  A $\gamma$-ray transient, Swift J0644.5-5111, has been claimed to be
associated with FRB 131104. The $\gamma$-ray energy output is estimated as
$E_\gamma \approx 5\times 10^{51}$\,erg at the nominal $z\approx 0.55$ redshift
implied by the dispersion measure of FRB 131104. However, a long-term radio
imaging follow-up observations only place an upper limit on the radio afterglow
flux of Swift J0644.5-5111. Applying the external shock model, we make a
detailed constraint on the afterglow parameters for the FRB 131104/Swift
J0644.5-5111 system. We find that for the commonly used microphysics shock
parameters (e.g., $\epsilon_e=0.1$, $\epsilon_B=0.01$ and $p=2.3$), if the
redshift value inferred from the DM value is correct to order of magnitude
(i.e., $z>0.1$), the ambient medium number density should be $\leq
10^{-3}~\rm{cm^{-3}}$, which is the typical value for a compact binary merger
environment but disfavors a massive star origin. Assuming a typical ISM
density, one would require that the redshift of the FRB much smaller than the
value inferred from DM value ($z\ll0.1$), implying a non-cosmological origin of
DM. The constraints are much looser if one adopts smaller $\epsilon_B$ and
$\epsilon_e$ values, as observed in some GRB afterglows. The FRB 131104/Swift
J0644.5-5111 association remains plausible. We critically discuss possible
progenitor models for the system.
