Epidemiological Modeling of Seasonal Influenza Through Periodic Mean-Reverting Stochastic Processes

Seasonal influenza follows annual cycles, but variation in epidemic magnitude and timing complicates preparedness. We applied periodic mean reverting stochastic differential equations (SDEs) to seven seasons of weekly Texas influenza surveillance data. Missing observations were imputed using MissForest and compared with four alternative methods. Stochastic variability was modeled using σₚX(t)ᵖ dW(t), with p = 1/2,2/3,5/6, and 1, representing square root, intermediate sublinear, and multiplicative diffusion. Each specification was evaluated for direct influenza modeling and within a susceptible, exposed, infectious , and recovered model incorporating waning immunity, where the SDE governed transmission. Parameters were estimated using least squares and approximate maximum likelihood for direct models and sample-proportion weighted least squares for stochastic SEIR models. Model-based parametric bootstrap assessed direct-model parameter uncertainty. For direct modeling, p = 1/2 achieved the lowest RMSE (712.8) and AIC (5383.0), with similar AIC for p = 2/3 (5383.2). For stochastic SEIR modeling, p = 1 achieved the lowest RMSE (0.02926), compared with 0.03506 for p = 1/2. The deterministic SEIR model yielded R₀ = 1.158. Transmission peaked approximately 10 weeks before infectious prevalence, while stochastic transmission generated variation in epidemic peak timing. Bootstrap analysis identified dependence between mean-reversion and diffusion parameters. Overall, the appropriate diffusion structure depended on the epidemiological quantity modeled, supporting flexible characterization of seasonal transmission variability while requiring prospective validation before operational forecasting.