The interaction of Io with the Jovian magnetosphere creates the best known and brightest satellite-controlled aurorae in our solar system. These aurorae are generated by the precipitation of electrons, which are accelerated by the Alfvén waves carrying the current between the satellite and the planet. A recent study computed the energy deposited on top of Jupiter's ionosphere due to the electron precipitation and retrieved the correct mean brightness of Io-related aurorae. The model developed in this study takes into account the acceleration mechanism and the Alfvén wave propagation effects. We use the same method to investigate the brightness variation of the different components of the Io footprint as a function of longitude. These observations are discussed in a companion paper. We identify several effects that act together to modulate the footprint brightness such as Alfvén wave reflections, magnetic mirroring of the electrons, the local interaction at Io and kinetic effects close to Jupiter. We identify the effects contributing the most to the modulation of the brightnesses of the three brightest components of the Io footprints: the main and reflected Alfvén wing spots and the transhemispheric electron spot. We show in particular that the modulation of the efficiency of the electron acceleration can be of greater importance than the modulation of the power generated at Io. We reproduce the average modulation of the spot brightnesses and present an extensive discussion of possible explanations for the observed features not reproduced by our model.