Pandemics strain healthcare systems worldwide, creating urgent challenges in allocating limited resources like hospital beds while controlling disease spread. Effective patient allocation during such crises is critical to minimizing unmet healthcare demand and ensuring equitable healthcare access across regions. This study addresses these issues by developing a mixed-integer nonlinear mathematical model that integrates Susceptible-Infected-Recovered-Vaccinated (SIRV) epidemic dynamics with patient transfer and allocation to improve patient distribution during outbreaks. Our approach also factors additional disease transmissions caused by the assignment of patients to different regions. Our model minimizes unmet demand per county for hospital beds and incorporates real-world data in the model parameters. We observe that in large metropolitan areas, where access to healthcare is critical, the vaccinated scenario exhibits significantly lower unmet hospital demand compared to the unvaccinated scenario, an outcome directly tied to the vaccines' success in lowering peak infection rates as demonstrated by the SIRV dynamics. The reduction in infection surges not only alleviates hospital capacity strain but also leads to fewer total patient transfers, underscoring the effectiveness of vaccination during pandemics. This research demonstrates the potential to enhance pandemic response strategies. The model provides policymakers and healthcare administrators with a robust, data-driven tool to make informed decisions, reducing strain on overburdened facilities and improving patient outcomes during pandemic scenarios.