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Properly representing the principle physical interactions of complex biological systems is paramount for building powerful, yet simple models. As an in depth look into different biological systems at different scales, multiple models are presented. At the molecular scale, an analytical solution to the (linearized) Poisson-Boltzmann equation for the electrostatic potential of any size biomolecule is derived using spherical geometry. The solution is tested both on an ideal sphere relative to an exact solution and on a multitude of biomolecules relative to a numerical solution. In all cases, the bulk of the error is within thermal noise. The computational power of the solution is demonstrated by finding the electrostatic potential at the surface of a viral capsid that is nearly half a million atoms in size. Next, a model of the nucleosome using simplified geometry is presented. This system is a complex of protein and DNA and acts as the first level of DNA compaction inside the nucleus of eukaryotes. The analytical model reveals a mechanism for controlling the stability of the nucleosome via changes to the total charge of the protein globular core. The analytical model is verified by a computational study on the stability change when the charge of individual residues is altered. Finally, a multiple model approach is taken to study bacteria that are capable of different responses depending on the size of their surrounding colony. The first model is capable of determining how the system propagates the information about the colony size to those specific genes that control the concentration of a master regulatory protein. A second model is used to analyze the direct RNA interference mechanism the cell employs to tune the available gene transcripts of the master regulatory protein, i.e. small RNA - messenger RNA regulation. This model provides a possible explanation for puzzling experimentally measured phenotypic responses. / Ph. D.

Exact characterization and performance analysis of wireless networks should incorporate dependencies or correlations in space and time, i.e., study how the network performance varies spatially and temporally while having prior information about the performance at previous locations and time slots. This spatio-temporal correlation in wireless networks is usually characterized by studying metrics such as joint coverage probability at two spatial locations/time slots or spatio-temporal correlation coefficient. While developing models and analytical expressions for studying the two extreme cases of spatio-temoral correlation - i) uncorrelated scenario and ii) fully correlated scenario are easier, studying the intermediate case is non-trivial. In this thesis, we develop realistic and tractable analytical frameworks based on random spatial models (using tools from stochastic geometry) for modeling and analysis of correlation in cellular networks. With an ever increasing data demand, caching popular content in the storage of small cells (small cell caching) or the memory of user devices (device caching) is seen as a good alternative to offload demand from macro base stations and reduce backhaul loads. After providing generic results for traditional cellular networks, we study two applications exploiting spatio-temporal correlation in cache-enabled cellular networks. First, we determine the optimal cache content to be stored in the cache of a small cell network that maximizes the hit probability and minimizes the reception energy for the two extreme cases of correlation. Our results concretely demonstrate that the optimal cache contents are significantly different for the two correlation scenarios, thereby indicating the need of correlation-aware caching strategies. Second, we look at a distributed caching scenario in user devices and show that spatio-temporal correlation (user mobility) can be exploited to improve the network performance (in terms of coverage probability and local delay) significantly. / Master of Science

The electrostatic properties of a molecule are often essential in determining its behavior; as such, the ability to approximate these electrostatic potentials computationally is often essential to obtaining a full understanding of how these molecules function. An approximate, analytical solution to the (linearized) Poisson-Boltzmann equation is proposed that is suitable for realistic biomolecules of virtually any size. A comparison with accepted numerical approaches on a large test set of biomolecular structures shows that the proposed method is considerably less expensive computationally, yet accurate enough to be considered as a possible alternative. The utility of the approach is demonstrated by computing and analyzing the electrostatic potential generated by full capsid of the tobacco ringspot virus (half a million atoms) at atomic resolution. The details of the potential distribution on the molecular surface sheds light on the mechanism behind the high selectivity of the capsid to the viral RNA. These results are generated with the modest computational power of a desktop PC. The applicability of the analytical approximation as an initial guess for traditional numerical methods as a means of improving the convergence of iterative solutions is investigated and found to be quite promising. / Master of Science

