  The high-temperature superconducting state in sulfur trihydride
($T_{C}=203$~K) has been investigated in the context of the non-adiabatic and
anharmonic effects. The Migdal-Eliashberg equations and the extended Eliashberg
equations, which include the lowest-order vertex corrections, have been solved
numerically in the self-consistent way. For $R3m$ crystal structure, the
lowest-order vertex corrections decrease the value of the Coulomb
pseudopotential from $0.123$ to $0.108$. The anharmonic effects work
antagonistically in relation to the vertex corrections shifting the value of
$\mu^{\star}$ to $0.156$. The studies conducted for the structure
$Im\overline{3}m$, where the Eliashberg function includes both the
non-adiabatic and anharmonic effects, prove the even higher value of
$\mu^{\star}=0.185$. Independently of the assumed method of the analysis, the
nearly identical no mean-field dependence of the order parameter on the
temperature was obtained: $2\Delta(0)/k_{B}T_{C}\sim 4.7$ - due to the
significant strong-coupling and retardation effects: $\lambda\sim 2$ and
$k_{B}T_{C}\slash \omega_{\rm \ln}\sim 0.15$-$0.19$. It means that the
classical equations of Migdal-Eliashberg can be treated as a correct effective
model for the superconducting state in $\rm H_{3}S$. This paper has shown that
the McMillan or Allen-Dynes formulas substantially lower the value of the
critical temperature in relation to the result obtained with the Eliashberg
equations.
