  Since the ($\beta$-deformed) Hurwitz Kontsevich model corresponds to the
special case of affine Yangian of ${\mathfrak{gl}}(1)$. In this paper, we
construct two general cases of the $\beta$-deformed Hurwitz Kontsevich model.
We find that the $W$-operators of these two models can be represented by the
generators $e_k,\ f_k,\psi_k$ of the affine Yangian of ${\mathfrak{gl}}(1)$,
and the eigenstates (the symmetric functions $Y_\lambda$ and 3-Jack
polynomials) can be obtained from the 3D Young diagram representation of affine
Yangian of ${\mathfrak{gl}}(1)$. Then we can see that the $W$-operators and
eigenstates are symmetric about the permutations of coordinate axes.
