Search for Rare Decays of ${D}_{s}^{+}$ to Final States ${\pi }^{+}{e}^{+}{e}^{-}$, ${\rho }^{+}{e}^{+}{e}^{-}$, ${\pi }^{+}{\pi }^{0}{e}^{+}{e}^{-}$, ${K}^{+}{\pi }^{0}{e}^{+}{e}^{-}$, and ${K}_{S}^{0}{\pi }^{+}{e}^{+}{e}^{-}$

Using $7.33\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data collected by the BESIII detector at center-of-mass energies in the range of $\sqrt{s}=4.128–4.226\text{}\text{}\mathrm{GeV}$, we search for the rare decays ${D}_{s}^{+}\to {h}^{+}\left({h}^{0}\right){e}^{+}{e}^{-}$, where $h$ represents a kaon or pion. By requiring the ${e}^{+}{e}^{-}$ invariant mass to be consistent with a $\varphi \left(1020\right)$, $0.98