Fig. 1.

Different effect of acting on the α or the β parameter. The SIR Eqs. (1) are numerically integrated and I(t) plotted in arbitrary units for given initial conditions and α, β parameters (solid), the maximum I_* being reached at $t=t_{*}$. Then they are integrated for the same initial condition but raising β by a factor $\vartheta =3/2$ (dashed) with maximum $I_{\beta }=r{I}_{*}$ reached at time $t_{\beta }={\sigma }_{\beta }{t}_{*}$; and lowering α by the same factor $\vartheta =3/2$ (dotted) with maximum $I_{\alpha }={\mathcal{I}}_{\beta }$ reached at time $t_{\alpha }={\sigma }_{\alpha }{t}_{*}$. Time unit is one day, $\alpha =(4/3)*{10}^{-8},$$\beta =1/7$; these parameters arise from our fitting of data from the early phase of COVID epidemics in Northern Italy [7]; the population of the most affected area in the initial phase is about 20 million, that of the whole Italy is about 60 million. The numerical simulation is ran with $N=6*{10}^7$; it results $r=0.74,$${\sigma }_{\alpha }=1.63,$${\sigma }_{\beta }=1.09,$$I_{*}=3.08*{10}^7,$$t_{*}=$ 26.4; note that ${\sigma }_{\alpha }/{\sigma }_{\beta }=3/2=\vartheta$.