  The two-photon-exchange (TPE) contribution in $ep\rightarrow ep\pi ^0$ with
$W=M_{\Delta}$ and small $Q^2$ is calculated and its corrections to the ratios
of electromagnetic transition form factors $R_{EM} =
E_{1+}^{(3/2)}/M_{1+}^{(3/2)} $ and $R_{SM} = S_{1+}^{(3/2)}/M_{1+}^{(3/2)}$,
are analysed. A simple hadronic model is used to estimate the TPE amplitude.
Two phenomenological models, MAID2007 and SAID, are used to approximate the
full $ep\rightarrow ep\pi ^0$ cross sections which contain both the TPE and the
one-photon-exchange (OPE) contributions. The genuine the OPE amplitude is then
extracted from an integral equation by iteration. We find that the TPE
contribution is not sensitive to whether MAID or SAID is used as input in the
region with $Q^2<2$ GeV$^2$.
  It gives small correction to $R_{EM}$ while for $R_{SM}$, the correction is
about -10\% at small $\epsilon$ and about $1\%$ at large $\epsilon$ for
$Q^2\approx2.5$ GeV$^2$. The large correction from TPE at small $\epsilon$ must
be included in the analysis to get a reliable extraction of $R_{SM}$.

Nuclear Theory;Nuclear Experiment