Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and $\Psi$ a virtual support function on $\Sigma$. Let $(X_{\Sigma},D_{\Psi})$ be the associated toric variety and $\mathbb{T}$-Cartier divisor defined over $K$. A toric model of $(X_{\Sigma},D_{\Psi})$ is a triple $({\mathcal{X}},D,e)$, where ${\mathcal{X}}$ is a toric model over $S$ of $X$, $D$ is a $\mathbb{T}$-Cartier divisor on ${\mathcal{X}}$ and $e>0$ is an integer such that the isomorphism $\iota\colon X_{\Sigma}\to{\mathcal{X}}_{\eta}$ that extends the identity of $\mathbb{T}_{K}$ satisfies $\iota^{\ast}(D)=eD_{\Psi}$. When $e=1$, the toric model $({\mathcal{X}},D,1)$ will be denoted simply by $({\mathcal{X}},D)$. A toric model will be called proper whenever the scheme ${\mathcal{X}}$ is proper over $S$.