The availability of location estimates of mobile devices (MDs) is vital for several important applications such as law enforcement, disaster management, battlefield operations, vehicular communication, traffic safety, emergency response, and preemption. While global positioning system (GPS) is usually sufficient in outdoor clear sky conditions, its functionality is limited in urban canyons and indoor locations due to the absence of clear line-of-sight between the MD to be localized and a sufficient number of navigation satellites. In such scenarios, the ubiquitous nature of cellular networks makes them a natural choice for localization of MDs. Traditionally, localization in cellular networks has been studied using system level simulations by fixing base station (BS) geometries. However, with the increasing irregularity of the BS locations (especially due to capacity-driven small cell deployments), the system insights obtained by considering simple BS geometries may not carry over to real-world deployments. This necessitates the need to study localization performance under statistical (random) spatial models, which is the main theme of this work. In this thesis, we use powerful tools from stochastic geometry and point process theory to develop a tractable analytical model to study the localizability (ability to get a location fix) of an MD in single-tier and heterogeneous cellular networks (HetNets). More importantly, we study how availability of information about the location of proximate BSs at the MD impacts localizability. To this end, we derive tractable expressions, bounds, and approximations for the localizability probability of an MD. These expressions depend on several key system parameters, and can be used to infer valuable system insights. Using these expressions, we quantify the gains achieved in localizability of an MD when information about the location of proximate BSs is incorporated in the model. As expected, our results demonstrate that localizability improves with the increase in density of BS deployments. / Master of Science / Location based services form an integral part of vital day-to-day applications such as traffic control, emergency response, and navigation. Traditionally, users have relied on the global positioning system system (GPS) for localizing a device. GPS systems rely on the availability of clear line-of-sight between the devices to be localized and a sufficient number of navigation satellites. Since it is not possible to have these line-of-sight links, especially in urban canyons and indoor locations, the ubiquity of cellular networks makes them a natural choice for localization. Typically, localization using cellular networks is studied using simulations, which are carried out by fixing the network configuration including the geometry of the base stations (BSs) as well as the number of BSs that participate in localization. This limits the scope of the results obtained since a change in the network configuration would mean that one must do another set of time consuming simulations with the new network parameters. This motivates the need to develop an analytical model to study the impact of fundamental system-design factors such as BS geometries, number of participating BSs, propagation effects, and channel conditions on localization in cellular networks. Such analysis would make it convenient to infer how changing these system parameters affects localization. In this thesis, we develop a general analytical model to study the localizability (ability of get a location fix) of a device in a cellular network. In particular, we study how information about the location of BSs in the proximity of the device to be localized affects localizability. We derive expressions for metrics such as the localizability probability of a device. Our results help quantify the gains achieved in localizability performance when information about the location of BSs in the vicinity of the device to be localized is available at the device. Our results concretely demonstrate that including this additional information significantly improves the localizability performance, especially in regions with dense BS deployments.

We study the problem of classifying all Poisson-Lie structures on the group Gy of local diffeomorphisms of the real line R¹ which leave the origin fixed, as well as the extended group of diffeomorphisms G₀_∞ ⊃ G_∞ whose action on R¹ does not necessarily fix the origin. A complete classification of all Poisson-Lie structures on the group G_∞ is given. All Poisson-Lie structures of coboundary type on the group G₀_∞ are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G_∞ of G_∞, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra structures of coboundary type on the Lie algebra Go_∞ of Go_∞ which is the Witt algebra. A large class of Poisson structures on the space V_λ of λ-densities on the real line is found such that V_λ becomes a homogeneous Poisson space under the action of the Poisson-Lie group G_∞. We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G_∞ and G₀_∞. / Ph. D.

