Uncovering Epistatic Interactions in SARS-CoV-2 Evolution Through Hidden Markov Models

Understanding epistatic interactions, where mutations collectively influence viral fitness, is critical for predicting pathogen evolution. We present a hidden Markov model (HMM) framework that captures the temporal dynamics of epistatic relationships in SARS-CoV-2, addressing limitations of static network-based approaches. Our method models single amino acid variant pairs as a two-state system (linked/unlinked), with emission probabilities derived from linkage disequilibrium theory and transition probabilities optimized via the Baum–Welch algorithm. We implement permutation-based validation with temporal order noise reduction (>80% agreement across five iterations) to distinguish biological signals from stochastic noise. Applied to 2,192,008 spike protein sequences from the United States (March 2020–December 2021), our approach identified three classes of epistatic dynamics: permanent (0.3%), transient (0.3%), and oscillating (0.7%) linkages. Analysis of Alpha variant positions revealed 78% epistatic linkage compared with 1.3% across all spike protein position pairs, with 60% exhibiting oscillating patterns suggestive of frequency-dependent selection. We detected all 17 previously reported epistatic pairs plus 18 novel interactions, including critical connections between positions 69–70 and other functional sites. Notably, Alpha variant epistatic networks were detectable as early as April 2020, months before widespread circulation. Our framework scales to variant-wide analysis, revealing distinct patterns across variants: Delta (96% linkage, 71% oscillating) and Omicron (87% linkage, 56% oscillating). The computational pipeline, implemented with parallelized HMM training and Viterbi decoding, processes hundreds of thousands of position pairs efficiently. By transforming epistasis detection from static to temporal analysis, this work provides computational tools for early variant detection and demonstrates how probabilistic modeling can capture evolutionary dynamics in real-time genomic surveillance systems.