key: cord-0982823-6vr00nyt
authors: Cherednik, I.
title: Modeling the spread of Covid-19 under active management
date: 2020-11-23
journal: nan
DOI: 10.1101/2020.11.20.20235903
sha: b8781b15b583c32ab4f9033f1095da301d3ce72a
doc_id: 982823
cord_uid: 6vr00nyt

Classical approaches to modeling the spread of epidemics are based on two assumptions: the exponential growth of the total number of infections and the saturation due to the herd immunity. With Covid-19, the growth is essentially power-type, especially during the middle stages, and the saturation is currently mostly due to the protective measures. Focusing on these features and the role of epidemic management, we obtain differential equations for the total number of detected cases of Covid-19, which describe the actual curves in many countries almost with the accuracy of physics laws. The two-phase solution we propose works very well almost for the whole periods of the spread practically in all countries we analyzed that reached the saturation during the first waves. Bessel functions play the key role in our approach. Due to a very small number of parameters, namely, the initial transmission rate and the intensity of the hard and soft measures, we obtain a convincing explanation of the surprising uniformity of the curves of the total numbers of detected infections in many different areas. This theory can serve as a tool for forecasting the epidemic spread and evaluating the efficiency of the protective measures, which is very much needed for epidemics. As its practical application, the computer programs aimed at providing projections for late stages of Covid-19 proved to be remarkably stable in many countries, including Western Europe, the USA and some in Asia.

Introduction. The evidence is strong that the exponential growth of the total number of detected infections of denoted by u(t) in this report, can be detected only during very short periods in any countries, especially during the middle stages. The corresponding curves are in fact of power type: u(t) ∼ Ct c in terms of the time t from the beginning of the spread and for some C, c. Here c, C depend on time; the exponent c approaches 1 near the turning point of the spread and C becomes small near the saturation. The saturation is not the end of the epidemic; generally, it is followed by a period of modest linear growth of the total number of infections.

The usage of the basic reproduction number R 0 is questionable for Covid-19 and other epidemics of power growth: u(t) is not like ∼ CR t 0 beyond very short periods. It is defined as the initial average number of people infected by one person who contracted the virus and it is commonly used; see [CJLP, Co, CD, DHB, He, HL] .

Even R = 1.1 or so would quickly begin to contradict the actual growth of infections of Covid-19, but significantly higher/lower R are constantly reported. For instance, Robert Kox Institute periodically provides the R-numbers for Germany sometimes reaching 0.7 and 2. This can be used qualitatively to see the current trend of Covid-19 (it is used!), but not for forecasting. The formula const R t for the total number of infections will stop working very quickly.

We model the power-type growth of the total number of detected positive cases with variable exponent c using Bessel functions. Our theory was posted in the middle of April, when the saturation of the spread was observed only in several countries; they were mostly in phase one. Later, our equations proved to be fully applicable to phase two in many countries, almost any in Western Europe. If Ct c represents the initial growth, then it becomes of type ∼ t c/2 cos(d log(t)) for some d at phase 2. The passage from the Bessel-type curves for phase 1 to those in phase 2 can be clearly seen in many countries; it is a switch to less aggressive management due to relatively low numbers of daily infections. Though the Bessel-type formulas worked well almost till the saturation in some countries too, for instance, in the first waves in Austria and Israel. This was when the hard measures were continued almost till the saturation.

The spread of Covid-19 in the USA was mathematically quite a challenge for us; the results of our efforts are systematically analyzed in [Ch1] . The first wave in the USA went through several stages, more than with any countries we considered. Our understanding is that it was so mostly because the hard protective measures were relaxed in the USA on the first signs of improvements, in contrast to Europe and several countries in Asia. It was similar in UK, but the latter eventually reached phase 2 and the saturation of the first wave.

The costs and consequences of hard measures, especially lockdowns, are huge for any country. Moreover, the saturation due to the hard measures is of unstable nature; the recurrence of the epidemic is quite likely if they are reduced or abandoned. Our theory generally provides the way to control the efficiency of protective measures, but this is quite a challenge even if advanced mathematical means are used.

There is increasing number of works where the power growth of the total number of infections is considered for modeling Covid-19. Let us mention at least [Ch1, MBS, MH, TKH] .

To begin with, in [CD] (well before , an ambiguity with the definition and practical calculation of R 0 is mentioned: "It is reassuring to know, however, that the sign of R 0 − 1 is independent of the decomposition used and that the prediction of exponential growth or decay is therefore correctly made by any of the counting schemes." Mostly assuming the exponential growth in this paper, the "resonances", when R 0 is not a strictly dominant eigenvalue, are considered at the end, which result in the power growth of the number of infections. Let us quote: "As far as we know, little can be said in general about the exceptional case that R 0 is not strictly dominant".

