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Accurate modeling of lockdown effects on SARS-CoV-2 epidemic evolution is akey issue in order e.g. to inform health-care decisions on emergencymanagement. The compartmental and spatial models so far proposed use parametricdescriptions of the contact rate, often assuming a time-invariant effect of thelockdown. In this paper we show that these assumptions may lead to erroneousevaluations on the ongoing pandemic. Thus, we develop a new class ofnonparametric compartmental models able to describe how the impact of thelockdown varies in time. Exploiting regularization theory, hospitalized dataare mapped into an infinite-dimensional space, hence obtaining a function whichtakes into account also how social distancing measures and people's growingawareness of infection's risk evolves as time progresses. This permits toreconstruct a continuous-time profile of SARS-CoV-2 reproduction number with aresolution never reached before. When applied to data collected in Lombardy,the most affected Italian region, our model illustrates how people behaviourchanged during the restrictions and its importance to contain the epidemic.Results also indicate that, at the end of the lockdown, around 12% of people inLombardy and 5% in Italy was affected by SARS-CoV-2. Then, we discuss how thesituation evolved after the end of the lockdown showing that the reproductionnumber is dangerously increasing in the last weeks due to holiday relaxespecially in the younger population and increased migrants arrival, reachingvalues larger than one on August 1, 2020. Since several countries still observea growing epidemic, including Italy, and all could be subject to a second waveafter the summer, the proposed reproduction number tracking methodology can beof great help to health care authorities to prevent another SARS-CoV-2diffusion or to assess the impact of lockdown restrictions to contain thespread.

Tracking the time course of reproduction number and lockdown's effect during SARS-CoV-2 epidemic: nonparametric estimation