Io's plasma interaction creates an electromagnetic coupling between Io and Jupiter through Alfvén waves triggering the generation of auroral footprints in Jupiter's southern and northern hemispheres. The brightness of Io's footprints undergoes periodic variations that are primarily modulated by Io's local plasma interaction through the Poynting flux radiated away from the moon. The periodic pattern with two maxima near 110<SUP>∘</SUP> and 290<SUP>∘</SUP> Jovian longitude where Io crosses the dense plasma sheet is generally understood. However, some characteristics, like the 2-4 times stronger brightening of the southern footprint near Jovian longitude 110<SUP>∘</SUP> or the lack of response to Io's eclipse passage, are not fully understood. We systematically study variations in Io's plasma interaction and the Poynting flux using a 3D magnetohydrodynamic model, performing a series of simulations with different upstream plasma conditions and models of Io's atmosphere. Our results indicate that the strong Jovian magnetic field near 110<SUP>∘</SUP> plays a more important role than previously estimated for the strong brightening there. We find that the Poynting flux is not fully saturated for a wide range of possible atmospheric densities (6 ×10<SUP>18</SUP> - 6 ×10<SUP>21</SUP> m<SUP>-2</SUP>) and that density changes in the atmosphere by a factor of &gt; 3, as possibly happening during Io's eclipse passage, lead to a change of the Poynting flux by &gt; 20%. Assuming that these expected changes in Poynting flux also apply to the footprints, the non-detection of a dimming in the footprint during the eclipse by Juno-UVS suggests that Io's global atmospheric density decreases by a factor of &lt; 2.5. We show that for smaller atmospheric scale heights (i.e. a more confined atmosphere), changes in the atmospheric density have less effect on the Poynting flux. The missing response of the footprint to the eclipse hence might also be consistent with a density decrease by a factor of &gt; 3, if the effective atmospheric scale height is small (&lt; 120 km). Finally, we provide new analytical approximations that can be used for analyzing the effect of the local interaction responsible for the footprint variability in future studies.