Browsing by Subject "Stochastic Modeling"

The research studies outlined in this thesis are all geared toward improving our understanding of infectious diseases and their impact on public health. Utilizing statistical frameworks, computational methods, and deterministic and stochastic mathematical models, these studies investigate the mean ICU stay during the COVID-19 pandemic to optimize hospital resources, the decay of viral load and distribution of lifetimes of infected cells in SHIV infected rhesus macaques, and the impact of various immune effector functions in effectively managing upper respiratory viral infections. The thesis combines methodologies to develop strategies to analyze infectious disease dynamics at both the individual and population level.

The first study establishes a statistical framework for estimating occupancy rates in intensive care units, incorporating variables such as hospital bed occupancy and SARS-CoV-2 test positivity rates. Using an immigration-death model, this research enables dynamic estimation of patient influx and efflux from ICUs, a crucial element in healthcare planning during a pandemic. This methodological approach is empirically evaluated using data from the University of California, Irvine Health and Orange County, California. The second study provides an efficient algorithm to compute transition probabilities for branching processes, which are commonly used in modeling ecological and epidemiological dynamics. Our proposal introduces a new method that uses variable splitting, leading to updates in a closed form through an efficient ADMM algorithm. It is important to note that no matrix multiplications, let alone inversions, are needed during any part of the process. This results in a significant decrease in computational costs by several orders of magnitude compared to current methods. In addition, the resulting algorithm can be easily parallelized and shows a high level of robustness to changes in tuning parameters. This method is compared to prior work by applying it to two scenarios that involve models of blood cell production and transposon evolution.

The third study introduces a modification of the Gompertz model to depict the dynamics of viral load post initiation of antiretroviral therapy. This simple model is applied to the infant rhesus macaque SHIV.C.CH505 infection data set and extended using a stochastic differential equation formulation. This model aligns well with the data and suggests that contemplating a continuous distribution of infected cell lifespans may yield a more nuanced insight into viral decay trends. In the final study, a mathematical model is proposed to improve the understanding of interactions between hosts and pathogens in upper respiratory infections. This model goes beyond traditional frameworks by encompassing an array of immune effector functions, thus offering a more thorough analysis of early infection phases. Although it concentrates on SARS-CoV-2, the model’s principles have broad relevance, highlighting the potential for diverse applications.

Collectively, these studies improve the knowledge base on infectious disease modeling and may help shape a well-informed approach to public health interventions.

Storage systems play a vital part in modern IT systems. As the volume of data grows explosively and greater requirement on storage performance and reliability is put forward, effective and efficient design and operation of storage systems become increasingly complicated.

Such efforts would benefit significantly from the availability of quantitative analysis techniques that facilitate comparison of different system designs and configurations and provide projection of system behavior under potential operational scenarios. The techniques should be able to capture the system details that are relevant to the system measures of interest with adequate accuracy, and they should allow efficient solution so that they can be employed for multiple scenarios and for dynamic system reconfiguration.

This dissertation develops a set of quantitative analysis methods for modern storage systems using stochastic modeling techniques. The presented models cover several of the most prevalent storage technologies, including RAID, cloud storage and replicated storage, and investigate some major issues in modern storage systems, such as storage capacity planning, provisioning and backup planning. Quantitative investigation on important system measures such as reliability, availability and performance is conducted, and for this purpose a variety of modeling formalisms and solution methods are employed based on the matching of the underlying model assumptions and nature of the system aspects being studied. One of the primary focuses of the model development is on solution efficiency and scalability of the models to large systems. The accuracy of the developed models are validated through extensive simulation.

The proper modeling of uncertainties in constitutive models is a central concern in mechanics of materials and uncertainty quantification. Within the framework of probability theory, this entails the construction of suitable probabilistic models amenable to forward simulations and inverse identification based on limited data. The development of new manufacturing technologies, such as additive manufacturing, and the availability of data at unprecedented levels of resolution raise new challenges related to the integration of geometrical complexity and material inhomogeneity — both aspects being intertwined through processing.

In this dissertation, we address the construction, identification, and validation of stochastic models for spatially-dependent material parameters on nonregular (i.e., nonconvex) domains. We focus on metal additive additive manufacturing, with the aim of closely integrating experimental measurements obtained by collaborators, and consider the randomization of anisotropic linear elastic and plasticity constitutive models. We first present a stochastic modeling framework enabling the definition and sampling of non-Gaussian models on complex domains. The proposed methodology combines a stochastic partial differential approach, which is used to account for geometrical features on the fly, with an information-theoretic construction, which ensures well-posedness in the associated stochastic boundary value problems through the derivation of ad hoc transport maps.

We then present three case studies where the framework is deployed to model uncertainties in location-dependent anisotropic elasticity tensors and reduced Hill’s plasticity coefficients (for 3D printed stainless steel 316L). Experimental observations at various scales are integrated for calibration (either through direct estimators or by solving statistical inverse problems by means of the maximum likelihood method) and validation (whenever possible), including structural responses and multiscale predictions based on microstructure samples. The role of material symmetries is specifically investigated, and it is shown that preserving symmetries is, indeed, key to appropriately capturing statistical fluctuations. Results pertaining to the correlation structure indicate strong anisotropy for both types of behaviors, in accordance with fine-scale observations.