Waves of the dynamics of the rate of increase in the parameters of Covid-19 in Russia for 03/25/2020-12/31/2020 and the forecast of all cases until 08/31/2021

In applied mathematics and statistics, only linear equations are still used. The article proposes the sum of asymmetric wavelets with variable amplitudes and periods of oscillation. As a result, the behavior of any object or subject is given by the sum of vibrations. Using the identifi cation method based on statistical daily data on four indicators of the dynamics of the rate of Covid-19, quanta of the pandemic behavior in the territory of the Russian Federation from March 25 to December 31, 2020 were identifi ed. It is shown that the rates are infected, cured, died, and “all cases = infected + cured + died” in Russia got two superimposed bulges. Based on the computational capabilities of CurveExpert-1.40, 4-5 components were jointly identifi ed with an overall correlation coeffi cient above 0.86 for infected and over 0.99 for all cases. It has been proven that the spread of the virus has the form of a set of fi nite-dimensional wavelets with variable amplitudes and, as a rule, with a decreasing oscillation period. By modeling the standard deviation by the serial numbers of the wavelets, it was proved that the parameters of the Covid-19 pandemic have fractal distributions. For the velocity parameter “died”, the main bulge does not reach its maximum. And the second member of the trend peaked at 164 deaths on 06/18/2020, and it will leave the scene from 03/23/2021. The third member of the model, aimed at countering mortality, at the beginning of the time series on 03/25/2020 received a fl uctuation period of 355 days. By the date of December 31, 2020, the fl uctuation period became equal to 278 days. More often with constant half-periods of 3.5 and 16.1 days, fl uctuations occurred. In this case, the 70th term gives a constant oscillation period, even 1.88 days. The average relative modeling error in modulus is equal for speeds: 1) died 2.09; all cases 3.22; cured 17.17 and infected 29.91%. In this case, the range of error values changes in the following intervals: 1) died from -18.93 to 11.95%; all cases from -31.37 to 20.20%; cured from -248.8 to 396.0%; infected from -1934.0 to 779.7%. According to the distributions of the relative error after 1%, the following rating was obtained: 1) the correlation coeffi cient of 0.9807 for the speed died; 2) at 0.9768 the rate of all cases; 3) 0.8640 has been cured; 4) 0.8174 infected. The fractality coeffi cient is equal to the ratio of the standard deviations of the linear model to the last component: for infected 3572.76 / 310.97 = 11.5; cured 5.8; died 24.3 and all cases 9.6. Further, due to the high range of relative error, the rates of cured and infected are excluded from forecasting. The forecast for the rate of deaths was carried out until 02/14/2021. The right border at the forecast horizon was adopted due to the fact that negative values appear from 15.02.2021. For a longer time interval from 01.01.2021 to 31.08.2021 the model allows predicting the rate of change of all cases. To reduce the relative modeling error, it is recommended to re-identify the model of the dynamics of the parameters died and all cases every three weeks. The identifi cation method is applicable to any statistical series, and not only to dynamic ones. Research Article

