Research Interests

My research focuses on central problems in stochastic and deterministic models for complex biological systems. For example, I have been addressing the following questions:

• How do we efficiently analyze a nonlinear dynamical model despite the huge number of parameters, variables, and equations?
• How does 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 developed span the fields of probability, queueing theory, Bayesian inference, and dynamical systems. Beyond the mathematical and statistical topics, I work closely with biologists and medical doctors in order to address biological problems related to cognitive impairment or COVID-19. I described my interests in more details in the following.

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. In particular, I am working on efficient algorithm to find finite-dimenional linear representation that approximates an original nonlinear dynamical system. Also, I am also interested in purely algebraic, sufficient or necessary conditions that allows a given nonlinear dyamical system to admit an exact finite-dimensional representation.

Chemical Reaction Network Theory

A biological/biochemical system consists of biological/chemical species and reactions describing interactions among the species. Chemical reaction network theory (CRNT) is a discipline of applied mathematics in which we model a system using a directed graph representation and infer the dynamical property based on the strucural properties of the system. In the CRNT, a stochastic system is modeled by means of a continous-time Markov chain (CTMC). I am working on analytic derivation of stationary distributions for the CTMC, the steady-state solution of the Kolmogorov forward equation (i.e., chemical master equation), associated with a stochastic CRN. The analytic formula provides long-term information of the system such as sensitivity, robustness, and also a likelihood function for Bayesian Inference.

Bayesian Inference for parameters in non-Markovian models of dynamical systems

Not all reactions in a biochemical system can be experimentally measured simultaneously. To analyse such system, replacing unobserved reactions with a single random time delay has been widely adopted as it significantly reduce the number of variables and parameters. However, the resulting process is non-Markovian (i.e., state evolution depends on not only present but also past), making it difficult to infer kinetic and delay parameters. Based on the knowledge from stochastic process (e.g., queueing theory) I have developed Bayesian Markov-chain Monte Carlo (MCMC) method to inter the system parameters. These studies provide an effective tool to analyze a non-Markovian system.

Collaborative works

Beyond the mathematical and statistical topics, I am closely working with biologists and medical doctors to solve biological problems. Some of the problems that we want to address (or have addressed) are the followings. 

 (1) Which features of activity pattern of human measured by wearable devices are altered by cognitive impairment? Can we provide a prediagnosis for cognifive impairment using activity patterns? We are using fractal analysis based on the detrended fluctuation analysis to find such features. 

 (2) Back in 2020, we had no idea where the COVID-19 pandemic will go. During that time, most of mathematical models focused on the near-future prediction of the diseases, and it worked well; it provided an effecitve guide for non-pharmaceutical interventions. However, to meet the end of the disease, we had to know what the "end" is. Thus, we modeled a COVID-19 extinction and endemicity based on two immunities with different longevities: long-lived severity-preventing immunity due to T-cell functioning and short-lived infection-preventing immunity due to antibody functioning. Our analysis demonstrates that high viral transmission unexpectedly reduces the rates of progression to severe COVID-19 during the course of endemic transition despite increased numbers of infection cases. It also shows that high viral transmission amongst populations with high vaccination coverages paradoxically accelerates the endemic transition of COVID-19 with reduced numbers of severe cases.