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

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