We propose a model for breathing in the form of a PDE system consisting of a mass balance equation and a momentum balance equation. Lungs are represented by a viscoelastic porous body with degenerate hysteretic pressure-saturation relation. We prove that if a periodic force is applied on the boundary, then the resulting system admits a periodic solution under suitable technical assumptions. The proof relies on uniform bounds for the pressure based on a time-periodic variant of the Moser-Alikakos iterations. This is a joint work with M. Eleuteri, E. Ipocoana, and J. Kopfova.