The purpose of the article is to use the example of time series from March 25 to December 31, 2020 to conduct a wavelet analysis and identify the quanta of behavior of the ergatic system "Covid-19 + Russian health care system" based on the speed dynamics of four pandemic parameters (infected, cured, died, cases = infected + cured + died), then the formation of a forecast model and verifi cation according to data from January 1 to January 25, 2021.
Epidemiologists predicting the spread of Covid-19 must use climate modeling techniques to make predictions more reliable, say computer scientists, who have spent months auditing one of the most infl uential pandemic models [5]. In this article, we have applied the method of wavelet analysis from quantum meteorology [6] to identify the sum of solitary waves in the process of pandemic behavior.
Further David Adam notes the following [7]. Many of the disease models are unique to the individual academic groups that have developed them over the years. But the mathematical principles are similar. They are based on attempts to understand how people move between three main conditions and how quickly: people are either susceptible to the virus; got infected (I); and then either recover (R) or die. It is assumed that the R group is immune to the virus, so it can no longer transmit infection. People with natural immunity would also belong to this group [7].
For the fi rst three months of 2020, according to statistical data, a dynamic epidemiological model was proposed to recognize the initial stage of Covid-19 infection among countries in the modern world [8]. The article [9] also notes that the epidemiological curves show an exponential relationship of growth.
Such exponential models are applied not only in epidemiology. For example, to study the dynamics of the distribution of investment schemes in this population, an epidemiological model was created, in which the rate is included in the model as an exponentially distributed random variable that gives distribution of losses [10].
Based on the concept of vibrational adaptation in nature and the use of trigonometric sums [11] of the cosine, a method of sequential identifi cation [1] of the general formula for an asymmetric wavelet signal was proposed to obtain a set in the form of a sum of stable wave patterns from tables of quantitative data, especially from tables of statistical data [2] in time.
Then each component of the trigonometric sum is a wavelet signal, and each wavelet in the CurveExpert-1.40 software environment is constructed from the so-called Hilbert bricks [2]. As a result, from such general invariants in the form of wavelets, a polynomial algebraic equation is constructed, sometimes containing more than 200 wavelets. Thus, we have proved the possibility of forming one single algebraic equation according to Descartes directly without solving differential and integral equations.
Thus, we do not address the detailed epidemiological reasons for the formation of statistical time series. For wavelet analysis, the data is accepted as it is, that is, without any consideration of measurement errors. Moreover, no grouping of data is allowed. The quality factor of the initial data is checked as follows: the more wavelets the dynamic series will be decomposed with the minimum value of the maximum relative error of the residuals (the difference between the fact of measurements and the calculation by formulas) after the last fl uctuation, the higher the quality of the initial tabular data. With an increase in the length of the dynamic series, the set of wavelets also increases, but there is a possibility of lengthening the forecast horizon for the future.
The proposed identifi cation method made it possible, for example, to carry out a wavelet analysis up to the measurement error of the maximum temperature in Central England [12].
Each component of the trigonometric sum is a quantized signal about the behavior of the object of study, for example, in the article [6] in quantum meteorology about the behavior of the temperature of the surface air layer. Table 1 shows a fragment of daily data from the site https:// стопкоронавирус.рф/information/. The origin of coordinates on the abscissa axis is taken for the date 03/25/2020, and for four time series, the end is December 31, 2020. In table 1, the following conventions are adopted, person/ day: S I -Speed infected; S cu -Speed cured; S D -Speed died; S ca -

Initial data
The speed or pace of daily occurrences.

Identifi cation method
The process of identifying stable patterns is presented as a sequential process of solving the problem of statistical where y -the index (dependent factor), i-the number  The signal can always be generated, but its reception by a person is not required. For example, the average annual temperature in Central England has been known since 1659, but until now, as a bunch of signals in the form of asymmetric bursts, this dynamic series for 360 years has not been deciphered. For the maximum temperature [3], the same heuristic is required.

Quantum of behavior as a signal
Any physical process or part of it can be a signal. It turns out that the change in the set of unknown signals has long been known by measurement methods, for example, in the form of temperature series, and at many meteorological stations.
However, there are still no statistical models of the dynamics of weather and climatic parameters at each meteorological station.    Only the fi rst two types of wavelets are used in the identifi cation process.

Comparison of quanta of pandemic and temperature behavior
In this article, we confi rm the conclusion of David Adam that climate modeling methods should be used to predict the spread in time of Covid-19 [5]. However, the wavelet analysis of behavior invariants from quantum meteorology [6] showed that almost half of the components in the general model (1) contain infi nite-dimensional oscillations.
At the same time, all four parameters of Covid-19 in Russia gave the sums of trigonometric equations, all of which are fi nite-dimensional equations. Hence, the fi rst conclusion suggests itself that the dynamics of the pandemic are not affected by space and climatic cycles. The second fi nding is that the parameters of Covid-19 appear to be seasonal (monthly cycles).
Apparently, the infl uence of cycles with a frequency of several years is possible, for example, the infl uence of a twoyear cycle of plant productivity. To prove such cycles, data on the dynamic series of the pandemic over several years are needed. So far, we can only conclude that Covid-19 is a phenomenon accompanying the behavior of people in various countries.

