How to reduce epidemic peaks keeping under control the time-span of the epidemic

One of the main challenges of the measures against the COVID-19 epidemic is to reduce the amplitude of the epidemic peak without increasing without control its timescale. We investigate this problem using the SIR model for the epidemic dynamics, for which reduction of the epidemic peak $I_P$ can be achieved only at the price of increasing the time $t_P$ of its occurrence and its entire time-span $t_E$. By means of a time reparametrization we linearize the equations for the SIR dynamics. This allows us to solve exactly the dynamics in the time domain and to derive the scaling behaviour of the size, the timescale and the speed of the epidemics, by reducing the infection rate $\alpha$ and by increasing the removal rate $\beta$ by a factor of $\lambda$. We show that for a given value of the size ($I_P$, the total, $I_E$ and average $\hat I_P$ number of infected), its occurrence time $t_P$ and entire time-span $t_E$ can be reduced by a factor $1/\lambda$ if the reduction of $I$ is achieved by increasing the removal rate instead of reducing the infection rate. Thus, epidemic containment measures based on tracing, early detection followed by prompt isolation of infected individuals are more efficient than those based on social distancing. We apply our results to the COVID-19 epidemic in Northern Italy. We show that the peak time $t_P$ and the entire time span $t_E$ could have been reduced by a factor $0.9 \le 1/\lambda\le 0.34$ with containment measures focused on increasing $\beta$ instead of reducing $\alpha$.