  Let $\mathcal{C}$ be a weakly idempotent complete extriangulated category. In
contrast with the Hovey correspondence of admissible model structures on weakly
idempotent complete exact categories from two complete cotorsion pairs, we give
a construction of model structures on $\mathcal{C}$ from only one complete
cotorsion pair. Our main result not only generalizes the work by
Beligiannis-Reiten and Cui-Lu-Zhang, but also provides methods to construct
model structures from silting objects of $\mathcal{C}$ and co-$t$-structures in
triangulated categories.
