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The aim of this thesis is to develop a quantitative measure of postural defects known as lordosis and kyphosis. The measurement of these is an important part of their identification and treatment.

The aim of this thesis is to develop a quantitative measure of postural defects known as lordosis and kyphosis. The measurement of these is an important part of their identification and treatment.

In any population of cells, individual cells grow for some period of time and then divide into two or more parts, called daughters. To describe this process mathematically, we need to specify functions describing the growth rate, size at division, and proportions into which each cell divides. In this thesis, it is assumed that the growth rate of a cell can be determined precisely from its size, but that both its size at division and the proportions into which it divides may be described stochastically, by probability density functions whose parameters are dependent on cell size and age (or birth-size). Special cases are also considered where all cells with the same birth-size divide at the same size, or where all cells divide exactly in half. We consider a population of cells growing and dividing steadily, such that the total cell population is increasing, but the proportion of cells in any size class remains constant. In Chapter 1, equations are derived which need to be solved in order to deduce the shape of the steady size distribution (or steady size/age or size/birth-size distributions) from any given growth rate and probability distributions describing the division rate and division proportions. In the general case, a Fredholm-type integral equation is obtained, but if the probability of cell division depends on cell size only (i.e. not age or birth-size), and all cells divide into equal-sized daughters, then we obtain a functional differential equation. In two special cases, the resulting equations simplify considerably, and it is these cases which are explored further in this thesis. The first is where the probability of a cell dividing in any instant of time is a constant, independent of cell age or size. In Chapter 2, the functional differential equation resulting when cells divide into equal-sized daughters is solved for the special case where the growth rate is constant, and in an appendix the case where the growth rate is described by a power law is dealt with. The second case which simplifies is where the time-independent part of the growth rate of a cell is proportional to cell size. This case is particularly important, as it is a good first-order approximation to the real cell growth rate in some structured tissues, and in some bacteria. The special case in which this leads to a functional differential equation is discussed in Chapter 3, and the integral equation arising in the general case is dealt with in Chapter 4. Finally, the conditions under which the integral operator in Chapter 4 will be both square-integrable and non-factorable are discussed in Chapter 5. It is shown that if these conditions are satisfied then a unique, stable, steady size distribution will exist.

<div>Public health policies increasingly rely on complex models that need to approximate epidemics realistically and be consistent with the available data. Choosing appropriate simplifying assumptions is one of the critical challenges in disease modeling. In this thesis, we focus on some of these assumptions to show how they impact model outcomes. </div><div>In this thesis, an ODE model with a gamma-distributed infectious period is studied and compared with an exponentially distributed infectious period. We show that, for childhood diseases, isolating infected children is a possible mechanism causing oscillatory behavior in incidence. This is shown analytically by identifying a Hopf bifurcation with the isolation period as the bifurcation parameter. The threshold value for isolation to generate sustained oscillations from the model with gamma-distributed isolation period is much more realistic than the exponentially distributed model.</div><div><br></div><div>The consequences of not modeling the spectrum of clinical symptoms of the 2014 Ebola outbreak in Liberia include overestimating the basic reproduction number and effectiveness of control measures. The outcome of this model is compared with those of models with typical symptoms, excluding moderate ones. Our model captures the dynamics of the recent outbreak of Ebola in Liberia better, and the basic reproduction number is more consistent with the WHO response team's estimate. Additionally, the model with only typical symptoms overestimates the basic reproduction number and effectiveness of control measures and exaggerates changes in peak size attributable to interventions' timing.</div><div><br></div>