Non-linear trend
Such a trend is formed from equation (1) when the oscillation period tends to infi nity, that is, under the condition . A special case is the condition under which the measurement period is many times less than the oscillation period of the studied process.
For the climatic factor of air temperature, due to the infl uence of space and climatic cycles in the general trigonometric formula, for example, according to the average annual temperature of Central England since 1659 (the measurement cycle is more than 360 years), 188 wavelets were obtained, of which a considerable part belong to infi nitedimensional vibrations. For such long series of dynamics, the main trend is Mandelbrot's law.
Here, for various natural and natural-anthropogenic processes, the fi rst term of the trend is the modifi ed Mandelbrot's law, and the second term is the biotechnical law [1].
For the dynamics of the Covid-19 parameters over 282 days, when only fi nite-dimensional wavelets are present in the general algebraic formula, a trend of the form was obtained

Identifi cation of the trend
To identify formulas (2)

Wave equation identifi cation (1)
Students begin to master the method of identifying simple patterns such as Mandelbrot's law from the second year of the bachelor's degree, and some students identify the parameters of model (1) in the master's degree.
To identify formula (1), we have published two teaching guides for students. There are only scientifi c articles on modeling the time series of air temperature and the content of pollutants, and even more so the parameters of Covid-19.
When identifying a set of wavelets in a temperature series, we found that the length of the data series signifi cantly affects the transition from the trend to the fi rst oscillation. When the duration of the temperature series exceeds 175 years, a complex relationship arises between the fi rst components of the general formula (1).
The technology for identifying formula (1) according to the principle from simple to complex includes the following basic procedures: 1) For an infi nite-dimensional wavelet, the simplest wave The identifi cation process in CurveExpert-1.40 environment is laborious. Therefore, according to our scenarios, we need to create a specialized software environment for partial automation of the identifi cation process. This will make it possible to search for wavelets with a measure of adequacy in terms of the correlation coeffi cient much less than 0.05.

Velocity dynamics models
After structural-parametric identifi cation of formulas (3) and (1), the parameters of the obtained models of the velocity dynamics for the four parameters of Covid-19 in the Russian Federation are given in Table 2.  Table   2, it turns out that the components No.

Speed infected S I
For speed infected, the fi rst three terms form a trend The third term of the trend, due to the negative sign, displays the inverse bulge of the infected velocity dynamics graph. As  With a constant period, as, for example, in quanta of behavior # 6 and # 7, 28 wavelets were identifi ed, which is 100 28/57 49.12% of their total number.
The rest of the characteristic terms from Table 2  According to the computing capabilities of the CurveExpert-1.40 software environment, fi ve components were jointly identifi ed. Each wavelet was further revealed separately from the fi rst fi ve members. As can be seen from the graphs in Figure 1, the fi nite-dimensional fourth and fi fth terms affect the future, the fi fth oscillation gives a small dying infl uence, and the sixth component was in the past time in the spring of 2020. Therefore, such wavelets are not involved in the compilation of the predictive model.
In Figure 2

Speed cured S Cu
The composite trend receives the sum of two biotechnical laws ( Table 2)

Speed of all cases S Ca
Together with the permanent member, only eight components were received. The graphs of the components and the distribution of residues after the eighth component are shown in Figure 5.
From the residuals in Figures 4,5, it can be seen that the spread of values has increased since autumn 2020, that is, the pandemic spreads seasonally (two bulges in the location of points in the residuals after the eighth component in Figure   5). As a result, the data on Covid-19 gets the character of a scedastic distribution, rather than a uniform one.

Quantization of behavior
In nature, there are quanta of state (structure) and quanta of behavior (functioning). Quantums of state are known in the quantum physics of elementary particles. However, structure quanta also exist for macro-objects, for example, quantization of a state occurs in crystals and rock crystalline formations. Therefore, quantum physics of state exists for macro-objects as well.
Behavioral quanta are inherent in living matter (a term according to V.I.Vernadsky) and functioning quanta refer to inert matter repeatedly processed in the Universe by living matter, including technical means created by living organisms, including humans.

Modeling errors
The graphs of the relative error for all four time series without their truncation are shown in Figure 6. The graph of the relative error in infected shows the presence of sharply deviating points, which indicates negligence in recording some values of the pandemic parameter.
The other three parameters received high values of relative error at the beginning of the series, which indicates instability in the behavior of the Russian healthcare system at the beginning of the response to the Covid-19 pandemic.
For modeling, you need to cut off the chaotic origin for all four parameters. It turned out that for the Russian Federation, stable trends in the dynamics of Covid-19 parameters were observed from 04/07/2020 ( Table 3).
The relative error Δ was obtained by dividing the residuals (absolute error ) after the last component of the model from