Lyme disease is the most significant vector-borne disease in the United States. Its southward advance over the last several decades has been quantified, and previous research has examined the potential role of climate change on the disease's expansion, but no research has considered the role of future land cover patterns upon its distribution. This research examines Lyme disease risk in the southeastern United States based on estimated land cover projections under four different Intergovernmental Panel on Climate Change Special Report Emissions Scenarios (IPCC-SRES) A1B, A2, B1, and B2. Results are aggregated to census tracts which are the basic unit of analysis for this study. This study applied previously established relationships between Lyme disease and land cover in Virginia to the projected land cover layers under each scenario. The study area, the southeastern United States, was defined from Level III Ecoregions that are present in Virginia and extend throughout the Southeast. Projected land cover data for each scenario were obtained from the USGS. The projected land cover datasets are compatible with the National Land Cover Dataset (NLCD) categories and had seventeen land cover categories. The raster datasets were reclassified to four broad land cover types: Water, Developed, Forest, and Herbaceous areas and the relationship between certain landscape configurations were analyzed using FRAGSTATS 4.2. Significant variables established in previous research were used to develop a spatial Poisson regression model to project Lyme disease incidence for each decade to the year 2100. Results indicated that potential land cover suitability for Lyme disease transmission will increase under two scenarios (A1B and A2) while potential land cover suitability for Lyme disease transmission was predicted to decrease under the other two scenarios (B1 and B2). Total area under the highest category of potential land cover suitability Lyme disease transmission was calculated for each year under each scenario. The A2 scenario experiences the most rapid acceleration of potential land cover suitability for Lyme disease transmission, with an average increase of 16,163.95 km² per decade, while the A1B scenario was projected to show an average increase of 3,458.47 km² per decade. Conversely, the B1 scenario showed an average decrease of 595.7 km² per decade and the B2 scenario showed the largest decrease of potential land cover suitability for Lyme disease transmission with an average decrease of 2,006.83 km² per decade. This study examined the potential spatial distribution of potential land cover suitability for Lyme disease transmission in the southeastern United States under four different future land cover scenarios. The results indicate geographic regions of the study area that are at greatest risk of potential land cover suitability for Lyme disease transmission under four different predictive scenarios developed by the IPCC. The A1B and A2 land cover projections are predicted to have an overall increase in areas where the Lyme disease transmission cycle will be enhanced by 2100 and the scenarios have a primary focus on economic development. Economic concerns outweigh environmental concerns for the A1B scenario, in addition to a high standard of living. The A2 scenario describes rapid population growth which results in high rates of land cover conversion to developed land; in addition, this scenario describes a reduction of environmental protection. The B1 and B2 land cover projections are predicted to have an overall decrease in areas of high Lyme disease transmission by 2100 and these scenarios have a central focus on environmental sustainability. The B1 scenario is characterized by a high environmental awareness which results in lower demand for forest products. A common theme for the B1 scenario is restoration and forest protection. Finally, the B2 scenario is described as improving local and regional environmental value which results in a high demand for biofuels and repossession of degraded lands, and an overall increase of forest cover. This study was the first to predict potential land cover suitability for Lyme disease risk and geographic distribution using projected land cover in the southeastern United States, and the results of this research can aid in the reduction of Lyme disease as it continues to expand in the south. / Master of Science / Lyme disease is the most significant vector-borne disease in the United States, recognized for its southward advance over the past several decades. Previous research has examined the potential role of climate change on the disease’s continued expansion at the northern extent of its distribution, but no studies have considered the role of future land cover scenarios upon its southward advance, despite a strong association between land cover and Lyme disease emergence. This research examines potential land cover suitability for Lyme disease transmission under projected land cover scenarios provided by the United States Geological Survey under four Intergovernmental Panel on Climate Change Special Report Emissions Scenarios: A1B, A2, B1, and B2. Based on previous research completed in Virginia, developed and herbaceous land cover, and the edges between both herbaceous and forested land in addition to the edges between herbaceous and developed land are all statistically associated with human Lyme disease occurrence. We use a similar statistical model developed in the previous research to quantify potential land cover suitability in the same level III ecoregions present in Virginia projected to their full extent to the south under different land cover scenarios in decadal increments from 2020 to 2100. Results demonstrated variation in potential suitable land cover for Lyme disease transmission depending on the specific scenario. Broadly, if future land cover patterns follow the A1B or A2 scenarios, an increase of suitable areas are to be expected for the Lyme disease transmission cycle. Conversely, if future land cover patterns follow the B1 or B2 scenarios, a decrease of suitable areas are to be expected for enhanced Lyme disease transmission. The results of this research can provide information to public health officials in these areas as the disease continues to expand to the south in the following decades.

