@Guy2007; @Conley2008; @Burns2011]. Here we will exploit some features of the two most common ones: the Multicolor Light-Curve Shape (MLCS2k2) [@Jha2007] one and the Spectral Adaptive Light curve Template (SALT2) [@Guy2007].

In a nutshell:

-   The MLCS2k2 fitting model [@Jha2007] describes the variation among SNe Ia light curves with a single parameter ($\Delta$). Excess color variations relative to the one-parameter model are assumed to be the result of extinction by dust in the host galaxy and in the Milky Way. The MLCS2k2 theoretical magnitude, observed in an arbitrary filter $Y$, at an epoch $\gamma$, is given by $$m_{Y,\gamma}^{\rm th}=M_{Y',\gamma}+p_{Y',\gamma}\Delta
    +q_{Y',\gamma}\Delta^2
    +K_{Y'Y,\gamma}+\mu +X_{Y',\gamma}^{\rm host} +X_{Y,\gamma}^{\rm MW},
    \label{mlcs2k2model}$$ where $Y' \in \{U, B, V, R, I\}$ is one of the supernova rest-frame filters for which the model is defined, $\Delta$ is the MLCS2k2 shape-luminosity parameter that accounts for the correlation between peak luminosity and the shape/duration of the light curve. Furthermore, the model for the host-galaxy extinction is $X^{\rm host}_{Y',\gamma}=\zeta_{Y',\gamma}(a_{Y'}+b_{Y'}/R_V)A_V$, where $\zeta_{Y',\gamma}:=X^{\rm host}_{Y',\gamma}/X^{\rm host}_{Y',0}$, and $a_{Y'}$, $b_{Y'}$ are constants; as usual, $A_V$ is the $V$ band extinction, at $B$ band peak ($a_V=1$, $b_V=0$), and $R_V:=A_V/E(B-V)$, the ratio of $V$ band extinction to color excess, at $B$ band peak. Finally, $X^{\rm MW}_{Y,\gamma}$ is 