Monte Carlo and Markov Chain Monte Carlo Methods for Identifiability and Data Analysis of Dynamical Systems in Biology

This dissertation develops new statistical methodologies for analyzing complex biological systems, with applications to single-molecule fluorescence experiments and epidemiological modeling.

In the first part, we develop a Bayesian nonparametric framework for the analysis of fluorescence time series obtained from single-molecule photobleaching experiments. We formulate a Bayesian model to characterize the fluorescence traces and implement four Markov Chain Monte Carlo (MCMC) samplers of increasing complexity. We first show that the novel sampling strategies incorporated into our most advanced sampler are essential for fast convergence and accurate reconstruction of trace signals. We then thoroughly validate this sampler using synthetic data generated to mimic a variety of experimental conditions and demonstrate strong performance across varying noise levels and stoichiometries. Lastly, we compare our sampler with existing methods and show that it excels in both computational performance and accuracy. Taken together, these advances, combined with the fact that our approach is model-free and does not rely on restrictive assumptions about molecular dynamics, make it an efficient and versatile tool for a wide range of experimental applications.

In the second part, we investigate the practical identifiability of epidemic models, with a focus on Monte Carlo methods. Standard implementations add independent Gaussian noise to the deterministic solution of the ODE model, an assumption that fails to capture the structured stochasticity inherent in epidemic processes. By analyzing continuous-time Markov chain (CTMC) simulations of the Susceptible–Infected–Recovered (SIR) model, we reveal variability patterns and temporal dependence that Gaussian noise cannot reproduce. We propose a hybrid approach that introduces time- and amplitude-dependent variability into ODE trajectories, achieving coverage results consistent with CTMC simulations while remaining computationally efficient.

Together, these contributions expand the statistical toolkit for biological data analysis, demonstrating how careful modeling of uncertainty can improve inference in both molecular biophysics and infectious disease dynamics.