key: cord-0746569-5cplv093 authors: Ouadfeul, S.-A. title: Multifractal behavior of SARS-CoV-2 COVID-19 pandemic spread, case of: Algeria, Russia, USA and Italy. date: 2020-09-18 journal: nan DOI: 10.1101/2020.09.16.20196188 sha: 9ee3a5a51ba1b00a6f361b6a3ad12ce6efe38ba9 doc_id: 746569 cord_uid: 5cplv093 Here, the multifractal behavior of the SARS-CoV-2 COVID-19 pandemic daily and death cases is investigated through the so-called Wavelet Transform Modulus Maxima lines (WTMM) method, data available via the World Health Organization (WHO) dashboard of Algeria, Russia, USA and Italy are analyzed. Obtained results show the multifractal behavior of the COVID-19 pandemic data with different spectra of singularities. Keywords: Multifractal behavior, daily and death cases, WTMM, COVID-19 pandemic data Takagi model description, while the section 4 is devoted the application of the WTMM method to daily and death cases to investigate the mulifractal behavior. The case ofAlgeria, Russia, USA and Italy is studied. We end the paper by a conclusion. Here we review some of the important properties of wavelets, without any attempt at being complete. What makes this transform special is that the set of basis functions, known as wavelets, are chosen to be well-localized (have compact support) both in space and frequency (Arneodo and Bacry, 1995) . Thus, one has some kind of "dual-localization" of the wavelets. This contrasts the situation met for the Fourier Transform where one only has "monolocalization", meaning that localization in both position and frequency simultaneously is not possible. The CWT of a function s(z) is given by Grossmann and Morlet (1985) as: Each family test function is derived from a single function ) (z ψ defined to as the analyzing wavelet according to [19] : Where a * + ∈ R is a scale parameter, b R ∈ is the translation and ψ * is the complex conjugate of ψ. The analyzing function ) (z ψ is generally chosen to be well localized in space (or time) and wavenumber. Usually, ψ(z) is only required to be of zero mean, but for the particular purpose of multiscale analysis ψ(z) is also required to be orthogonal to some low order polynomials, up to the degree n−1, i.e., to have n vanishing moments : The 1D Wavelet Transform Modulus Maxima lines (WTMM) method is a wavelet based multifractal analysis formalism introduced by Arneodo et al (1995) , the algorithm of the WTMM is composed with five steps which are (Ouadfeul, 2020 4-Estimation of the spectrum of exponents ( ). For more details about the 1D WTMM method we invite readers to the paper of Arneodo et al (1996) or Ouadfeul and Aliouane (2011) . One the popular Pandemic models is the Takagi curve, construct the curve with the aid of the function θ: R → R which measures the distance from every point x to its closest integer: We compute the graph of an approximation of the Takagi function for an n great enough in figure 1. This graph is a classical example of a fractal. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020 . . https://doi.org/10.1101 with c varying between (−2 , 2) we obtain new functions. The graph in figure 1 is obtained for computing the function τ( x ) for100 linearly spaced points in the interval [0,1], with c randomly chosen in (−2 , 2) at each iteration. Figure 2 shows an example of the randomized Takagi function (Pecurar and Necula, 2020). The first case to be analyzed is the daily cases for Algeria, data form Health World Organization (HWO) COVID-19 dashboard are used, figure 3.a shows the graph of the daily cases, the date of first case observed is 25 th of February 2020, the number of days since the first case is 192. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020 . . https://doi.org/10.1101 Where + = 152, ! = 675 R(t) is the random component. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. . https://doi.org/10.1101/2020.09.16.20196188 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. . https://doi.org/10.1101/2020.09.16.20196188 doi: medRxiv preprint 5. Case of USA Figure 5 .a shows the daily cases in USA, the number of infection is higher that Algeria (this is due to the higher number of population). Figure 5 .b shows the modulus of the continuous wavelet transform in (t-f) domain, while figure 5.c shows the graph of the spectrum of exponents, which confirm the multifractal behavior of the daily cases in this country. Figure 5 .d shows the spectrum of singularities versus the Holder exponents, we can observe that the Holder exponents are very low (<0.53), which confirm the presence of the high frequency components only in the daily cases signal. The presence of random character in the SARS-CoV COVID-19 pandemic spread in USA is due to the non-respect of confinement in this country. The first observed case was in 20 th of January 2020, we observe also that the SARS-CoV2 coronavirus has the maximum aggressiveness between June and September 2020. The propagation looks like will become more stable in the future. Figure 6 .a shows the graph of the daily death in USA, we can observe that the maximum death number was during the period of April-June 2020 this may be due to the version the SARS-CoV2 coronavirus that was very dangerous and fatal in this period, since the virus is far from stability and the equilibrium and undergoes a lot of mutation process (Raoult, 2020; Mandal et al, 2020; Ouadfeul, 2020) . