UCLA

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.