## 1 Introduction  
The logistic equation is defined as  
\[u^{\prime}(t)=\frac{s}{u_{max}}\,u(u_{max}-u),\quad u(0)=u_{0}. \tag{1}\]  
where \(t\) is time, \(u=u(t)\) is the unknown function and the parameters \(s\)-steepness and \(u_{max}\)-saturation level are constants. The integral curve \(u(t)\) fulfilling condition \(0<u(t)<u_{max}\) is known as the logistic function. After solving (1) we obtain the logistic function in the form  
\[u(t)=\frac{u_{max}}{1+e^{-s(t-t_{0})}}, \tag{2}\]  
where \(t_{0}\) is the inflection point, which is related to the initial condition \(u(0)=u_{0}=\frac{u_{max}}{1+e^{st_{0}}}\), therefore \(t_{0}=\frac{1}{s}\log\left(\frac{u_{max}-u_{0}}{u_{0}}\right)\). Putting \(t=t_{0}\) in (2) we see that \(u(t_{0})=u_{max}/2\). The logistic function finds applications in many fields, including biology, biomathematics, chemistry, demography, economics, physics, probability, sociology, statistics, and artificial neural networks. The logistic function and the logistic equation, as well as some of their generalizations, have also been widely used in epidemiology to describe various phenomena with a sigmoid trend (see for example papers by [PERSON] et al. [1] and [PERSON] et al. [2], and the references therein). [PERSON] et al. [3] used a generalization of the logistic function for forecasting the number of individuals reported to be infected with COVID-19 in different countries. Wavelet analysis is now frequently used to extract information from epidemiological and other time series. [PERSON] et al. [4] introduced wavelet analysis for characterizing non-stationary epidemiological time series. [PERSON] et al. [5] used the Morlet wavelets for applications in epidemiology. [PERSON] et al. [6] proposed a two-phase procedure (combining discrete graphs and Meyer wavelets) for constructing true COVID-19 epidemic growth. [PERSON] and [PERSON] [7] used the reverse biorthogonal wavelet pairs for modeling the rotavirus epidemic dynamics. They also mentioned that they obtained similar results when using the Haar wavelet. [PERSON] et al. [8] apply the Daubechies wavelet (db2) and the Coiflets wavelet (coif1) for modeling of pertussis incidence in China from 2004 to 2018. [PERSON] et al. [9] and [PERSON] et al. [10] use the wavelet analysis to investigate the correlation between the incidence of dengue and weather conditions. [PERSON] [11] applies the Haar wavelet transform and the simplex forecasting method to the dataset of hepatitis A in the United States to give predictions of its incidence. [PERSON] et al. [12] modeled the disease dynamics caused by Mycobacterium tuberculosis in Russia using a sum of two logistic functions (2) (so-called bi-logistic model). Similar method was used by [PERSON] and [PERSON] [13] for predicting the end date of COVID-19 disease in Italy. From a slightly broader point of view, the logistic Equation (1) can be considered in the context of the following non-linear first-order autonomous differential equation  
\[\frac{du}{dt}=f(u), \tag{3}\]  
where \(f(u)\) is a real, continuous function of \(u\), representing a non-linear part of the equation. Several exact forms for \(f(u)\) have been studied by [PERSON] [14] and by [PERSON] and [PERSON] [15]. An advanced theory with important applications of Equation (3) is provided by [PERSON] and [PERSON] [16]. The connections with Lie algebra, Bose-Einstein state, and quantum theory, are put there in evidence. The method of separated variables gives a first stage of the solution of (3) in terms of \(t(u)\). This function is explicit only if the integral of the reciprocal of \(f(u)\) is exactly available. The second stage of the solution depends on if \(t(u)\) can be inverted in a closed form. Most of the important equations are not exactly solvable in terms of \(t(u)\) and we need to solve them with numerical integration and inversion.