This study presents a mathematical framework to analyze the transmission dynamics of the Ebola Virus Disease (EVD) using an extended SEIRVH model. The model incorporates vaccinated and hospitalized compartments, addressing critical factors such as vaccination efficacy, healthcare interventions, and natural disease progression. Differential equations describe the transitions between six population compartments. The study evaluates model stability and bifurcation through well-posedness, positivity, and boundedness analyzes, ensuring realistic and biologically valid solutions. The basic reproduction number, R0, derived from the next generation matrix, serves as a threshold for outbreak control. Local and global stability analyzes of disease-free and endemic equilibria reveal critical insights into epidemic thresholds and long-term dynamics. Furthermore, sensitivity analysis highlights key parameters that influence R0, emphasizing the importance of vaccination and hospitalization in mitigating EVD outbreaks. Numerical simulations validate theoretical findings, underscoring the model's utility in informing effective public health strategies, such as vaccination campaigns and hospitalization measures, for controlling EVD transmission. This research provides a robust analytical and computational tool for understanding and managing the spread of Ebola and similar infectious diseases.