Stochastic control of influenza spread: A Lévy-driven SDE and branching process approach.

Background: Forecasting influenza outbreaks remains a significant challenge due to the complexity of disease transmission and the influence of environmental and behavioral factors. Traditional models based solely on the basic reproduction number Methods: In this study, we employ a seasonally adjusted SEIRT model incorporating stochastic differential equations (SDEs), including Brownian motion and Lévy jump processes, to simulate random and abrupt fluctuations in transmission. A branching process approximation is used to evaluate the probability of an epidemic under the influence of seasonal variability and stochastic perturbations. The model is calibrated using weekly influenza case data from Mexico, with noise components estimated from publicly available CDC [1] and WHO [2] surveillance data. Results: Simulation results show that the inclusion of stochastic effects and periodic transmission rates significantly enhances the model's accuracy in reflecting real-world epidemic dynamics. Numerical comparisons between deterministic, Brownian-based, and Lévy-based scenarios reveal that both the initial state of the exposed or infectious subpopulation and the seasonal transmission patterns are critical to determining outbreak probabilities. Results indicate that seasonal transmission rates and stochastic effects significantly alter epidemic probabilities, with Lévy processes capturing abrupt outbreak dynamics more accurately than deterministic models. Conclusions: The findings underscore that deterministic models may underestimate epidemic risk when they overlook random and sudden changes in contact rates or disease introduction. The proposed stochastic modeling framework yields a deeper understanding of influenza transmission dynamics by incorporating uncertainty and seasonal variability, thereby supporting more informed and effective public health decision-making.