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. . https://doi.org/10.1101/2020.09.16.20196188 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. . https://doi.org/10.1101/2020.09.16.20196188 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. which confirm the dominance of medium to low frequencies components in the daily cases signal. This is due to the progressive decrease in the pandemic daily cases. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020 . . https://doi.org/10.1101 is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. . https://doi.org/10.1101/2020.09.16.20196188 doi: medRxiv preprint 7. Case of Italy Figure 9 .a shows the COVID-19 daily cases in Italy, the first observed case was in the 29 th if January 2020, the maxima of pandemic was between February-Mai 2020, after that the situation become stable until the beginning of the month of August where a second wave of the pandemic seems to be in the horizon. Figure 9 .b shows the modulus of the continuous wavelet transform, while figure 9.c shows the spectrum of exponents versus q which is demonstrating the multifractal behavior of SARS-CoV2 COVID-19 epidemic in Italy. Figure 9 .d shows the spectrum of singularities, we can observe the dominance of the high Holder exponents, which means the absence of the high frequency components in this signal. The origin of absence of the high frequency components is the progressive increase of the daily cases. Figure 10 .a shows the daily death of COVID-19 pandemic in Italy, the maximum death number was between Mars and Mai 2020, after that the number of death was very low. This due to the current version of SARS-CoV2 coronavirus which becomes less fatal. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020 . . https://doi.org/10.1101 is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020 September 18, . . https://doi.org/10.1101 September 18, /2020 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020. . https://doi.org/10.1101/2020.09.16.20196188 doi: medRxiv preprint We have analyzed the daily and death cases of SARS-CoV2 COVID-19 pandemic using the wavelet transform modulus maxima lines method, data of the World Health Organization are used. The case of Algeria, USA, Federation of Russia and Italy are studied, the multifractal behavior in all time series is demonstrated. The dominance of the high/low Holder exponents in the spectrum of singularities is controlled by the random respect of distancing and confinement protocols imposed by the governments, while the dominance in the daily death cases is mainly controlled by the health condition of the infected patients. Decrease and low values of the daily death number in Italy is most probably due to the current version of SARS-CoV-2 coronavirus (it undergoes mutation) which currently less dangerous and fatal. COVID-19 daily cases in Algeria fellow a randomized Gaussian model, while the daily death in Federation of Russia fellows the randomized Takagi function. A second wave of the pandemic seems to be in the front in Italy by consequence a new protocol is imposed. SARS-CoV2 COVID-19 pandemic spread in the world fellows a variety of models, modeling them is a challenge, each model depends on the confinement/distancing protocol and protocol respect, traditions of peoples and architectures of the cities. Arneodo, A., Bacry, E., 1995,Ondelettes ,multifractal is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 18, 2020 . . https://doi.org/10.1101 The thermodynamics of fractals revisited with wavelets Data as received by WHOfromnational authorities by 10:00 CEST Nonlinear Advection-Diffusion-Reaction Phenomena Involved in the Evolution and Pumping of Oil in Open Sea: Modeling, Numerical Simulation and Validation Considering the Prestige and Oleg Naydenov Oil Spill Cases Complexity in SARS-CoV-2 genome data: Price theory of mutant isolates Multifractal analysis revisited by the continuous wavelet transform applied in lithofacies segmentation from well-logs data Multifractal Analysis of SARS-CoV-2 Coronavirus genomes using the wavelet transform An analysis of COVID-19 spread based on fractal interpolation and fractal dimension Le Covid-19 présente une «surmutation», affirme le professeur Didier Raoult