Your dualism between model and world is nearly Cartesian. The model itself isn't separate from the world but produced materially (by ideology, sociality, naturally, etc.).
To me this is like drawing a circuit diagram on a piece of paper and trying to convince someone that, "Really there is electricity flowing through it."
Models are relations between signifiers. There exists a transformation between the signified relations and the relations of the signifiers, but they are, in fact, two separate categories and the transformation isn't bijective. ie it doesn't form an isomorphism.