  In this study, cosmological models are considered, where dark matter and dark
energy are coupled and may exchange energy through non-gravitational
interactions with one other. These interacting dark energy (IDE) models have
previously been introduced to address problems with the standard $\Lambda$CDM
model of cosmology (which include the coincidence problem, Hubble tension and
$S_8$ discrepancy). However, conditions ensuring positive energy densities have
often been overlooked. Assuming two different linear dark energy couplings, $Q
= \delta H \rho_{\rm{de}}$ and $Q = \delta H \rho_{\rm{dm}}$, we find that
negative energy densities are inevitable if energy flows from dark matter to
dark energy (iDMDE regime) and that consequently, we should only seriously
consider models where energy flows from dark energy to dark matter (iDEDM
regime). To additionally ensure that these models are free from early time
instabilities, we need to require that dark energy is in the `phantom'
($\omega<-1$) regime. This has the consequence that model $Q=\delta H
\rho_{\rm{dm}}$ will end with a future big rip singularity, while $Q = \delta H
\rho_{\rm{de}}$ may avoid this fate with the right choice of cosmological
parameters.
