Research

My research interests lie in probability theory and stochastic processes, with a particular focus on stochastic models of biological interaction networks. My research has two distinct areas of emphasis: (i) numerical simulation of stochastic models, with a focus on coupling methods utilized for variance reduction, and (ii) exploring the connection between stochastic models and their deterministic counterparts, with a focus on determining when explicit formulas for the time-dependent distribution of the stochastic model can be derived.

Numerical Simulation and its application

Numerical simulations have been adopted extensively to understand the underlying dynamics of the stochastic models. It relies on simulation, which generates statistically exact trajectories. These sample trajectories can then be used in conjunction with Monte Carlo methods to estimate any statistics of interest. 

In [1], We constructed efficient estimators for expectations and parametric sensitivities that produced up to a thousand-fold increase in efficiency. These estimators took advantage of an efficient simulation algorithm for coupled time-inhomogeneous stochastic processes, and were particularly useful in the numerical computation of parametric sensitivities and fast estimation of expectations via Multilevel Monte Carlo methods. Specifically, it produced pairs of trajectories with exceptionally low variance, often thousands of times lower than the variance between the paths produced by the other couplings, as shown in Figure below. 

In [3], we performed numerical analysis pertaining to finite difference methods in the context of parametric sensitivity analysis. Earlier work assumes globally Lipschitz intensity functions, which was only applicable to a small percentage of the models found in the literature. Our analysis includes the situation of locally Lipschitz and/or time-inhomogeneous intensity functions, which extend the region of validity for the coupling methods to the vast majority of systems considered in the literature. 

In [4] we considered Togashi-Kaneko model (TK model), which is a stochastic reaction network that displays discreteness-induced transitions (DIT) between meta-stable patterns. The system state switches between patterns where a few species are abundant and the remaining species are almost absent. (Some example trajectories can be viewed below on the left. ) We studied a constrained Langevin approximation (CLA) of this model, which is an obliquely reflected diffusion process on the positive orthant and hence it respects the constrain that chemical concentrations are never negative. Our simulation suggest that the CLA well captures the stationary distribution in all parameter regimes, and when the volume of the system in which all the reactions take place is large, the CLA is also a good approximation to the TK model in terms of transition times between patterns. 

Stochastic Epidemic model

In an ongoing project, a novel stochastic spatial model for both SIR and SIS epidemic models is considered, where the rate of disease transmission is proportional to virus concentration in the underlying environment. Direct analysis of the stochastic spatial model is challenging, hence numerical simulations are used to explore interesting propeties of the proposed model.  Dynamics of the stochastic model exhibits wave-like behavior, where a single trajectory of the stochastic simulation is included in the left figure below. Simulation further indicates that wave speed of the stochastic spatial model  converges to the wave speed to its associated PDE model, which also possess traveling wave solutions. 

Explicit solution to the forward equation

Another approach to understand the underlying dynamics focuses on solving or approximating the solution to the Kolmogorov forward equation, also termed the chemical master equation in much of the chemistry and biology literature. This system of equations describes the time evolution of the probability distribution of the stochastic model. In general, it is analytically intractable to solve the forward equation (except when we have simple linear dynamics), given that there is one equation for each state of the system. We derive a necessary and sufficient condition for non-linear stochastic reaction networks to possess a time-dependent product-form Poisson distribution. The condition found is closely related to properties of the trajectory of the corresponding deterministic model. Here are some nice visualizations of the result! 

Slow mixing for positive recurrent stochastic processes

In a near completion project, we consider the mixing time of Markov chains and how it is affected by different choice of initial conditions. In particular, we proposed continuous time Markov chains whose rate of convergence to the stationary distribution may vary significantly under different choice of initial conditions, and we establish a lower bound of the mixing time. ß

Publications and Preprints :

5. Wai-Tong (Louis) Fan, Jinsu Kim, Chaojie Yuan,  Boundary-induced slow mixing for Markov chains and its application to stochastic reaction networks, submitted, 2024

4. Wai-Tong Louis Fan, Yifan Johnny Yang, Chaojie Yuan, Constrained Langevin approximation for the Togashi Kaneko model of autocatalytic reactions, Mathematical Biosciences and Engineering, , 2023

3. David F.Anderson and Chaojie Yuan, Variance of finite difference methods for reaction networks with non-Lipschitz rate functions , Volume 58, Number 6, 3125-3143, SIAM Journal on Numerical Analysis, 2020 

2. David F. Anderson, David Schnoerr, and Chaojie Yuan,  Time-dependent product-form Poisson distributions for reaction networks with higher order complexes , Journal of Mathematical Biology, Volume 80, 1919-1951, 2020.

1. David F. Anderson and Chaojie Yuan, Low variance couplings for stochastic models of intracellular processes with time dependent rate functions, Bulletin of Mathematical Biology, Vol. 81, Issue 8, 2902 - 2930, 2019

Thesis : 

1. Compuational and Analytical Methods for Stochastic Reaction Network Models, Ph.D thesis.