![]() ![]() The points of this space adequately reflect all the characteristic features of the COVID-19 pandemic in the time period we are interested in. The set of points M in R 2 with Cartesian coordinates v( t) = ( N( t + 1) − N( t))/ N m, N m = 10 6, and a( t) = ( v( t + 1) − v( t)) will be referred to as the phase space of the COVID-19 pandemic. In our case, it is formed by the function v( t) and its finite difference derivative a( t). Ī powerful tool for studying dynamical systems and processes is the use of phase space (PS). 3, but calculated from the statistical data. As initial data, we used the data of world statistics of COVID-19 for the time interval from Januto July 6, 2021, indicated below as and. Section 5 summarizes the results and perspectives. Section 4 is devoted to modeling the multifractal dynamics of the main reproduction number of COVID-19. Section 3 simulates the multifractal dynamics of the COVID-19 pandemic. Section 2 gives the main definitions of multifractal dynamics with piecewise linear trends and the fractal dimension of the phase space of the COVID-19 pandemic. Second, we estimate the variations in the basic reproduction number of COVID-19 using the equations of the discrete delay model or the reduced Susceptible-Infected-Removed (SIR) model, to clarify short-term forecasts and identification of new trends in the dynamics of the COVID-19 pandemic, , developed within both the traditional and fundamentally new approaches and software tools, ,, ,. First, we calculate the fractal dimensions of the phase space of the COVID-19 pandemic and various segments of the daily incidence in the world according to the global COVID-19 statistics. We start the present work basing on our experience in the field. We believe that the extension of the multifractal approach to the COVID-19 pandemic dynamics will allow a fresh look at this topical problem and attract the attention of researchers in this field. In this case, the multifractality means that the entire temporal interval of observing the process is divided into subintervals, so that the fractal dimension has a certain value for each of these intervals. In this model, the fractal properties of the curve v( t) allow describing the pandemic dynamics using no assumptions and hypotheses about the structure of the disease process. īased on the original Mandelbrot “Multifractal Walk down Wall Street” model, a model of multifractal dynamics (MFD) was developed and tested. Alternative descriptions of COVID-19 dynamics using fractional derivatives have been developed, e.g., in Refs. Within the approaches based on differential or integro-differential equations, the description of such jumps is not possible. Then this number fell to preceding values during a day. ![]() On this day, the daily incidence increased by 2.4 times as compared to the preceding time. For example, on Decema substantial jump of COVID-19 daily incidence in the world had been observed. The description of v( t) evolution is greatly complicated by the presence of jumps. Therefore, the applicability of differential and integro-differential equations to the pandemics description becomes problematic. From this fact it follows that the curve v( t) is not differentiable. As shown by calculations of the fractal dimensions D for various segments of v( t) during the time interval of interest for us, the values of D lie within an interval from 1.0708 to 1.4118. The function v( t) constructed basing on the world statistical data for COVID-19 pandemic determines a complex multifractal curve. In many papers, the function v( t) is determined by solving differential and integro-differential equations. This function is determined using relevant statistical data. The quantitative description of pandemics is based on the function v(t), which determines the time dependence of the number of daily diseases in the world associated with specific pandemics. The relevance of constructing and applying such models has been greatly enhanced by the COVID-19 pandemic. At present, one of the most demanded issues of mathematical modeling is developing adequate mathematical models of pandemics.
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