Figure 1

XI representation of COVID-19 outbreaks. (a) Model illustration. The closed phase-space expression \(I=I(X)\) of actual infected cases I as a function of total infected cases X, as given by Eq. (3), is shown for two cases: \(\alpha _X=0\) (no control, red line) and \(\alpha _X=10\) (long-term control, blue line) for an intrinsic reproduction factor of \(g_0=3\). The number of infections is maximal at \(I_{\mathrm{peak}}\) (open circle), after starting at \(X=I=0\), with the epidemic ending when the number of actual cases drops again to zero. At this point the number of infected reaches \(X_{\mathrm{tot}}\). The peak \(X_{\mathrm{peak}}=2/3\) of the uncontrolled case, \(\alpha _X=0\), is sometimes called the ‘herd immunity’ point. The final fraction of infected is \(X_{\mathrm{tot}}=0.94\). (b) Model validation for a choice of four countries/regions. The model (lines) fits the seven-day centered averages of COVID-19 case counts well. For South Korea data till March 10 (2020) has been used for the XI-fit, at which point a transition from overall control to the tracking of individuals is observable. (c) Data collapse for ten countries/regions. Rescaling with the peak values \(X_{\mathrm{peak}}\) and \(I_{\mathrm{peak}}\), obtained from the XI fit, maps COVID-19 case counts approximately onto a universal inverted parabola. (d) Robustness test. The often strong daily fluctuations are smoothed by n-day centered averages. Shown are the Bergamo data (dots, \(n=7\)) and XI-fits to \(n=1\) (no average), \(n=5\) and \(n=7\). Convergence of the XI-representation is observed.