Short-term forecasting of epidemics like COVID-19 at a city-wide scale presents challenges to traditional SIR models. These include difficulties in estimating the number of unreported infected individuals - who may account for a significant portion of transmissions - variability in incubation periods, and the significant delay between infectious contacts and the appearance of symptoms. To support the Bologna health authority, we developed a compartmental model based on Distributed-delay Differential Equations, as in the presence of significant delays these have been shown to capture well the physics of infection spreading. A dedicated U compartment represents Unreported infected individuals, their permanence time in the compartment being modeled probabilistically by an empirical distribution and a time-dependent sociability parameter modulates infection rates, capturing variations in social activity due to government interventions or natural behavioral changes. Adjusting on a weekly basis the sociability parameter we were able to interpolate the clinical time series of new positives. Due to the delays present in the model, the regression procedure implies an adjustment on the past values of social activity, increasing the need to provide the sociability value in real time. To address this, we construct a proxy for sociability based on aggregated open mobility data, which proves to be reliable, up to a shift, for considerable durations of time, corresponding to the periods where other factors affecting transmission could be considered roughly unchanged. By combining this with a linear response analysis, identifying the moments where it is crucial to accurately estimate social activity rates, we are able to show that mobility data allows DDE-based models to be effective in forecasting epidemic events on the short and medium term at the metropolitan scale.