## 3 Results and Discussion
### Isolate Fast from Slow Is Better Than Mingle: Adding and Mixing
The 12 th and 14 th row shows \(t_{inf}\) and \(N_{max}\equiv N_{max}^{\prime}\) for the 'fast' and'slow' group that are well isolated from each other. The factor \(m>1\) is a measure for the statistical inhomogeneity (\(m=3/2\) is almost homogeneous). One scenario per row in Figure 5a-h shows the difference between adding and mixing. The red lines are for mixing. \(N\) is for'mixing' (full red line) and the sum of \(N\) of fast and slow or 'adding' (full black line). The three dotted lines are for \(\tau\times N^{\prime}\): slow group in blue, fast in black line and red dotted for'mixing.' The independent contributions to the epidemic curve are denoted as \(\tau\times N_{slow}^{\prime}\); \(\tau\times N_{fast}^{\prime}\) in Figure 5a,c. Figure 5 simulates scenarios with the limit value \(M=5\times 10^{4}\) and four m-values. No change is assumed in risks during the interval \(0<t/\tau<20\). The area under the lin-lin curves in Figure 5b,d,f,h is the total number of cases and remains constant because a constant value of \(M\) is chosen in the simulation. The dynamics and height of maxima can be quite different. Figure 5b,d,f,h shows the addition of \(\tau\times N^{\prime}\) for fast and slow to compare with mixing in lin-lin scales and in log-lin scales. Isolation between fast and slow (adding) always results in lower peaks in epidemic curves compared to mixing of fast and slow groups (red dotted lines). Figure 5a,b shows almost homogeneous, with \(m=3/2\), and describes a scenario of a slightly faster subgroup, \(2/3\) of the population with a 1.5 higher risk than in the slow group. Because \(m\) is low, only one peak is visible. Figure 5b compares the same epidemic curve in lin-lin with log-lin for mixing and adding. The ratio inferred from the analysis of the blue and black tangent lines gives: \(\tau_{rec}/\tau_{grow}\cong 2\times m=3\)\(for\)\(m<2\). Such high \(n\)-values are also found in real data, as presented in Figures 2 and 3. Figure 5c,d, with \(m=2\), represents subgroups equal in number. Yet, the fast group has a twice lower \(\tau\)-value than the slow group. In Figure 5d, the two peaks in \(\tau\times N^{\prime}\) are visible. Figure 5e,f, with \(m=3\), shows the simulation of a minority (\(1/3\) of the population) fast group, with a 3 times lower \(\tau\)-value than the slow main (\(2/3\) of the population) group. Figure 5g,h, simulates with \(m=4\), which represents a subgroup of \(25\%\) weak elderly people with strong interaction between their caretakers, nurses all together having a four times higher risk than the rather well-isolated large (\(75\%\)) slow group outside in the nursing home. In this model, all peaks \(\tau\times N_{fast}^{\prime}\) of the fast group have the same height independent of \(m\). For an almost statistical homogeneous group with \(1<m<2\), the two peaks in the adding scenario melt together and result in \(2<\tau_{re}/\tau_{grow}<4\). A stronger heterogeneity in the population with \(m\geq 3\) results in two distinct peaks in the epidemic curve in the case of adding (isolation between the high- and low-risk group). Figure 6 shows the effect of imposing rules at different times, \(t_{r}\) in the exponential growth phase and the strong effect of a 75% vaccination degree. At time \(t_{r}\), rules are imposed lowering \(M\) and increasing \(\tau\). We ignore the dead time due to the relatively long incubation time for COVID-19, and we assume a homogeneous population and use \(V_{n2}\) with \(N_{0}\
eq 1\) but the value on N at \(t_{r}\). The relative times: \(t_{inf}/\tau\), \(t_{r}/\tau\), the \(N_{0}\)-value and the improve factor f for \(\tau\) and \(M\) are shown in Figure 6a for three scenarios to compare with 'not imposing rules change' (red line):  
1. The blue line represents an early and moderated change at \(t_{rule}/\tau=7.1\), at about 55% of the inflection time, \(t_{inf}\), as indicated in the table of Figure 6a.