Research
My research focuses on central problems in stochastic and deterministic models for complex biological systems. I address questions such as:
- How do we efficiently analyze a nonlinear dynamical model despite the huge number of parameters, variables, and equations?
- How do biochemical systems show homeostasis despite fluctuating environmental conditions?
- How do we find a closed form of solutions for dynamical systems?
- How can we accurately estimate unknown parameters in a model with realistic assumptions using Bayesian approaches?
Recently, after joining UW–Madison as a postdoc, I have been expanding my interest to a more general dynamical system that is not necessarily biological. Specifically, I am working on Koopman theory, which offers a (possibly infinite-dimensional) linear representation of a nonlinear dynamical system. The techniques I have developed span the fields of probability, queueing theory, Bayesian inference, and dynamical systems.
Koopman Theory
Koopman theory is a mathematical framework for representing nonlinear dynamical systems using an infinite-dimensional linear operator. This operator acts on a space of measurement functions of the system's state, allowing for a globally linear representation of nonlinear dynamics. I am working on efficient algorithms to find finite-dimensional linear representations that approximate an original nonlinear dynamical system, as well as purely algebraic conditions that allow a given nonlinear dynamical system to admit an exact finite-dimensional representation.
Chemical Reaction Network Theory
Chemical reaction network theory (CRNT) is a discipline of applied mathematics in which we model a biological/biochemical system using a directed graph representation and infer dynamical properties based on its structural properties. I work on analytic derivation of stationary distributions for the continuous-time Markov chain (CTMC) associated with a stochastic CRN — the steady-state solution of the chemical master equation — which provides long-term information such as sensitivity, robustness, and a likelihood function for Bayesian inference.
Bayesian Inference for Non-Markovian Dynamical Systems
Not all reactions in a biochemical system can be experimentally measured simultaneously. Replacing unobserved reactions with a single random time delay significantly reduces the number of variables and parameters, but the resulting non-Markovian process makes parameter inference difficult. Based on knowledge from stochastic processes (e.g., queueing theory), I have developed Bayesian MCMC methods to infer system parameters in such non-Markovian systems.
Collaborative Works
(1) Cognitive Impairment & Wearable Devices. We use fractal analysis based on detrended fluctuation analysis to find features of activity patterns — measured by wearable devices — that are altered by cognitive impairment, with the goal of enabling pre-diagnosis.
(2) COVID-19 Modeling. We modeled COVID-19 extinction and endemicity based on two immunities with different longevities: long-lived severity-preventing immunity (T-cell) and short-lived infection-preventing immunity (antibody). Our analysis shows that high viral transmission unexpectedly reduces severe COVID-19 rates during endemic transition, and paradoxically accelerates endemic transition with reduced severe cases in highly vaccinated populations.