## 3 Results  
We model the cumulative number of persons reported to be infected by COVID-19 virus, using a sum of logistic functions (multilogistic function) of the form  
\[f(x)=\sum\limits_{j=1}^{k}\frac{u_{j,\ max}}{1+\exp(-\frac{x-b_{j}}{a_{j}})}, \tag{20}\]  
where \(k\) is the number of all considered logistic waves. Denote by \(y_{n}^{*}\) the total cumulative number of individuals reported to be infected up to \(n\)th day in a country or a region and by \(y_{n}\) the 7-day moving arithmetic average for the sequence \(y_{n}^{*}\), i.e.,  
\[y_{n}=\frac{1}{7}\sum\limits_{j=-6}^{0}y_{n+i}^{*}.\]We will look, in the sequence \((y_{n})\), for points (days \(n\)) corresponding to zeros of the second or the third derivative of the logistic function. This is equivalent to detect points, where the sequence of second differences,  
\[\Delta^{2}y_{n}=y_{n+1}-2y_{n}+y_{n-1},\]  
takes a value close to zero or a maximum, respectively. In order to calculate the continuous wavelet transform (CWT) coefficients for \((\Delta^{2}y_{n})\), we implemented \(\psi_{2}(x)\) into Matlab's Wavelet Toolbox giving the following definition of the Logistic wavelet:  
function [psi,t] = logist(LB,UB,N,\(\sim\))  
% _LOGISTIC Logistic wavelet._  
% _[PSI,T] = LOGIST(LB,UB,N) returns values of_  
% _the Logistic wavelet on an N point regular_  
% _grid in the interval [LB,UB]._  
% _Output arguments are the wavelet function PSI_  
% _computed on the grid T._  
% _This wavelet has [-7 7] as effective support._  
% _See also WAVEINFO._  
% _Compute values of the Logistic wavelet._  
t = linspace(LB,UB,N); % wavelet support. psi = (exp(- 2* t)-exp(-t))./(1+exp(-t)).\({}^{\wedge}\)3;  
end  
Observations show that successive, relatively large, logistic waves arise at fairly long time intervals. Therefore, usually, the second differences corresponding to all previous waves are small as the next wave unfolds. However, the largest wave (with relatively high values of both the saturation level \(u_{max}\) and the parameter \(a\)), may be an exception. Therefore, at least for some countries, we perform the next wavelet analysis and calculate the CWT coefficients after subtracting the largest wave. Hence, we calculate the second differences and perform their wavelet analysis for the following new time series  
\[z_{n}=y_{n}-\frac{u_{i,\,max}}{1+\exp(-\frac{x-b_{i}}{a_{i}})},\]  
where for \(j=i\), the parameters \(a_{i}\), \(b_{i}\), \(u_{i,\,max}\) correspond to the largest wave in (20). Moreover, if necessary, we also carry out the wavelet analysis for shorter periods of time. We find the above mentioned points and parameters either directly by observing the sequence of second differences \((\Delta^{2}y_{n})\) or read them from the scalograms of the CWT transform of Matlab's Wavelet Toolbox by using the Logistic wavelet. From the considerations in Section 2.1 and from (9) it follows that parameter \(b\) should be determined as that point where the sequence \((\Delta^{2}y_{n})\) changes sign. Parameter \(a\) should be chosen in such a way that the distance between the zero and the maximum of \((\Delta^{2}y_{n})\) was approximately \(1.319a\) (see (9)). Thus, we obtain two parameters defining a logistic function approximating the time series \((y_{n})\). It remains to determine the third parameter of the wave, i.e., its saturation level \(u\).