  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 $\alpha$.
