## 3 Results and Discussion  
The aim of our analysis and simulations of pandemic curves is to quantitatively understand the effect of isolation between subgroups against the spreading of COVID-19. Isolation and vaccination are among the most crucial influencers in the dynamics of pandemic curves. Even with isolation rules,'spontaneous' outbreaks caused by returning travellers from abroad can result in a spatiotemporal dynamic that provokes plateaus or recurrence of multi-small peaks visible on log-lin scales in the overall picture. From the model follows that plateaus are a succession of small outbreaks with small intervals of about \(t/\tau=3\) to 5 and \(\tau\) is about 10 \(d\). Hence, several peaks in the log-lin plots from different countries [1] are analysed. For simplicity, we choose arbitrarily five countries in a semi-logarithmic presentation: two small neighbouring countries (Belgium, the Netherlands) with different confinement rules; a large country (USA) to show the advantage of using semi-logarithmic plots, and a typical country with a very low population density (Finland) to show the problem of low noisy number (between 1 and 0) and Portugal, interfacing only one country. Websites such as Financial Times and Our World in Data present confirmed cases and deaths of COVID-19 in cumulative numbers and in epidemic curves (per day, \(d\)) in logarithmic-linear (log-lin) scales from several countries [1]. Cumulative numbers and epidemic curves are issued daily at \(t_{i}\) and in integers as, \(N_{i}\) and \(\Delta N/\Delta t\) in discrete curves. For large numbers and for \(\tau>1\)\(d\), in the run up holds:  
\[N(t)\propto\frac{dN}{dt}\propto e^{t/\tau_{group}};for\;t/\tau<1\;holds\;e^{t/ \tau_{group}}\approx 1+t/\tau, \tag{2}\]  
and the curves are analysed and simulated as continuous functions. For convenience, an exponential growth of, e.g., \(\tau_{group}=10\)\(d\), is equivalent with a reproduction number \(R=1.1\) because [3]:  
\[R\approx 1+1d/\tau, \tag{3}\]or more precisely:  
\[R=e^{1d/\tau}, \tag{4}\]  
which is equivalent to a doubling time, \(T_{2}\), expressed as [3]:  
\[T_{2}=\tau_{grow}\times ln2=6.9\;d. \tag{5}\]  
The R-value is also applicable for non-exponential growth and decay and must not be translated into a \(\tau\)-value, which is only applicable for an exponential dependence. The typical monotonic increasing cumulative curves presented in the log-lin are convenient to visualize the situation in one plot for large and small countries and go asymptotically to the limiting value of \(M\). The exponential growth is in log-lin, a straight line as shown by the tangent lines in Figure 1 for USA, Portugal, Belgium, the Netherlands and Finland. The slope analysis of the tangent line in Figure 1 follows the characteristic time, \(\tau_{grow}\), with the time difference, \(t_{high}-t_{low}\), expressed in days, \(d\), and ln as the natural logarithm [6]. For confirmed cases of COVID-19, in most countries, it holds that for the run up to the first peak: \(\tau_{grow}\approx 5\;day\). The surprising result is verified, among others, for Ireland, United Kingdom, Italy and Germany, not shown in Figure 1 for clarity. The small difference, e.g., \(\tau_{grow}=5.68\) d for Italy and \(\tau_{grow}=4.73\) d for the Netherlands, can point to better handling by the Italian authorities [3]. The tangent line for the USA indicates a steeper exponential growth with \(2.7\;d<\tau_{grow\;conf\;cases}<3.3\;d\). The epidemic curves in Figure 2 for new confirmed cases per day, \(d\), in log-lin for Belgium and the Netherlands [1] show an exponential growth in the run up. In less than one year, the Belgium and the Netherlands curves show four to five peaks. Figure 2 shows  
Figure 1: Cumulative number of confirmed COVID-19 cases in absolute numbers [1]. The advantage of a log–lin plot is that small and large numbers are visible. The typical S-shape of the curve in the lin–lin plot is lost in the log–lin presentation. The blue tangent line is for Portugal, Netherland and Belgium.