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A new reactive ionization region model (R-IRM) is developed to describe the reactive Ar/O2 high power impulse magnetron sputtering (HiPIMS) discharge with a titanium target. It is then applied to study the temporal behavior of the discharge plasma parameters such as electron density, the neutral and ion composition, the ionization fraction of the sputtered vapor, the oxygen dissociation fraction, and the composition of the discharge current. We study and compare the discharge properties when the discharge is operated in the two well established operating modes, the metal mode and the poisoned mode. Experimentally, it is found that in the metal mode the discharge current waveform displays a typical non-reactive evolution, while in the poisoned mode the discharge current waveform becomes distinctly triangular and the current increases significantly. Using the R-IRM we explore the current increase and find that when the discharge is operated in the metal mode Ar+ and Ti+ -ions contribute most significantly (roughly equal amounts) to the discharge current while in the poisoned mode the Ar+ -ions contribute most significantly to the discharge current and the contribution of O+ -ions, Ti+ -ions, and secondary electron emission is much smaller. Furthermore, we find that recycling of atoms coming from the target, that are subsequently ionized, is required for the current generation in both modes of operation. From the R-IRM results it is found that in the metal mode self-sputter recycling dominates and in the poisoned mode working gas recycling dominates. We also show that working gas recycling can lead to very high discharge currents but never to a runaway. It is concluded that the dominating type of recycling determines the discharge current waveform.