Optimal control of pandemic dynamics using a piecewise fractional order SVIR model

Modeling the long-term dynamics of the COVID-19 pandemic is challenged by evolving public behavior and interventions. We propose a novel piecewise fractional-order (SVIR) model incorporating vaccination and education controls. The model uniquely employs a classical derivative for the initial, memoryless phase of the epidemic. It then transitions to a Caputo-Fabrizio fractional derivative to capture long-term collective memory effects on transmission. We establish the model's mathematical well-posedness and derive the basic reproduction number ( $R_0$). Under our baseline parameterization, the reproduction number is $R_0\simeq 4.95$. An optimal control problem is formulated to determine the ideal implementation of time-varying vaccination and education. Numerical simulations validate the distinct crossover dynamics produced by our piecewise approach. Results demonstrate that a synergistic strategy combining vaccination and education is highly effective, reducing the peak of infected individuals by over 90% compared to the uncontrolled scenario, and significantly outperforms isolated interventions. This study offers a flexible tool for understanding and controlling epidemics.