Dynamic analysis of a fractional-order SEAIR model for influenza transmission with optimal control and stochastic stability

Understanding the dynamic characteristics of infectious disease transmission is key to designing effective control strategies. To this end, we proposed a Caputo fractional-order SEAIR model to study the transmission of influenza. First, the existence, uniqueness, and non-negativity of the model's solution were proven, ensuring its biological plausibility. Then, the basic reproduction number $R_0$ was derived using the next-generation matrix method, the existence and stability of equilibrium points were analyzed, and bifurcation phenomena in the model were proven. Next, vaccination and isolation were considered control strategies, and the existence of optimal solutions was discussed. The optimal control strategies were derived using the Pontryagin maximum principle. Through numerical simulations, the dynamic behavior of the controlled model under different initial values and $\eta$ values was further analyzed. In addition, the stability of the stochastic model when $\eta =1$ was studied, as well as the dynamic characteristics of the stochastic model under different influenza mortality rates ( $d$). Finally, real influenza patient data was used for parameter estimation to validate the accuracy and predictive capability of the deterministic model. The research results indicated that early detection and treatment helped significantly reduce the transmission range, thereby facilitating better control of influenza. In areas with high population density, a single control strategy may not be sufficient to effectively curb the spread, thus requiring a combination of multiple strategies to achieve better control outcomes.