## 3 Results and Discussion
### Isolate Fast from Slow Is Better Than Mingle: Adding and Mixing  
The simulations in Figure 5a-h compares the effect of isolation between a fast and slow group. The two groups with different \(\tau\) and sizes are considered, either living rather isolated (no travelling) or mixed (no isolation). The total effect of the fast group with higher risk isolated from the slow group is compared with the intermingling. The two scenarios are denoted by 'adding' and for not isolated by'mixing.' Figure 5: _Cont._  
The statistical inhomogeneity stems from subgroups with different risks, \(1/\tau\) on infection, with \(\tau\) the characteristic time. The elderly people with lower immunity and confined in a retirement house show a higher risk for COVID-19 and have a lower \(\tau\). The peak of the epidemic curve is proportional to the growth limit, \(M\). In Figure 5a-h, the 'adding' of a fast with an isolated slow group is compared with strong'mixing.' In this simulation, only a fast and slow group are involved mostly unequal in number, \(\tau\) and \(M\). By 'adding,' we assume that the 'fast' group is well isolated from the'slow' group with \(\tau_{slow}\equiv\tau\). The risk for the fast group is a factor \(m>1\) higher than for the slow group, and their limit value is \(M/m\) a fraction m of the total \(M\), given by:  
\[1/\tau_{fast}=\ m/\tau;\ \tau_{fast}=\tau/m, \tag{15}\]  
and the population in the groups are proportional, e.g., \(M_{fast}=M/m\). The slow group is defined as:  
\[\tau_{slow}\equiv\tau\ ;\ M_{slow}=M(m-1)/m\ \text{and}\ M_{fast}+M_{slow}=M;\ m \geq 1. \tag{16}\]  
Figure 5: Difference between adding and mixing without a change in parameters: \(\tau\) and \(M\) during the outbreak. (**a,b**) Shows almost homogeneous, with \(m=3/2\). (**c,d**) Simulations with \(m=2\), representing subgroups equal in number. (**e,f**) Simulation with \(m=3\) and (**g,h**) simulation with \(m=4\). The fast group gives an earlier, sharper and higher peak than the slow group. Mixing always results in a higher and sharper peak than adding. Improving isolation by, e.g., avoiding social gatherings, specifically, between fast and slow groups, is more efficient than by strengthening general rules. The parameter choice considers that: 'the larger the deviation in \(\tau\) the rarer.' This inversely proportional dependence agrees with a large number of statistics, as in [PERSON]'s power law [14; 15]. The consequence of the chosen dependence of \(\tau_{fast}\) and \(M_{fast}\) on the factor \(m>\)1 makes \(\dot{N}_{max}\)\(m\) independent. The consequence of the chosen dependence on the factor \(m>\)1 for \(\tau_{fast}=\tau/m\) and \(M_{fast}=M/m\) is that \(\dot{N}_{max}\) for the fast group is \(m\)-independent. Mixing is the worst-case scenario. The fast and slow group become a 'homogeneous group' with harmonic averaged \(\tau\). In the scenario for intermingling, we propose the harmonic average \(\tau\)-value for the mixed group. The general average, \(\tau_{a\ av}\), is defined as:  
\[\tau_{a\ av}=\left(\frac{1}{k}\sum_{i}^{k}\tau_{i}^{a}\right)^{1/a}. \tag{17}\]  
For a = 1, 2 and \(-1\); \(\tau_{a\ av}\) is the well-known arithmetic mean, root mean square (rms) mean and harmonic average, respectively [16]. The harmonic average is used because the average risk is used for the mixing group, and \(1/\tau\) represents the risks. For the intermingling average, \(\tau_{-1\ av}\) holds:  
\[\tau_{-1\ av}=\left(\frac{1}{M}\left(\frac{M/m}{\tau/m}+\frac{M(m-1)/m}{\tau} \right)\right)^{-1}=\frac{m}{2m-1}\tau\rightarrow\tau_{fast}<\tau_{-1\ av}<\tau. \tag{18}\]  
Table 14 shows the relations and numerical values used in the simulations of Figure 5a-h to show the differences between mixing and adding. The total result of adding and mixing are denoted by the subscript 'add' and'mix.' The 9 th row in Table 14 shows the characteristics: \(t_{inf}\) and \(\dot{N}_{max}\equiv N_{max}^{\prime}\) for mixing towards a 'homogeneous' group by a strong interaction.