Clarifying the relationship between regular earthquakes and slow fault slip is essential for understanding the mechanisms behind seismic activity. We hypothesize that the background seismic activity around the Guerrero seismic gap in Mexico is partially triggered by interplate slow-slip events (SSEs). Consequently, we present an extension of the spatio-temporal epidemic-type aftershock sequence (ETAS) model, which incorporates background seismicity as a piecewise constant function over time. In this study, Global Navigation Satellite System (GNSS) data is employed to identify the occurrence periods of SSEs, thereby delineating the intervals during which changes in background seismicity may occur. Due to the technical complexity of performing inference with an inhomogeneous ETAS model, this work employs a penalized maximum likelihood inference method using the Expectation-Maximization (EM) algorithm. This approach also permits the inference of the branching process for aftershocks, thus enabling the estimation of the genealogy between earthquakes. This information could be utilized to decluster earthquakes. This study elucidates how the background seismicity increases during the periods of the Guerrero SSEs, which allows for a more comprehensive understanding of seismic activity and the relationship between slow and fast earthquakes in Mexico. Our new model can be applied not only in Mexico but also at plate boundaries worldwide to quantify the impact of SSEs on seismic activity.