By "Martin Löf type theory with identity types", I was thinking of the fragment without Π and Σ. Then by Thm 3.1. of arxiv.org/pdf/0709.0248.pdf, I think finitely completeness and weak factorization system is enough to model Martin Löf type theory with identity types. Or isn't the stability condition satisfied?

Could you please make it precise? I think model categories can interpret Martin Löf type theory with identity types. (By "dependent type theory", I meant Martin Löf type theory with identity types.) Is my understanding correct?