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The subject of this paper is to investigate the weak regime covariantscalar-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 fieldsis converted into $\Lambda(t),G(t)$ models. Time evolution of the cosmologicalparameters for a family of viable models have been investigated. Numericalresults with the cosmological data have been adjusted. We've introduced a modelfor dark energy (DE) density and cosmological constant which involves firstorder derivatives of Hubble parameter. To extend this model, correction termsincluding the gravitational constant are added. In our scenario, thecosmological constant is a function of time. To complete the model,interactionterms between dark energy and dark matter (DM) manually entered inphenomenological form. Instead of using the dust model for DM, we have proposedDM equivalent to a barotropic fluid. Time evolution of DM is a function ofother cosmological parameters. Using sophisticated algorithms, the behavior ofvarious quantities including the densities, Hubble parameter, etc. have beeninvestigated graphically. The statefinder parameters have been used for theclassification of DE models. Consistency of the numerical results withexperimental data of $SneIa+BAO+CMB$ are studied by numerical analysis withhigh accuracy.

Observational constraints on models of the Universe with time variable Gravitational and Cosmological constants along MOG