Asian Journal of Probability and Statistics

This study aims to develop and apply a perturbation-theoretic framework to evaluate the stability and sensitivity of discrete-time Markov chains derived from epidemiological data. The study uses norm-based perturbation bounds and sensitivity analysis to assess model robustness. Total variation distance measures changes in the stationary distribution, the Dobrushin coefficient evaluates stability, and first-order sensitivity approximations capture the impact of small transition matrix perturbations. Although transition probabilities obtained from finite data are inherently vague, analytically derived Markov chains are widely used to characterize stochastic systems. Three epidemiological states were distinguished based on the weekly incidence of Lassa fever in Makurdi, Nigeria, increase, decrease, and no change. To assess the degree of perturbation due to sampling variability, we computed the transition matrix from surveillance data using perturbation analysis.   The results show that, while the increase and decrease states were slightly affected by perturbation, the stable "No Change" state dominates the stationary distribution. The Markov chain exhibits a 28% stability margin, indicating it remains well within the ergodic regime despite perturbations. Although the theoretical worst-case bound is 0.429, the observed change is only 0.013 (about 3% of the bound), demonstrating strong numerical stability of the stationary distribution. These results provide a useful method for robust stochastic modeling under finite-sample uncertainty by confirming that empirically calculated Markov chains can consistently reflect endemic stability and transient oscillatory behavior in epidemic data.

The methodology should be extended to Ebola virus disease, Cholera, and Measles, where outbreak spikes, environmental variability, and vaccination effects can be modeled as structural perturbations. Future studies should perform cross-disease spectral comparisons to identify which infections are more unstable under small transition changes, thereby developing an epidemiological stability classification framework. Additionally, agencies such as the Nigeria Centre for Disease Control should incorporate spectral gap monitoring into routine surveillance, using shifts in transition probabilities as early-warning indicators of potential regime change.