In this brief work we present a novel approach to the logistic dynamics of populations and epidemic spreading that can take into account of the complex nature of such a process in several real situations, where due to different agents the dynamics is no longer characterized by a single characteristic timescale, but conversely by a distribution of time scales, rendered via a time-dependent growth rate. In detail, a differential equation containing a power-law time dependent growth rate is proposed, whose solution, named Stretched Logistic Function, provides a modified version of the usual logistic function. The model equation is inspired by and applied to the recent spreading on COVID-19 disease in Italy, showing how the real dynamics of infection spreading is characterized by a time dependent dynamics. A speculative discussion of the Stretched Logistic Function in relation to diffusion processes is attempted.