Gauss's law in error distributions
The graphs are given in Figure 7. For infected, 31 measurement points were excluded from the dynamic range.
Then the statistical representativeness of the series is 100

Fractal distribution of wavelets
The arithmetic mean for the rate of the pandemic is not modeled, therefore the zero rank was adopted for the linear model of the type y a bx   (Table 4)  the graphs. In this case, the constant term is not taken into account here, since it is not a Mandelbrot fractal. The largest series of 70 members was formed for the wavelet analysis of the velocity parameter died.
The fractality coeffi cient was adopted by Mandelbrot equal to 2. However, in nature, apparently, such an ideal multiplicity does not exist. As a rule, all relationships between the levels of fractals according to Table 4 are non-multiple. As can be seen from the data in Table 4

Drawing up a forecast model
The wavelet forecasting method consists in selecting from the entire set of model members (1) those components that affect the future. Of the 70 members of the model (1), the speed died, 35 components affect the future, that is, 50% of their total number. For speed, all cases of eight terms, the second term of the trend has no effect on the future, so only seven components are taken into account in the forecast model.

Calculations of the parameters of the pandemic for the future
The parameters of the model with fi ve signifi cant digits given in Table 2

The speed died for 07/04/2020-14/02/2021
The forecast was carried out before 02.14.2021, that is, for 44 days from 01.01.2021, which is a share of 44/269 = 0.1636 of the length of the forecast base, that is, the forecast horizon will be equal to more than one sixth of the forecast base.
Such a right border of the forecast horizon was adopted due to the fact that from 15.02.2021 negative values of the velocity appear (and continue rhythmically after several days), which is physically unacceptable (those who died from Covid-19 cannot be resurrected).
Charts of models for four and 70th components are shown in Figure 9. The vertical line separates the base and the forecast horizon.
Initially, according to the fi rst four fl uctuations from Table   2  Comparison of the graphs shows their high similarity, so it turns out that it is possible to quickly identify the number of deaths by only four components.
Then the speed died allows to predict only a month and a half.
At the same time, the daily data on deaths from Covid-19 in Russia is very accurate: "The daily summary of the federal headquarters on morbidity, discharge and mortality for the previous day. It is used for prompt management decisions to assess the load on the health care system. Only promptly confi rmed cases with an unambiguous diagnosis of death from coronavirus are taken into account "(according to data from the Russian website https://стопкоронавирус.рф/information/).

The forecast of the speed of all cases until 08/31/2021
For a longer period from 01.01.2021 to 31.08.2021, that is, for eight months, the rate parameter of all cases can be predicted ( Figure 10). In this case, the trigonometric formula (1) contains only seven terms, which signifi cantly reduces the complexity of identifying asymmetric wave equations. According to Table  3, the average relative error in absolute value is only 3.22%. All cases have low stochasticity in comparison with died, which can be explained by the different vector orientation of the meaningful meaning of all four parameters taken into account.
Forecasts always depend on the measures taken by the system of government of a particular country. It may turn out that a seasonal bulge could appear in the spring of 2021. Therefore, we recommend a monthly iterative mode of identifi cation by statistical series died and all cases with additions to the actual data for the month.