In [MBS] , the authors comment on the power growth: "the nature is full of surprises". In [TKH] : "this new contamination regime is hard to explain by traditional models". In our one: "power law of epidemics must be the starting point of any analysis if we want our mathematical models to be up to date". See also article [Ray] and works mentioned there concerning a potential usage of small-world interaction network, where individuals are assumed to contact (mostly) local neighbors and have occasional long-range connections.

Paper [BBT] and some prior works argue that the levels of herd immunity sufficient to impact the spread of Covid-19 can be significantly lower than the "classical" 60% or so: as low as 40% in some areas due to the population heterogeneity. From this viewpoint we make the next step: our main assumption is that local herd immunity is present almost from the very beginning of epidemics and results in the power growth of the total number of infections. This is actually similar to the small-world approach.

The main problem with modeling is actually not the power law of epidemics itself, which is a starting point for us. This law alone is insufficient for forecasting. The exponent c and the corresponding scaling coefficient C heavily depend on the time passed from the beginning of the spread of the infection. An exact mathematical model of this time . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101 https://doi.org/10. /2020 dependence is necessary, which was proposed in [Ch1] using the Bessel functions.

The approach of [MBS] to the power growth is of experimental kind. Since the corresponding exponents depend very much on the considered periods, the data in Figure 1 in this paper and in similar papers mostly show that the growth is no greater than polynomial. Our c are different from their exponents. Mathematically, the authors suggest the usage of the SEIR model (Susceptible-Exposed-Infectious-Recovered), which does not result in the power growth, though "small world" is mentioned there as a possibility.

Paper [TKH] is based on the Poissonian small-world network. This approach results in the linear growth (c ≃ 1) of the number of infections. The linear growth is clearly present at some stages, but it is far from linear during the first phase and closer to the saturation everywhere. For instance, const t c for c = 3.75 models very well the total number of detected cases in India for about 6 months (!) during the period 3/20-10/7; see Fig. 1 .

In this graph, the main parameters were determined on 08/03. This forecast was posted on 10/07. It was till 11/06 (the maximum of the u-function), and it matched ideally the actual curve of detected cases. As always, a linear growth is expected after the top of the Bessel-type curve u(t), which can be seen in the graph. Generally, it can described by the formula for the second phase.

Anyway, the explanation of the linear growth and the saturation in [TKH] , is very different from what we proposed in [Ch1] .

Saturation due to hard measures. For us, the saturation, including some modest linear growth of the total number of infection after (if) it is reached, is a result of active protective measures, mostly hard ones. They are imposed by authorities in charge, but self-restrictions are equally important. The key is detection-isolation-tracing, which includes closing the places where the spread of infection is the most likely. The societal cost of hard measures is huge, but they proved to reduced the spread efficiently.

It is not disputed that the saturation of the first waves of Covid-19 in many countries (almost all Western Europe) was not due to herd immunity. The latter probably requires about 40%-60% of all susceptible population to be infected and recovered [BBT] , which was far from . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10. 1101 these levels during the first waves. Thus, the saturation mechanisms of SIR-type models are not applicable here, at least for the first waves.

The timing and the intensity of the second waves clearly confirm the validity of our approach to modeling, based on the prime role of protective measures, mostly the hard ones. Recurrence of epidemics is quite frequent; see e.g. [HL] . However the second waves of Covid-19 begin unusually quickly, sometimes even on the top of the unfinished first waves, as in the USA. The relaxation of hard measures closer to the end of the first waves seems the only logical explanation for this. The summer vacations (and closed schools) in Western Europe actually worked in the same way as the hard measures. At the end of August, the second waves began almost everywhere in Europe, and the number of new detected infections began to grow (again) in the USA from the middle of September.

Mathematically, our exponent "c", which we call the initial transmission rate, appeared comparable for the first and the second waves in many countries, though with a clear tendency to increase. This parameter is one of the main on our theory; it reflects the virus strength and the "initial" number of contacts in the areas. So by reducing the protective measures, especially the hard ones, c is essentially back to the beginning of the epidemic. By contrast, the second key parameter of our theory, the intensity a of protective measures, dropped very significantly for the second waves.

Qualitatively the duration of the wave is 1/ √ a; quantitatively, Bessel functions must be used here. So, mathematically, we essentially repeat the first waves, but now with significantly lower levels of hard measures, potentially longer periods of intensive infections, and accordingly higher magnitudes of the total numbers of infections.

