Speaker: Dr. Fahad Mostafa, Arizona State University

Title: Periodic Mean-Reverting Stochastic Models for Seasonal

Epidemiological Time Series

Abstract:

Seasonal time series are pervasive in biology, epidemiology, and environmental science, arising in applications ranging from temperature and rainfall patterns to infectious disease dynamics. Statistical modeling of such processes often requires balancing deterministic seasonal structure with stochastic variability. Classical approaches, such as seasonal autoregressive integrated moving average and state-space models, can capture serial dependence but typically impose linear dynamics and lack direct biological interpretation. Recent data-driven methods in epidemiology, while flexible, often fail to incorporate interpretable mechanisms such as periodicity, environmental variability, mean reversion or stochastic forcing. We propose a class of periodic mean-reverting stochastic differential equation models that explicitly ac-

count for both deterministic periodicity and random fluctuations in seasonal time series. The general model is given by

dX(t) = r(β(t)−X(t)) dt+ dβ(t) + σXp(t) dW(t),

where r,σ > 0, p ∈(0,1/2,2/3,5/6,1), and β(t) is a periodic mean function. This framework unifies several well-known processes: the Ornstein–Uhlenbeck diffusion (p= 0), the Cox–Ingersoll–Ross process (p=1/2), and geometric Brownian motion (p = 1). Analytical results show that when β(t) is periodic, higher-order moments of the CIR- and GBM-type models also exhibit periodicity. The periodic mean function β(t) is first estimated by nonlinear least squares, followed by maximum likelihood estimation of (r,σ). Asymptotic confidence regions provide interpretable uncertainty quantification, and simulation studies confirm the efficiency and robustness of the estimators. Missing observations in seasonal data are addressed using a newly developed seasonal-MissForest algorithm that preserves both temporal autocorrelation and periodic dependencies, ensuring consistent parameter estimation. Applications to weekly influenza and temperature data show that periodic mean-reverting stochastic models effectively separate deterministic seasonal trends from stochastic variability, yielding improved forecasts over classical seasonal models while maintaining biological interpretability. An extension with a periodic mean-reverting transmission rate and waning immunity captures realistic multi-seasonal epidemic dynamics. Overall,

the proposed framework offers a unified, computationally tractable, and statistically principled approach for modeling seasonality in complex epidemiological systems, integrating stochastic modeling, likelihood-based inference, and data-driven imputation within a coherent statistical structure.