Structured population models are fundamental in the fields of biology, ecology, and social sciences, as they provide both theoretical insights and practical applications. Different structured population models range from modeling cellular population proliferation and population dynamics to simulating disease spread on social networks. However, there has been little work on modeling populations across different scales that could link individual behavior to population dynamics. Additionally, for existing mathematical models on structured populations, several computational challenges arise as how to develop efficient numerical solvers to simulate those models and to control the dynamics of those models.

Overall, my dissertation covers three related topics: modeling structured populations, developing efficient numerical solvers to simulate these models, and developing control algorithms to control population dynamics. Specifically, my dissertation focuses on modeling and devising algorithms for two types of structured populations: i) age, size, or added size-structured cell population for describing cellular proliferation and ii) the structured infected-time- or number-of-contact-based human population for describing disease spread.

Regarding the structured cellular population, we derive mathematical models at both the macroscopic population dynamics level and microscopic individual behavior level, leading to structured partial differential equation (PDE) models for cellular proliferation with different structure variables such as cellular age, size, or added size.

Next, we develop an efficient adaptive spectral method for numerically solving spatiotemporal PDEs, which was inspired by simulating the blowup behavior in the unbounded-domain PDE model for cellular populations. In addition to the structured population models, the adaptive spectral method proves efficient and accurate in solving a wide range of spatiotemporal PDEs in unbounded domains such as the Schr�dinger equations in quantum mechanics.

Regarding the structured human population, we introduce an infected-time-structured PDE model and a number-of-contact-structured ODE model for simulating disease spread, e.g., COVID-19, in the population. Then, for the number-of-contact-structured ODE model, we develop classic Pontryagin-maximum-principle-based and reinforcement-learning-based optimal control algorithms. These two algorithms can effectively mitigate the spread of disease by appropriately allocating limited test kits or vaccination resources.

In this dissertation, we use birth-death-immigration systems of ordinary differential equations to study the dynamics of human naive T-cell populations in healthy aging and disease of the immune system. We derive a model that tracks both total cell counts and counts of clones (groups of genetically identical cells) of a particular size; from the latter, we compute the total clone count, or ``diversity", which provides a quantitative measure of the extent to which the T-cell pool can cope with invading pathogens. We first formulate a nonautonomous model of T-cell birth and replenishment throughout an individual's lifetime, and use it to assess the relationship between the immune tissue damage, T-cell loss and dysfunction, and weakened immune response all observed simultaneously in aging. We identify tissue loss in the thymus as a fundamental cause of diminished T-cell counts, and also that diversity loss can only underlie weakened immune effectiveness in aging if small clones are ineffective against pathogen. Using a short-time, autonomous version of the same ODE, we then study changes to the T-cell pool during an instance of acute thymic atrophy and recovery. We identify equilibrium solutions that arise at different rates of T-cell production, and derive analytic approximations to the eigenvalues and eigenvectors of the linearization around the equilibria. From the forms of the eigenvalues and eigenvectors, we are able to estimate rates at which different size-segregated groups of clones converge to equilibria--that is, ``adjust" to the changing rate of T-cell production. Finally, we formulate a total cell count model of populations of HIV-immune and HIV-susceptible cells of the T-cell lineage in the bone marrow, thymus, and peripheral blood after transplant of HIV-immune bone marrow into an HIV-infected individual. We show that independent of assumptions about the homeostatic mechanism in the periphery, the ratio of HIV-immune to HIV-susceptible T-cells in each immune region should asymptotically approach the ratio of HIV-immune to HIV-susceptible stem cells that results after bone marrow transplant with no infection present. We show that with virus introduced into the system, the asymptotic ratio of HIV-immune to HIV-susceptible peripheral T-cells is highly sensitive to parameters affecting the body's ability to clear the virus.

This dissertation deals with a range of mathematical research problems that have arisen out of the study of a neurophysiological phenomenon known as cortical spreading depression (CSD). CSD is a slow-moving wave of depolarization and subsequent electrical depression that occurs whenever the brain is in one or more of several different distressed states. CSD is of particular interest in migraine research, as it is thought to be responsible in migraine with aura, and in stroke research, where the number of CSD events is a strong determinant of patient outcome.

This dissertation presents a mathematical model for CSD that predicts the feed-forward and feed-back effects of vasoconstriction. The model predicts that the main constraint on recovery from CSD is metabolic - the facts that recovery from CSD is energetically expensive and oxygen flux is finite play a larger role than vasoconstriction in inhibiting CSD recovery. Yet, vasoconstriction and vasodilation both affect CSD.

In the laboratory, CSD is often studied using optical imaging techniques, where a need for extracting information out of images is needed. This dissertation first considers the more-general problem of object identification in images (segmentation), where a-priori shape-information is known in the form of an ensemble of templates. Using kernel-density-estimation, these templates are constructed into a prior probability distribution over the space of desired shapes. A nonlinear energy functional is defined where the desired segmentation is the minimum-energy state. This functional is minimized using a majorization-minimization algorithm where each iteration is solved quickly using graph cuts.

The image segmentation technique is then extended to the tracking of monotonically advancing boundaries like those observed in CSD. Here, a statistical model of boundary motion provides a set of templates that is used to regularize boundary segmentation. The boundary is modeled using Gaussian random fields, where boundary speed is expected to vary smoothly in space, and is expected to be locally correlated in space.

Finally, when studying CSD and other wave phenomenon, one often encounters inverse problems with partial differential equations. For example, one encounters inverse problems of the eikonal equation when studying CSD wave geometry. This dissertation provides a general framework for solving inverse problems involving partial differential equations. By connecting the method of Tikhonov-regularization to the method of Bayesian statistical regularization using a-priori Gaussian random fields, we develop a probabilistic model that is useful for both solving inverse problems, and for determining uncertainty. The mathematical framework is based on methods developed by the physics community to study the statistical properties of particles.