We construct a $B-L$ model with $A_4\times Z_3\times Z_4$ flavor symmetry which can accounts for %which successfully explains the recent $3+1$ active-sterile neutrino data. The tiny neutrino mass and the mass hierarchy are obtained by the type-I seesaw mechanism. The hierarchy of the lepton masses is satisfied by a factor of $v_H \left(\frac{v_l}{\Lambda}\right)^2 \sim 10^{-4}\, \mathrm{GeV}$ of the electron mass compared to the muon and tau masses of the order of $\frac{v_H v_l}{\Lambda} \sim 10^{-1}\, \mathrm{GeV}$. Th\Revised{e} $3+1$ active-sterile neutrino mixings are predicted to be $0.015 \leq|U_{\Revised{1} 4}|^2\leq 0.045$, $0.004 \leq|U_{\Revised{2} 4}|^2\leq 0.012$ and $0.004 \leq|U_{\Revised{3} 4}|^2\leq 0.014$ for normal hierarchy while $0.020\leq|U_{\Revised{1} 4}|^2\leq 0.045$, $0.008 \leq|U_{\Revised{2} 4}|^2\leq 0.018$ and $0.008\leq|U_{\Revised{3} 4}|^2\leq 0.022$ for inverted hierarchy. Sterile neutrino masses are predicted to be $0.7 \lesssim m_s \, (\mathrm{eV}) \lesssim 3.16$ for normal hierarchy and $2.6 \lesssim m_s \, (\mathrm{eV}) \lesssim 7.1$ for inverted hierarchy. For three neutrino scheme the model predicts $0.3401 \leq \sin^2\theta_{12}\leq 0.3415, \, 0.460 \leq \sin^2\theta_{23}\leq 0.540,\, -0.60 \leq \sin\delta_{CP}\leq -0.20$ for normal hierarchy and $0.3402 \leq \sin^2\theta_{12}\leq 0.3416,\, 0.434\leq\sin^2\theta_{23}\leq 0.610,\, -0.95 \leq \sin\delta_{CP}\leq -0.60$ for inverted hierarchy.