The increasing availability of molecular data means that phylogenetic studies nowadays often use datasets which combine a large number of loci for many different species. This leads to a trade-off. On the one hand more complex models are preferred to account for heterogeneity in evolutionary processes. On the other hand simple models that can answer biological questions of interest that are easy to interpret and can be computed in reasonable time are favoured. This thesis focuses on four cases of phylogenetic analysis which arise from this conflict. - It is shown that edge weight estimates can be non-identifiable if the data are simulated under a mixture model. Even if the underlying process is known the estimation and interpretation may be difficult due to the high variance of the parameters of interest. - Partition models are commonly used to account for heterogeneity in data sets. Novel methods are presented here which allow grouping of genes under similar evolutionary constraints. A data set, containing 14 genes of the chloroplast from 19 anciently diverged species is used to find groups of co-evolving genes. The prospects and limitations of such methods are discussed. - Penalised likelihood estimation is a useful tool for improving the performance of models and allowing for variable selection. A novel approach is presented that uses pairwise dissimilarities to visualise the data as a network. It is further shown how penalised likelihood can be used to decrease the variance of parameter estimates for mixture and partition models, allowing a more reliable analysis. Estimates for the variance and the expected number of parameters of penalised likelihood estimates are derived. - Tree shape statistics are used to describe speciation events in macroevolution. A new tree shape statistic is introduced and the biases of different cluster methods on tree shape statistics are discussed.

This thesis is a collection of the three projects I have worked on at Purdue. The first is a paper on thermoacoustic tomography involving circular integrating detectors that was published in Inverse Problems and Imaging. Results from this paper include demonstrating that the measurement operators involved are Fourier integral operators, as well as proving microlocal uniqueness in certain cases, and also stability. The second paper, submitted to the Journal of Inverse and Ill-Posed Problems, is much more of an application of sampling theory in to the specific case of thermoacoustic tomography. Results from this paper include demonstrating resolution limits imposed by sampling rates, and showing that aliasing artifacts appear in predictable locations in an image when the measurement operator is under sampled in either the time variable or space variables. We also show an application of a basic anti aliasing scheme based on averaging of data. The last project moves slightly away from microlocal analysis and considers the uniqueness in medical imaging of the restricted Radon transform in even dimensions. This is the classical interior problem, and we show a characterization of the range of the Radon transform, and from this are able to obtain a characterization of the kernel of the restricted Radon transform. We include figures throughout to illustrate results.

<div>Wildlife diseases are an increasingly growing concern for public health managers, conservation biologists, and society at large. These diseases may be zoonotic -- infective wildlife are able to spread pathogens to human populations. Animal or plant species of conservation concern may also be threatened with extinction or extirpation due to the spread of novel pathogens into their native ranges. In this thesis, I develop some mathematical methods for understanding the dynamics of vector-borne diseases in wildlife populations which include several elements of host and vector biology. </div><div><br></div><div>We consider systems where a vector-borne pathogen is transmitted to a host population wherein individuals either die to disease or recover, remaining chronically infective. Both ordinary differential equations (ODE) and individual based (IBM) models of such systems are formulated then applied to a specific system of wildlife disease: avian malaria in Hawaiian honeycreeper populations -- where some species endure disease-induced mortality rates exceeding 90\%. The ODE model predicts that conventional management methods cannot fully stop pathogen transmission.</div><div><br></div><div>Vector dispersal and reproductive biology may also play a large role in the transmission of vector-borne diseases in forested environments. Using an IBM which models dispersal and mosquito reproductive biology, we predict that reducing larval habitat at low elevations is much more effective than at higher elevations. The ODE model is extended to include distinct populations of sensitive and tolerant hosts. We find that the form which interaction between the hosts takes has a significant impact on model predictions.</div>

