Stochastic Modeling and Applications to Discrete and Continuous Dynamical Systems

Abstract

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.