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We have analyzed low-latitude ionospheric current responses to two intense (X-class) solar flares that occurred on 13 May 2013 and 11 March 2015. Sudden intensifications, in response to solar flare radiation impulses, in the <i>S</i>_<i>q</i> and equatorial electrojet (EEJ) currents, as detected by magnetometers over equatorial and low-latitude sites in South America, are studied. In particular we show for the first time that a 5 to 8 min time delay is present in the peak effect in the EEJ, with respect that of <i>S</i>_<i>q</i> current outside the magnetic equator, in response to the flare radiation enhancement. The <i>S</i>_<i>q</i> current intensification peaks close to the flare X-ray peak, while the EEJ peak occurs 5 to 8 min later. We have used the Sheffield University Plasmasphere-Ionosphere Model at National Institute for Space Research (SUPIM-INPE) to simulate the E-region conductivity enhancement as caused by the flare enhanced solar extreme ultraviolet (EUV) and soft X-rays flux. We propose that the flare-induced enhancement in neutral wind occurring with a time delay (with respect to the flare radiation) could be responsible for a delayed zonal electric field disturbance driving the EEJ, in which the Cowling conductivity offers enhanced sensitivity to the driving zonal electric field.