Phylogenetic models generally assume a homogeneous, time reversible, stationary process. These assumptions are often violated by the real, far more complex, evolutionary process. This thesis is centered on non-homogeneous, lineage-specific, properties of molecular sequences. It consist several related but independent studies. LineageSpecificSeqgen, an extension to the Seq-Gen program, which allows generation of sequences with changes in the proportion of variable sites, is introduced. This program is then used in a simulation study showing that changes in the proportion of variable sites can hinder tree estimation accuracy, and that tree reconstruction under the best-fit model chosen using a relative test can result in a wrong tree. In this case, the less commonly used absolute model-fit was a better predictor of tree estimation accuracy. This study found that increased taxon sampling of lineages that have undergone a change in the proportion of variable sites was critical for accurate tree reconstruction and that, in contrast to some earlier findings, the accuracy of maximum parsimony is adversely affected by such changes. This thesis also addresses the well-known long-branch attraction artifact. A nonparametric bootstrap test to identify changes in the substitution process is introduced, validated, and applied to the case of Microsporidia, a highly reduced intracellular parasite. Microsporidia was first thought to be an early branching eukaryote, but is now believed to be sister to, or included within, fungi. Its apparent basal eukaryote position is considered a result of long-branch attraction due to an elevated evolutionary rate in the microsporidian lineage. This study shows that long-branch estimates and basal positioning of Microsporidia both correlate with increased proportions of radical substitutions in the microsporidian lineage. In simulated data, such increased proportions of radical substitutions leads to erroneous long-branch estimates. These results suggest that the long microsporidian branch is likely to be a result of an increased proportion of radical substitutions on that branch, rather than increased evolutionary rate per se. The focus of the last study is the intriguing case of Mesostigma, a fresh water green alga for which contradicting phylogenetic relationships were inferred. While some studies placed Mesostigma within the Streptophyta lineage (which includes land plants), others placed it as the deepest green algae divergence. This basal positioning is regarded as a result of long-branch attraction due to poor taxon sampling. Reinvestigation of a 13- taxon mitochondrial amino acid dataset and a sub-dataset of 8 taxa reveals that site sampling, and in particular the treatment of missing data, is just as important a factor for accurate tree reconstruction as taxon sampling. This study identifies a difficulty in recreating the long-branch attraction observed for the 8-taxon dataset in simulated data. The cause is likely to be the smaller number of amino acid characters per site in simulated data compared to real data, highlighting the fact that there are properties of the evolutionary process that are yet to be accurately modeled.

Phylogenetic models generally assume a homogeneous, time reversible, stationary process. These assumptions are often violated by the real, far more complex, evolutionary process. This thesis is centered on non-homogeneous, lineage-specific, properties of molecular sequences. It consist several related but independent studies. LineageSpecificSeqgen, an extension to the Seq-Gen program, which allows generation of sequences with changes in the proportion of variable sites, is introduced. This program is then used in a simulation study showing that changes in the proportion of variable sites can hinder tree estimation accuracy, and that tree reconstruction under the best-fit model chosen using a relative test can result in a wrong tree. In this case, the less commonly used absolute model-fit was a better predictor of tree estimation accuracy. This study found that increased taxon sampling of lineages that have undergone a change in the proportion of variable sites was critical for accurate tree reconstruction and that, in contrast to some earlier findings, the accuracy of maximum parsimony is adversely affected by such changes. This thesis also addresses the well-known long-branch attraction artifact. A nonparametric bootstrap test to identify changes in the substitution process is introduced, validated, and applied to the case of Microsporidia, a highly reduced intracellular parasite. Microsporidia was first thought to be an early branching eukaryote, but is now believed to be sister to, or included within, fungi. Its apparent basal eukaryote position is considered a result of long-branch attraction due to an elevated evolutionary rate in the microsporidian lineage. This study shows that long-branch estimates and basal positioning of Microsporidia both correlate with increased proportions of radical substitutions in the microsporidian lineage. In simulated data, such increased proportions of radical substitutions leads to erroneous long-branch estimates. These results suggest that the long microsporidian branch is likely to be a result of an increased proportion of radical substitutions on that branch, rather than increased evolutionary rate per se. The focus of the last study is the intriguing case of Mesostigma, a fresh water green alga for which contradicting phylogenetic relationships were inferred. While some studies placed Mesostigma within the Streptophyta lineage (which includes land plants), others placed it as the deepest green algae divergence. This basal positioning is regarded as a result of long-branch attraction due to poor taxon sampling. Reinvestigation of a 13- taxon mitochondrial amino acid dataset and a sub-dataset of 8 taxa reveals that site sampling, and in particular the treatment of missing data, is just as important a factor for accurate tree reconstruction as taxon sampling. This study identifies a difficulty in recreating the long-branch attraction observed for the 8-taxon dataset in simulated data. The cause is likely to be the smaller number of amino acid characters per site in simulated data compared to real data, highlighting the fact that there are properties of the evolutionary process that are yet to be accurately modeled.