Implementation and design of national and regional feedback intervention strategies
We model the implementation of regional social distancing strategies by capturing their effects as a variation of the social distancing parameters, ρi in (7)–(8), in each region. Specifically, we assume each region adopts the following feedback control rule with hysteresis:ρi is set and kept equal to \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\underline \rho _i$$\end{document}ρ_i as long as the saturation of the regional health system, computed as the ratio between the number of the hospitalized requiring care in ICU (estimated as 0.1Hi) over the number of available ICU beds in the region, is above 20%, i.e., \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\rho _i = \underline \rho _i,{\mathrm{if}}\,\frac{{0.1H_i}}{{T_i^H}} \ge 0.20$$\end{document}ρi=ρ_i,if0.1HiTiH≥0.20
ρi is set and kept equal to \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\bar \rho _{\mathrm{i}}$$\end{document}ρ¯i as long as the saturation of the regional health system is below 10%, i.e., \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\rho _i = \bar \rho _i,{\mathrm{if}}\,\frac{{0.1H_i}}{{T_i^H}} \le 0.10$$\end{document}ρi=ρ¯i,if0.1HiTiH≤0.10
In our simulations, \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\underline \rho _i$$\end{document}ρ_i is set equal to the minimum estimated value in that region during the national lockdown (see Supplementary Table 4) and \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\bar \rho _{\mathrm{i}}$$\end{document}ρ¯i increased as a worst case to \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${\mathrm{min}}(1,3\underline \rho _i)$$\end{document}min(1,3ρ_i) so as to simulate the effect of relaxing the lockdown measures in each region. (The case where \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\bar \rho _{\mathrm{i}}$$\end{document}ρ¯i is set to a lower value equal to 1.5\documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\underline \rho _i$$\end{document}ρ_i is shown for the sake of comparison in Supplementary Figs. 5 and 6.)
Also, when a region is in lockdown, we assume all fluxes in and out of that region are reduced by 70% of their original values to better simulate the actual reduction in people’s movement observed during the lockdown in Italy (for further details see Supplementary Information). Such a reduction level was estimated qualitatively by considering the publicly available mobility data from Google (Google mobility data (https://www.google.com/covid19/mobility/)).
National lockdown measures are modelled by setting all ρi simultaneously to \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\underline \rho _i$$\end{document}ρ_i in all regions and reducing all fluxes by 70% while national reopening of all regions by setting all ρi simultaneously to \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\bar \rho _{\mathrm{i}}$$\end{document}ρ¯i and restoring inter-regional flows to their prelockdown level.
To model the increase in the COVID-19 testing capacity of each region the parameter αi in region i is multiplied by a factor 2.5, which corresponds to the average increase in the number of tests carried out nationally since the COVID-19 outbreak first started.