Understanding the temporal dynamics of the human microbiome is critical for deciphering host-microbe interactions and identifying responses to environmental or physiological disruptions. In this study, we develop a likelihood-based framework grounded in Ornstein-Uhlenbeck (OU) state-space modeling to analyze longitudinal microbiome data. By applying log-ratio transformations to genus-level relative abundances, we accommodate the compositional nature of microbiome profiles and enable biologically interpretable inference. We estimate the mean reversion rate of the OU process using maximum likelihood estimation and construct profile likelihood confidence intervals to assess temporal stability. Biologically, higher estimated mean-reversion rates correspond to stronger regulatory feedback and thus greater temporal stability of the gut microbiome, whereas lower rates reflect weaker stabilization and slower recovery from perturbations. In the datasets analyzed here, most estimated mean reversion rates were between 0.12 and 0.36, corresponding to a half-life in the range 2-6 days for the system to return to the equilibrium level. These mean-reversion rates are substantially lower than previous estimates obtained from OU models that did not account for measurement error. To capture structural shifts in microbial dynamics, we propose a statistical testing procedure that compares a single-parameter model to a split-parameter model across disrupted time periods. We also propose an OUSSM to jointly model multiple microbes, taking into account the special measurement error covariance structure for log-ratio transformed data.

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