Verifi cation of the parameters of the pandemic died and all cases
While the article was being prepared, new factual data appeared for 25 days of January 2021 (Table 5). This data turned out to be suffi cient to verify the forecasts ( Figure 11). In To the smallest relative error of 1.94%, the speed died before 15.01, the actual values are less than the calculated ones, and after 01/15/2021, and on the contrary, the actual values are higher than the calculated values. As a result, the 70-component model gives an outstripping decline in the number of deaths: Covid-19 in Russia is strongly resisting. Due to the permissible interval of 30%, it is impossible to predict the number of deaths beyond January 25, 2021. Then the periodicity of iterative modeling (with a shift in the beginning of the series from 03/05/2020, when in Russia all the velocity parameters did not begin to give sharp bursts) is three weeks.
For the speed of all cases, the picture is better, since the actual values decrease faster than the calculated ones. To reduce the relative modeling error, it is also recommended to carry out repeated identifi cations (1) every three weeks.
For four parameters of the daily dynamics of the Covid-19 pandemic in the Russian Federation, polynomial trigonometric equations were identifi ed, including from eight (for the parameter all cases) to 70 (for the parameter died) components. It has been proven that the dynamics of a pandemic can be decomposed into individual asymmetric wavelets, which are quanta of behavior.
The identifi cation method based on statistical daily data on four indicators of Covid-19 speed dynamics revealed quanta of pandemic behavior and responses from the healthcare system in the Russian Federation from March 25 to December 31, 2020.
The general equation (1) of an asymmetric wavelet is presented in the form of a solitary wave equation. It is shown that the speed is infected, cured, died and "all cases = infected + cured + died" in Russia got two superimposed bulges. At the same time, the large bulge will continue until the end of August 2021 for the parameter all cases. Based on the computational capabilities of the CurveExpert-1.40 software environment, a joint identifi cation of 4-5 components was carried out with a general correlation coeffi cient above 0.86 for infected and more than 0.99 for all cases.
It has been proven that the spread of the virus has the form of a set of fi nite-dimensional wavelets with variable amplitudes and, as a rule, with a decreasing oscillation period. This is how the dynamics of the pandemic differs from the behavior of natural and natural-anthropogenic objects, which have many and infi nite-dimensional wavelets with an amplitude in the form of Laplace's law (in mathematics), Mandelbrot (in physics), Zipf-Pearl (in biology) Pareto (in econometrics).
By modeling the standard deviation by the ordinal number of the wavelet, it has been proved that the parameters of the Covid-19 pandemic also have fractal distributions. Due to fi nite-dimensional wavelets, the dynamics of a pandemic depends on the behavior of each country's epidemiological system. The Russian medical system has proved to be at the proper height to counter the speed of the pandemic since April 7, 2020.
According to the computational capabilities of CurveExpert-1.40, it was found that the parameter of all cases receives the greatest adequacy with a correlation coeffi cient of 0.9971, in the second place with a correlation coeffi cient of 0.9878 is the speed died, in the third place -the speed is cured and in the fourth place -the speed is infected. Of greatest interest is the speed died.
For the velocity parameter, the fi rst term of the trend (main bulge) has died and does not reach its maximum. And the second component of the trend reached a maximum of 164 deaths on 06/18/2020, and it will disappear from the scene of the speed dynamics died approximately from 03/23/2021. The third component, aimed at countering mortality, at the beginning of the series on March 25, 2020, received a fl uctuation period of 355 days. By the date of December 31, 2020, the fl uctuation period became equal to 278 days. Then, the oscillatory adaptation of physicians in Russia to the fi ght against mortality from Covid-19 increased only 1.28 times.   More often, with constant half-periods of 3.5 and 16.1 days, fl uctuations of the dead occurred. In this case, the 70th term gives a constant oscillation period, even 1.88 days.
The average relative simulation error in modulus is equal for the speed of the parameters: 1) died -2.09; all cases -3.22; cured -17.17 and infected 29.91%. In this case, the range of error values among 269 values varies in the following intervals: 1) died from -18.93 to 11.95%; all cases from -31.37 to 20.20%; cured from -248.8 to 396.0%; infected from -1934.0 to 779.7%. According to the distributions of the relative error at intervals of 1%, the following rating was obtained: 1) the correlation coeffi cient of 0.9807 for the rate of the deceased; 2) at 0.9768 for the speed of all cases; 3) 0.8640 for the cured speed parameter; 4) 0.8174 for the speed of the infected.
The value of the standard deviation is taken as an indicator of the fractal distribution of quanta of behavior. In nature, apparently, there is no such ideal multiplicity of 2, as in Mandelbrot. As a rule, all relationships between fractal levels are non-multiple. The linear model gives the standard deviation of speeds, person/day: infected 3572.76; cured 4962.32; died 94.827 and all cases 5002.48.
Then the coeffi cient of total fractality will be equal to the ratio of the standard deviation of the linear model to the standard deviation of the last component, that is, to the border of quantum uncertainty. This general coeffi cient of fractality will be equal: for the parameter infected 3572.76 / 310.97 = 11.5; cured 5.8; died 24.3 and all cases 9.6.
Due to the high range of the relative error, the rates of cured and infected were excluded from the forecast. The range of error values among 269 values changes: 1) died from -18.93 to 11.95%; all cases from -31.37 to 20.20%.
The forecast for the rate of deaths was carried out until 02/14/2021, that is, for 44 days from 01/01/2021. The right border of the forecast horizon was adopted due to the fact that