Exploration of interacting dynamical dark energy model with interaction term including the equation-of-state parameter: alleviation of the $H_{0}$ tension

We explore a scenario of interacting dynamical dark energy model with the interaction term $Q$ including the varying equation-of-state parameter $w$. Using the data combination of the cosmic microwave background, the baryon acoustic oscillation, and the type Ia supernovae, to global fit the interacting dynamical dark energy model, we find that adding a factor of the varying $w$ in the function of $Q$ can change correlations between the coupling constant $\beta$ and other parameters, and then has a huge impact on the fitting result of $\beta$. In this model, the fitting value of $H_{0}$ is lower at the $3.54 \sigma$ level than the direct measurement value of $H_{0}$ . Comparing to the case of interacting dynamical dark energy model with $Q$ excluding $w$, the model with $Q$ including the constant $w$ is more favored by the current mainstream observation. To obtain higher fitting values of $H_{0}$ and narrow the discrepancy of $H_{0}$ between different observations, additional parameters including the effective number of relativistic species, the total neutrino mass, and massive sterile neutrinos are considered in the interacting dynamical dark energy cosmology. We find that the $H_{0}$ tension can be further reduced in these models, but is still at the about $3 \sigma$ level.