Reliable and efficient modelling of magnetic hysteresis in inhomogeneous and aniso-tropic media is an important step in developing a state-of-the-art closed-loop degaussing system for naval ships and submarines, to be developed by TNO and to be used by the Royal Netherlands Navy in an updated generation of naval vessels and submarines. Different models have been proposed to describe the nonlinear and history-dependent nature of ferromagnetic hysteresis at a material level. With a focus on three key differing aspects of models, namely linear versus nonlinear(hysteresis), isotropic versus anisotropic and homogeneous versus inhomogeneous, we attempt to discriminate between the performance of models on the basis of these criteria. More specifically, with increasing model complexity, we have combined Maxwell's equations with four different hysteresis models within the context of a prolate steel ellipsoid, whose ferromagnetic properties evolve under the influence of a uniform applied background field. Among other aspects, the hysteresis models differ in terms of physical motivation, complexity and parameter spaces. In this research, we have analysed four hysteresis models in more detail: The Induced - Permanent magnetization model, The Rayleigh} model, the Jiles-Atherton model and an Energy-Variational model, based on energy balances. The thus derived forward models have subsequently been inverted in order to estimate material hysteresis parameters. With increasing complexity also, twin experiments have been performed. This increasing complexity extit{temporally} stems from the fact that the hysteresis models named previously, are stated in increasing order of complexity, and can all be modified in order to model anisotropic material by generalizing model parameters to tensors. Spatially, the increase in complexity is caused by the fact that in special cases, namely of uniform ellipsoid magnetization, an analytical formula relating the magnetic field, the background field and the ellipsoid magnetization exists by solving the Poisson partial differential equation on an infinite domain using direct computation with Green's functions.