Characterization of striped surfaces

Let $Z$ be a non-compact two-dimensional manifold and $\Delta$ be a one-dimensional foliation of $Z$ such that $\partial Z$ consists of leaves of $\Delta$ and each leaf of $\Delta$ is a non-compact closed subset of $Z$. We obtain a characterization of a subclass of such foliated surfaces $(Z,\Delta)$ glued from open strips $\mathbb{R}\times(0,1)$ with boundary leaves along some of their boundary intervals.