Power Law of Epidemics. With such complex processes as epidemics, there can be of course multiple factors contributing to the power growth, biological ones included [CLL] . The "justification" from [Ch1] goes as follows. We first assume that infected people mostly transmit the disease to their (susceptible) neighbors, and that the population is distributed uniformly. The second assumption is that the wave of the infections expands linearly. The third one, local herd immunity, is that people "inside the infection zone" do not transmit the disease because they are surrounded by those already infected or recovered, i.e. the border of this zone mostly contributes to the spread . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10. 1101 of this disease. This readily gives that u(t) ∼ t 2 . This is indeed the lowest c-coefficient we observed; it was c = 2.2 for the USA.

Then, people from the infected zone do shopping, travel, visit friends. Higher dimensions are needed to incorporate such connections provided that the geometric distances between points representing people are essentially the numbers of links between them, i.e. reflect the intensity of the contacts.

So we embed the graph of contacts into some R N . The other assumptions remain the same: the uniform distribution of the points in R N representing people and the linear spread of the disease in R N . Then u(t) ∼ Ct c , where c is the "dimension" of the image of this graph, a number from 2 to N .

Next, we represent this u(t) as a solution of the differential equation du(t)/dt = cu(t)/t. This is standard when we need to add "external forces". The exponential growth is unsustainable, but the power growth is unsustainable long term too; so we "correct" it as follows.

The main ODE. Combining the initial power growth of the total number of detected infections u(t) with the impact of protective measures we obtain the following two systems of differential equations:

Here t is the time from the beginning of the massive growth, which is not always the very beginning of the spread of Covid-19, but is generally sufficiently close to it. System (1) describes the impact of hard measures under the most aggressive response. The second describes the impact of the soft measures: wearing the protective masks and social distancing are the key. We called these modes (A) and (B) in [Ch1, Ch3] .

When a = 0, d = 0, we obtain the power growth u(t) ∼ Ct c ; so c can be measured experimentally during the initial stages of Covid-19 and is supposed to be the same for (1) and (2). Mostly it is in the range 2.2 ≤ c ≤ 2.8, but reached c = 4.5, 5.2 in Brazil and India.

There is a variant of these systems, when the second equation in (1) is replaced by that from (2), called the transitional (AB)-mode in [Ch1] . It modeled reasonably the spread in the USA, UK, and Brazil, . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235903 doi: medRxiv preprint but the usage of (1) and (2) appeared mostly sufficient for our two-phase solution in many countries.

The function p(t) is the key here. Up to proportionality, p(t) = ( the total number of prevented infections from 0 to t) for 1, where we assume that each infected individual isolated at t • will not transmit the disease to const · (t − t • ) people. For 2, p(t) = const( the number of infected people who began wearing the masks before t), and similarly for the social distancing.

Related processes. Both systems are actually from [Ch2] , where they were used to describe the dynamic of the (relative) stock prices p(t) under news driven momentum trading. The function u(t) there was the news propagation triggered by some event. It is of power growth in terms of time t passed from the event, but the exponent c is generally significantly smaller than 1, especially when the "taken positions" are short term.

The arguments there were from behavioral finance. This is actually related; the behavioral aspects of epidemics are of obvious importance [St] . However news fades, and this happens quickly; this is very different for the spread of epidemics. System 1 described in [Ch2] profit taking in stock markets; the second one modeled the "usual" newsdriven investing.

As a matter of fact, these two systems are of very general nature. For instance, they are supposed to occur in any momentum risk taking. This concept, MRT for short, is from [Ch2] ; it is somewhat similar to Kahneman's "thinking-fast" [Ka] . Managing epidemics on the basis of the current data is very much momentum. As in stock markets, some change of data can be random or a new tendency, so it is risky not react promptly.

It was expected in [Ch2, Ch1] , though without biological evidence, that both systems of equations may describe real neural processes in our brain, where u(t) is the number of neurons involved in the analysis of a particular event at the moment t, counted from the event, and p(t) is the expected importance of this event vs. other ones and the corresponding expected brain resources needed for its analysis. I.e. p(t) is basically the expected allocation of resources (which are very limited in our brain). We do not know much about the ways our brain work, but the confirmation of the power laws and related saturations are solid in the stock markets and, as we demonstrate, in epidemics.

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The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10. 1101 We note that a significant part of [Ch2] is devoted to the discretization. Decision-making always requires some action potentials, i.e. it is discrete by its nature. With epidemics, this seems not really necessary; the usage of ODE worked very well so far, though potentially the discretization can become important for our approach too.

Two-phase solution. The solutions of (1) and (2) we need are (1, t) )). m+α+1) are Bessel functions of the first kind; [Wa] (Ch.3, S 3.1). The function u B is for d = √ b−c 2 /4 > 0; it will be used to model later stages of Covid-19.

