Vectorial models of focused beams are important to a variety of fields including microscopy, lithography, optical physics, and biomedical imaging. This has led to many models being developed, which calculate how beams of various profiles are focused both in free space and in the presence of stratified media. The majority of existing models begin with a vectorial diffraction formula, often referred to as the Debye-Wolf integral, which must be evaluated partially analytically and partially numerically. The complexity of both the analytic and numerical evaluations increases significantly when exotic beams are modeled, or, a stratified medium is located in the focal region. However, modern-day computing resources permit this integral to be evaluated entirely numerically for most applications. This allows for the development of a vectorial model of focusing in which the focusing itself, interaction with a stratified medium, and incident beam specification are independent, allowing for a model of unprecedented flexibility. We outline the theory upon which this model is developed and show examples of how the model can be used in applications including optical coherence tomography, high numerical aperture microscopy, and the properties of cylindrical vector beams. We have made the computer code freely available.