016

Climbing Up Cloud Nine: Performance Enhancement Techniques For Cloud Computing Environments, Mohamed Abusharkh

Electronic Thesis and Dissertation Repository

With the transformation of cloud computing technologies from an attractive trend to a business reality, the need is more pressing than ever for efficient cloud service management tools and techniques. As cloud technologies continue to mature, the service model, resource allocation methodologies, energy efficiency models and general service management schemes are not yet saturated. The burden of making this all tick perfectly falls on cloud providers. Surely, economy of scale revenues and leveraging existing infrastructure and giant workforce are there as positives, but it is far from straightforward operation from that point. Performance and service delivery will still depend on …


Joint Analysis Of Zero-Heavy Longitudinal Outcomes: Models And Comparison Of Study Designs, Erin R. Lundy Jul 2016

Joint Analysis Of Zero-Heavy Longitudinal Outcomes: Models And Comparison Of Study Designs, Erin R. Lundy

Electronic Thesis and Dissertation Repository

Understanding the patterns and mechanisms of the process of desistance from criminal activity is imperative for the development of effective sanctions and legal policy. Methodological challenges in the analysis of longitudinal criminal behaviour data include the need to develop methods for multivariate longitudinal discrete data, incorporating modulating exposure variables and several possible sources of zero-inflation. We develop new tools for zero-heavy joint outcome analysis which address these challenges and provide novel insights on processes related to offending patterns. Comparisons with existing approaches demonstrate the benefits of utilizing modeling frameworks which incorporate distinct sources of zeros. An additional concern in this …


X-Ray Populations In The Local Group: Insights With Hubble & Chandra, Neven Vulic Jun 2016

X-Ray Populations In The Local Group: Insights With Hubble & Chandra, Neven Vulic

Electronic Thesis and Dissertation Repository

X-ray observations provide a unique perspective on the most energetic processes in the Universe. In particular, Low-mass X-ray binaries (LMXBs) found in globular clusters have been shown to depend on the mass, radius, and metallicity of the cluster. This thesis focuses on the impact environmental parameters have on X-ray sources and the underlying physical explanations for them.

I studied the X-ray binary population in M31 using 1 Ms of Chandra ACIS data and 6-filter photometry from the Panchromatic Hubble Andromeda Treasury Survey. From a sample of 83 star clusters we found the brightest and most compact star clusters preferentially hosted …


The Corrosion Of Carbon Steel Under Deep Geologic Nuclear Waste Disposal Conditions, Shannon L.W. Hill Jun 2016

The Corrosion Of Carbon Steel Under Deep Geologic Nuclear Waste Disposal Conditions, Shannon L.W. Hill

Electronic Thesis and Dissertation Repository

The proposed disposal scenario for high-level nuclear waste (spent fuel) in Canada is emplacement within a sealed, deep geological repository (DGR) located in either granitic rock or sedimentary clay. Disposal is based on a multi-barrier approach, with the primary barrier being a sealed container which could be either dual-walled with a copper shell over an inner carbon steel vessel for granitic rock or a single thick-walled steel container for sedimentary clay. This study focuses on the corrosion behaviour of A516 Gr70 carbon steel as well as the corrosion products formed in a variety of groundwater compositions and concentrations expected within …


Joint Modelling In Liver Transplantation, Elizabeth M. Renouf Jun 2016

Joint Modelling In Liver Transplantation, Elizabeth M. Renouf

Electronic Thesis and Dissertation Repository

In the setting of liver transplantation, clinical trials and transplant registries regularly collect repeated measurements of clinical biomarkers which may be strongly associated with a time-to-event such as graft failure or disease recurrence. Multiple time-to-event outcomes are routinely collected. However, joint models are rarely used. This thesis will describe important considerations for joint modelling in the setting of liver transplantation. We will focus on transplant registry data from the United States. We develop a new tool for joint modelling in the context where a critical health event can be tracked in the longitudinal biomarker and often presents as a non-linear …