A linear combination of u 1,2 for the first phase, till a trend toward the saturation, and then the usage of u B for the second (last) phase is our two-phase solution, which proved to be quite exact for modeling the curves of total numbers of detected infections of Covid-19. For t ≈ 0:

The second fundamental solution of system 2 is with sin instead of cos. When the soft measures are modest, mathematically when D = c 2 /2 − b > 0, the leading fundamental solution is t r with r = c/2 + √ D, i.e. it diminishes from r = c in the beginning of the spread to r = c/2 (and then remains unchanged). This is of importance, but we will not touch this stage in this report.

India: 3/20-10/07-11/20. The starting number of detected cases was 191, which was subtracted. The power function 0.0125(t + 0.07) 3.65 is a very good approximation for a very long interval.

In this graph, the main parameters were determined on 08/03. The forecast posted on 10/07 was that the curve of the total number of detected infections would reach its technical saturation on November 6 with the number 8.25M of the cases. It matched the actual number of cases ideally. As always, a linear growth is expected after the top of the Bessel-type curve u(t), which can be seen in the graph. Generally, it can described by the formula for the second phase, and during this period our programs can be used.

Here y =cases/10K; similarly, y is the total number of cases divided by proper powers of 10 in the other charts we will consider. Say, divided . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. by 100K for the USA. The x-axis is always time in days from the beginning of the curve. The red-blue-black dots give the corresponding actual total numbers of the detected cases. The u-function for India is:

where c = 5.75, a = 0.035. There was no phase 2 till the middle of November there, so we have only u(t).

Italy: 2/22-5/22. Figure 2 . The starting point was 2/22, when the total number of infections was 17; we subtract this initial value when calculating our dots. One has: c = 2.6, u 1,2 (t) = 0.8 t (c+1)/2 J ± c−1 2 ( √ at), u(t) = u 1 (t) − u 2 (t), and u B (t) = 2.85 t c/2 cos(d log(M ax(1, t))), a = 0.2, d = 0.5.

We use here the solutions u 1,2 (t) of (1). Figure 3 . We began with the initial number of total infections 684 (subtracted). This was approximately the moment when a systematic management began. One has:

u B (t) = 2.95 t c/2 cos(d log (M ax(1, t) )), a = 0.35, d = 0.56.

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The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235903 doi: medRxiv preprint 

too smooth, which is not unusual, it is managed well by our 2-phase solution :

u B (t) = 3.15 t c/2 cos(d log (M ax(1, t) )), a = 0.3, d = 0.6.

The Netherlands: 03/13-5/22. See Figure 5 . The number of the total case was 383 on 3/13, the beginning of the intensive spread from our perspective; u 1 appeared sufficient: (1, t) )), d = 0.54.

UK: 03/16-06/13. This country was a challenge for us, though it "eventually" managed to reach phase 2. The u-function here is with the same a, c as for the Netherlands. Actually the red dots are modeled better with the transitional (AB)-mode. However, we prefer to stick to the "original" u(t) determined for the period till April 15. The twophase solution is a combination of two phases separated by a linear period, about 10 days. See Figure 6 . The parameter d = 0.465 is . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101 https://doi.org/10. /2020 (1, t) )), d = 0.465.

The USA: 06/16 -9/12. The two-phase solution worked well at least till the middle of September (2020) for the second wave in the USA. The accuracy is comparable with what we had above for the first waves in Japan, Italy, Germany, the Netherlands and UK. Upon subtracting 2.1M , the second phase matched well the following functions: (1, t) )), a = 0.06, d = 0.435.

We note that the initial transmission rate was c = 2.2 for the USA during the first wave. The parameters c, C and 0.6 in the first formula were determined for the period marked by red dots; the black dots form a control period. The projected saturation for u B is given by the formula t end = exp

) . Numerically, t end = 17.8463, which is . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 178 days from 06/16: December 11, 2020. See Figure 7 . Here the red dots were used to determine c, a.

Auto-forecasting (USA). We will provide the automated forecast for 50 states was based on the period 03/17-05/27; the data were from . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10. 1101 https://github.com/nytimes/covid-19-data . Every state was processed individually with the interaction; see [Ch1] . The interaction is of independent interest: we allow the curves for individual states to become decreasing as far as the total sum increases, which is motivated by physics.

Our program focuses on the last 20 days; however, the match with the total number of detected infections appeared perfect almost from 03/17 and remained so for further auto-forecasts for a sufficiently long period; see Figure 8 . The sum of the a ically found curves , where the interaction s allowed. The red dots: the infections .

