  We consider the quantum affine vertex algebra
$\mathcal{V}_{c}(\mathfrak{gl}_N)$ associated with the rational $R$-matrix, as
defined by Etingof and Kazhdan. We introduce certain subalgebras $\textrm{A}_c
(\mathfrak{gl}_N)$ of the completed double Yangian
$\widetilde{\textrm{DY}}_{c}(\mathfrak{gl}_N)$ at the level $c\in\mathbb{C}$,
associated with the reflection equation, and we employ their structure to
construct examples of quasi $\mathcal{V}_{c}(\mathfrak{gl}_N)$-modules.
Finally, we use the quasi module map, together with the explicit description of
the center of $\mathcal{V}_{c}(\mathfrak{gl}_N)$, to obtain formulae for
families of central elements in the completed algebra $\widetilde{\textrm{A}}_c
(\mathfrak{gl}_N)$.
