  Confinement properties of the $1+1$ Schwinger model can bestudied by
computing the string tension between two charges. It is finite (vanishing) if
the fermions are massive (massless) corresponding to the occurrence of
confinement (screening). Motivated by the possibility of experimentally
simulate the Schwinger model, we investigate here the robustness of its
screened and confined phases. Firstly, we analyze the effect of
nearest-neighbour density-density interaction terms, which -- in the absence of
the gauge fields -- give rise to the Thirring model. The resulting
Schwinger-Thirring model is studied, also in presence of a topological $\theta$
term, showing that the massless (massive) model remains screened (confined) and
that there is deconfinement only for $\theta=\pm\pi$ in the massive case.
Estimates of the parameters of the Schwinger-Thirring model are provided with a
discussion of a possible experimental setup for its realization with ultracold
atoms. The possibility that the gauge fields live in higher dimensions while
the fermions remain in $1+1$ is also considered. One may refer to this model as
the Pseudo-Schwinger-Thirring model. It is shown that the screening of external
charges occurs for $2+1$ and $3+1$ gauge fields, exactly as it occurs in $1+1$
dimensions, with a radical change of the long distance interaction induced by
the gauge fields. The massive (massless) model continues to exhibit confinement
(screening), signalling that it is the dimensionality of the matter fields, and
not of the gauge fields to determine confinement properties. A computation for
the string tension is presented in perturbation theory. Our conclusion is that
$1+1$ models exhibiting confinement or screening -- massless or massive, in
presence of a topological term or not -- retain their main properties when the
Thirring interaction is added or the gauge fields live in higher dimension.
