## 3 Results and Discussion
The analysis gives [6]: \(\tau_{grow}\cong 5.3\;d\). that the growth rate (\(1/\tau\)) in the first peak (steeper slope) is faster than for the second peak (slower slope), e.g., for Belgium and the Netherlands: first peak is \(\tau_{grow}\cong 5\,d\), and second peak is \(\tau_{grow}\cong 12\,d\). In general, \(\tau_{grow}\) in the growth phase is smaller than \(\tau_{rec}\) in the recovery, and the decay phase is sometimes hidden by a plateau. Clear plateaus between maxima are visible in, e.g., Finland at a low level from the beginning of October to the middle of November (2020), and USA (not shown) at a high level from June to the end of November (2020). Plateau levels are explained by overlapping outbreaks in the proposed model and simulations in Section 3.3. In Figure 2, \(a_{grow}\) and \(\alpha_{rec}\) are the angles of the tangent lines. To be specific, the new cases in Belgium show a ratio \(n=4.76\); \(1.6\) and \(4\) for the first, second and third peaks, respectively. The relation between \(\tau\), the scale factors on the axis, and the \(tg\) of the angle of the slope of the tangent line in a plot is given by:  
\[tg\alpha_{grow}=\frac{s_{y}}{s_{x}}\frac{log_{10\epsilon}}{\tau_{grow}}\to \tau_{grow}=\frac{s_{y}}{s_{x}}\frac{log_{10\epsilon}}{tg\alpha_{grow}}\to \frac{\tau_{grow2}}{\tau_{grow1}}=\frac{tg\alpha_{grow1}}{tg\alpha_{grow2}}, \tag{6}\]  
where the scale factor for the \(x\)-axis is \(S_{x}\) in mm/d and for the \(y\)-axis is \(S_{y}\), which is the distance in mm for a factor of 10 along the log scale \(y\)-axis. In the same plot, where the ratio \(s_{y}/s_{x}\) is constant, and the ratio of \(tg\) for the slopes between the first peak and the second peak are equal to the ratio of the \(\tau\)-values of the second peak over the first peak. Figure 3 shows the curves for deaths per day, \(d\), attributed to COVID-19 in a log-lin plot for a few countries in absolute numbers taken from Our World in Data [1]. We expect a delay between the curve for deaths and confirmed cases per day, \(d\). The comparison between the curves for new cases in Figure 2 and new deaths in Figure 3 shows a variable  
Figure 2: The epidemic curve in log-lin for new confirmed cases in Belgium and the Netherlands [1]. Results are taken over 7 days, \(d\) to smooth out the fluctuations and normalized on \(10^{5}\) to better compare countries with different populations. The two tangent lines and slope angles are an indication for the analysis in a ratio of \(\tau\)-values. Some peaks in the log–lin presentation at a level on a factor of 30 below the main peaks are hardly visible in a linear presentation. lag-time of about 2, 9 and 13 \(d\) between the corresponding peaks of cases and deaths for the first, second and third peak in Belgium. The increase in \(\tau_{grow}\) from 5 \(d\) for the first peak to 12 \(d\) for the second peak and the increase in lag-time delay for the first peak to the succeeding peaks may be explained by the faster transmission of infection for the age group with a higher risk (\(I/\tau\)) in the first peak compared to the (younger) age group with smaller risk in the second having slower transmission. The last peak can be the typical lowest transmission after imposing the strongest behaviour rules for the group in the third 'wave.' The reduction of risks can be a consequence of imposing behaviour rules in the meantime between the first and second peak or lower risk for the younger age group than in the first peak. Summarizing for new cases and deaths per day, \(d\), holds:  
\[1.6<n\equiv\tau_{recov}/\tau_{grow}=tg_{a\ grow}/tg_{a\ rev}<5. \tag{7}\]  
Daily new confirmed COVID-19 deaths  
Shown is the rolling 7-day average. Limited testing and challenges in the attribution of the cause of death means that the number of confirmed deaths may not be an accurate count of the true number of deaths from COVID-19. All epidemic plots in log-lin show a typical steep non-exponential dependence that is ignored to calculate \(\tau\) and is simulated and explained in Figure 4a.