  The subject of this paper is to investigate the weak regime covariant
scalar-tensor-vector gravity (STVG) theory, known as the MOdified gravity (MOG)
theory of gravity. First, we show that the MOG in the absence of scalar fields
is converted into $\Lambda(t),G(t)$ models. Time evolution of the cosmological
parameters for a family of viable models have been investigated. Numerical
results with the cosmological data have been adjusted. We've introduced a model
for dark energy (DE) density and cosmological constant which involves first
order derivatives of Hubble parameter. To extend this model, correction terms
including the gravitational constant are added. In our scenario, the
cosmological constant is a function of time. To complete the model,interaction
terms between dark energy and dark matter (DM) manually entered in
phenomenological form. Instead of using the dust model for DM, we have proposed
DM equivalent to a barotropic fluid. Time evolution of DM is a function of
other cosmological parameters. Using sophisticated algorithms, the behavior of
various quantities including the densities, Hubble parameter, etc. have been
investigated graphically. The statefinder parameters have been used for the
classification of DE models. Consistency of the numerical results with
experimental data of $SneIa+BAO+CMB$ are studied by numerical analysis with
high accuracy.
