Quasi modules for the quantum affine vertex algebra in type $A$

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 $\widetilde{\textrm{A}}_c (\mathfrak{gl}_N)$.