## 3 Results and Discussion
### Statistical Inhomogeneity  
The parameters used in the simulations are inspired by real data from five different countries in Figures 1-3, and \(M\) is arbitrary. Therefore, we use a relative time scale \(t/\tau\). Our simulations consider a statistical inhomogeneous population and entail that characteristics of subgroups are essential for a better understanding of the pandemic curve dynamics. Hence, Figure 4a-f presents the simulations of the partitioning scenario, where a population is divided into \(m\) equal isolated (no travelling) subgroups, either starting with the outbreaks at the same time or more realistically with outbreaks shifted in time. The proposed scenario is partitioning in equal subgroups to compare with a city situation. The effect on the epidemic curve of four outbreaks at different times and of vaccination is also shown in Figure 4a-f. The aim of the partitioning simulation is to compare the sum of \(m\) isolated subgroups (all equal) with a city situation. Well-isolated subgroups are crucial to fight against the epidemic. Partitioning in \(m\) equal groups is the simplest scenario. The isolated subgroups are supposed to live on \(m\) 'islands' or in \(m\) isolated areas. All subgroups have equal \(\tau\)- and limiting value \(M\)/\(m\) adapted to their size and the start of outbreaks at \(t/\tau=0\) with \(N_{0}=1\). \(M\) is the limiting value for the total group supposed to live in close contact in a city. Already from this simple simulation, we learn that the behaviour of all islands together shows an earlier peak in the epidemic curve. The equations for \(N\) in an island, \(N_{1/m}\), and the sum \(m\times N_{1/m}\) and \(t_{inf}\) are:  
\[N_{1/m}=\frac{(M/m)e^{t/\tau}}{\big{(}\sqrt{M/m}-1+e^{(t/2\tau)}\big{)}^{2}} \to m\times N_{1/m}=\frac{Me^{t/\tau}}{\big{(}\sqrt{M/m}-1+e^{(t/2\tau)}\big{)} ^{2}}\
eq N_{city}=\frac{Me^{t/\tau}}{\big{(}\sqrt{M}-1+e^{(t/2\tau)}\big{)}^{ 2}}, \tag{12}\]  
\[t_{\text{infisland}}=2\tau ln\Big{[}2\Big{(}\sqrt{M/m}-1\Big{)}\Big{]}\
eq t_ {\text{infcity}}=2\tau ln\Big{[}2\Big{(}\sqrt{M}-1\Big{)}\Big{]}, \tag{13}\]  
\[\tau\dot{N}_{\text{maxisland}}=4(M/m)/2\tau\to\tau\dot{N}_{\text{maxisland}}= \mu\tau\dot{N}_{\text{maxisland}}=4M/2\tau. \tag{14}\]  
The full lines in the simulations represent cumulative numbers, \(N(t/\tau)\equiv N\). The dotted lines represent the new cases per day, \(d\) (\(dN/dt\equiv\dot{N}\equiv N^{\prime}\)), multiplied by the characteristic time, \(\tau\), versus the relative time scale, \(t/\tau\). This makes results more general. If \(\tau=10\)\(d\), then the absolute time scale is found by multiplying the x-axis values by 10 and dividing normalized epidemic curve values by 10. The simulation in Figure 4a-d have an arbitrarily chosen value \(M=10^{5}\). This means that for \(N_{0}=1\), \(M\)/\(N_{0}=10^{5}\) and \(t_{inf}/\tau=12.86\). If the reference \(\tau\) of the slow group is \(10\)\(d\), and the epidemic curve is about new hospitalizations per day, then a value of \(\tau N^{\prime}=500\) should be read as \(500/10\) or \(50\) patients per day, \(d\). In addition, if people stay in the hospital for \(10\)\(d\) on average, then we need a capacity of \(500\) beds for the newcomers. A stationary state holds: 'population size' is equal to the 'birth rate' multiplied by the 'average lifetime.' The simulations in Figure 4a-f are without a parameter change during the outbreak. In Figure 4a, the sum of \(10^{3}\) islands is compared with the city situation (\(m=1\)). We assume that all outbreaks start at the same time. This is much earlier than for the city situation with the peak at \(t/\tau=12.9\). The maximum for one island, \(\tau N^{\prime}_{max(1/m)}\), is \(10^{3}\) lower than the sum and comes earlier than in the city situation. The \(t_{inf}/\tau\) and \(\tau N^{\prime}_{max(1/m)}\) for \(\text{m}=1\) and \(10^{3}\) are indicated in the inserted table. The blue arrow indicates that the sum of all subgroups starts at a higher \(N\) and \(\tau\times N^{\prime}\) at low \(t/\tau\) compared with the city situation.