Titre de l'écran-titre (visionné le 10 octobre 2023) / Ce mémoire étudie les impacts qu'a une mauvaise spécification de la loi du champ aléatoire latent dans un modèle spatio-temporel. Précisement les impacts sur les estimations et l'inférence d'une mauvaise spécification dans un modèle Poisson log-normal spatio-temporel ont été investigués. La mauvaise spécification correspond à la présence de valeurs très extrêmes pour une distribution Poisson log-normale. Un modèle pour tenir compte de ces valeurs extrêmes a été proposé. L'amélioration des estimations avec ce modèle alternatif est mise en évidence par une étude de simulation Monte Carlo. L'ajustement des modèles impliqués dans cette étude fait intervenir des intégrations en grandes dimensions pour évaluer la vraisemblance. Le package R TMB met en oeuvre une solution, en l'occurence l'approximation de Laplace, à ce problème. / This thesis studies the impact of a misspecification of the latent random field distribution in a spatio-temporal model. Specifically, the impact on estimates and inference of misspecification in a space-time log-normal Poisson model has been investigated. The misspecification corresponds to the presence of very extreme values for a log-normal Poisson distribution. A model to account for these extreme values was proposed. The improvement in estimates with this alternative model is demonstrated by a Monte Carlo simulation study. The fitting of the models involved in this study involves high-dimensional integrations to evaluate the likelihood. The R package TMB implements a solution to this problem: the Laplace approximation.

Pour une bonne modélisation mathématique de l'occurrence de phénomènes aléatoires, part fondamentale de la discipline actuarielle, il est nécessaire d'employer des distributions multivariées permettant de capturer adéquatement les relations de dépendance présentes entre les phénomènes. Celles qu'offrent les champs aléatoires markoviens, une famille de modèle probabilistes graphiques, répondent à ce besoin, les relations de dépendance qu'elles introduisent se calquant à un arbre ou à un graphe. Les champs aléatoires markoviens misent ainsi sur les riches possibilités de topologies d'arbres et de graphes pour offrir cette même richesse en termes de dépendance. Une nouvelle famille de champs aléatoires markoviens arborescents, c'est-à-dire se basant sur des arbres, est proposée. Les membres de cette famille se distinguent par le fait qu'ils ont des distributions marginales fixes de Poisson, « fixes » dans le sens que la dépendance introduite n'a pas d'impact sur elles. Des distribution marginales fixes sont inhabituelles pour un champ aléatoire markovien, bien que généralement désirables pour fins de modélisation. Cette caractéristique est possible par l'encapsulation, dans les arêtes de l'arbre, de la dynamique de propagation induite par l'opérateur d'amincissement binomial. Cela mène également à une représentation stochastique intuitive des champs aléatoires markoviens de la famille, à des méthodes simples de simulation et à des expressions analytiques pour leur fonction de masses de probabilités conjointe et leur fonction génératrice de probabilités conjointe, notamment. Quantités importantes dans un contexte actuariel, la somme des composantes du champ aléatoire markovien, interprétable comme le nombre total d'événement s'étant produits, et les contributions individuelles de ces composantes sont étudiées en profondeur. Cette analyse passe notamment par l'établissement d'ordres stochastiques. À cet effet, un nouvel ensemble partiellement ordonné est défini pour comparer des arbres aux topologies différentes selon la distribution qu'ils induisent pour la somme, ce qui est, à notre connaissance, novateur dans le contexte de modèles pobabilistes graphiques. Est offerte une comparaison de cet ensemble partiellement ordonné avec quelques autres en lien avec la théorie spectrale des graphes. / For adequate mathematical modeling of random phenomena's occurrences, it is necessary to employ multivariate distributions that appropriately capture the existing dependence relations between those phenomena. The multivariate distributions granted by Markov random fields, a family of probabilistic graphical models, answer to this need, by encrypting the dependence scheme they introduce on a tree or a graph. Markov random fields thus leverage on the rich possibilities of tree shapes and graph shapes to provide these possibilities in terms of dependence schemes. We propose a new family of tree-based Markov random fields, characterized by their Poisson marginal distributions. The marginal distributions are also fixed, meaning they are not affected by the introduced dependence. This fixedness is uncommon for Markov random fields, while being desirable for modeling purposes. It is obtained from the encapsulation, in the edges of the tree, of the propagation dynamic induced by the binomial thinning operator. This leads to an intuitive stochastic representation of Markov random fields from the proposed family, simple methods of simulation, and analytic expressions for their joint probability mass function and their joint probability generating function, notably. Important quantities in an actuarial context are the sum of the components of the Markov random field, interpreted as the total number of occurring phenomena, and the individual contributions of these components. They are thoroughly studied, notably via the use of stochastic order relations. We incidently design a new partially ordered set (poset) of trees, in order to compare trees of different shapes based on the distribution of the sum they respectively convey. To our knowledge, this approach is innovative in the context of probabilistic graphical models. We provide comparisons of the newly defined poset with some other posets of trees fetched from spectral graph theory.