The B-type curves were found on the basis of the ed well . This means that the forecasting remained very stable for 2 months. Such a high level of stability is actually rare in any forecasting, which made the chances good to reach the saturation around 9/19, the projection based on Fig. 8 . The saturation does not mean the end of the spread; normally, it is followed by a period of modest linear growth of the total number of infections. No country is isolated and new clusters of infection are always possible.

However, the hard measures were significantly reduced in the USA at the end of May practically in all 50 states. As a result, the number of states that reached phase 2 dropped from about 22 at 5/27 to 8 at 7/12. Then, in the second half of June, the USA entered the second wave.

Auto-forecasting (Europe). The situation was quite stable in Europe in summer. We provide a sample forecast our automated system . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101 https://doi.org/10. /2020 produced for Western Europe till the end of July, to be exact, for the following 45 countries: Albania, Andorra, Austria, Belgium, Bosnia and Herzegovina, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Faeroe Islands, Finland, France, Germany, Gibraltar, Greece, Guernsey, Hungary, Iceland, Ireland, Isle of Man, Italy, Jersey, Kosovo, Latvia, Liechtenstein, Lithuania, Luxembourg, Macedonia, Malta, Monaco, Montenegro, Netherlands, Norway, Poland, Portugal, Romania, San Marino, Serbia, Slovakia, Slovenia, Sweden, Switzerland, Vatican. See Figure 9 .

Here and above the main source of Covid-19 data we used was: https://ourworldindata.org/coronavirus. The "curve average" is the maximum and the corresponding value of the average of 9 last curves u B (t) for the sums of the curves of total cases and the forecast curves over the countries above. I.e. it is the moving average. The 9-day average is the simple average of the corresponding maxima.

As of July 8, the following countries had clear second phases: Albania, Bosnia and Herzegovina, Bulgaria, Croatia, Czech Republic, Greece, Kosovo, Luxembourg, Macedonia, Montenegro, Romania, Serbia, Slovakia, Slovenia. Sweden, Poland, Portugal and some other countries did not reach phase 2 at that time. Nevertheless, the forecasts were sufficiently stable. Such stability changed this fall due to the end of the vacation periods and the beginning of the school year.

We use is a brand new approach to modeling the total number of infections during epidemics. The key hypotheses are their power growth, which has solid confirmations, and understanding that the saturation of the spread (so far) was mostly due to the protective measures. The later is not unique for Covid-19, but the range and intensity of the measures employed reached unprecedented levels with Covid-19, at least in the recent history of epidemics. The model we propose is based on Bessel functions, and their asymptotic periodicity, one of the deepest results in their theory. It is very different from the classical approaches based of SID, SIR, SIER models and their variants. We use very limited number of parameters (actually 3 for our twophase model), which is much more rigid than in any other approaches. Nevertheless, the match we reach is practically for the whole periods of the first waves of Covid-19 in many countries; it seems almost ideal for such stochastic processes as epidemics. Since our theory was created in the middle of April, we had a unique opportunity to determine our parameters during relatively early stages of Covid-19 and then to . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) preprint

The copyright holder for this this version posted November 23, 2020. Figure 9 . A sample auto-forecast for Europe as of 7/14.

test them extensively for sufficiently long control periods. This is quite similar to routine testing the quality of the models used for forecasting share-prices in stock markets, where no approach can be accepted without real-time runs and carefully crafted historic experiments that exclude any "usage of future". Testing our system during the control periods, including automated forecasting programs we developed, is an important part of papers [Ch1, Ch3] . We acknowledge support by NSF grant DMS-1901796 and the Simons Foundation.

We address the following features of Covid-19. The growth of the curves of total numbers of detected infections is of powertype, where the initial exponent diminishes over time, and the saturation is currently mostly because of the protective measures, not due to the herd immunity. Our theory seems the first when the active management is considered the leading factor. It results in differential equations depending only on the initial transmission rate and the intensities of the hard and soft measures. The actual curves in many countries are described uniformly and with surprisingly high accuracy. The saturation due to the active management is of unstable nature;

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The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10. 1101 its modeling and forecasting requires sharp mathematical tools. Bessel functions are the key in our approach.

Starting with the power law of epidemics, we demonstrate that Besseltype functions describe very well long periods of the growth of the total number of detected cases in many countries. Mathematically, they provide the passage from t c to t and then to the saturation, which appeared exactly what is needed to model Covid-19, where c is the initial transmission rate, which can be captured at relatively early stages of the epidemic. We obtain them via ODE describing the active epidemic management, especially the impact of hard measures.

The saturation due to active protection measures is of unstable nature. To control the efficiency of the management and for forecasting Covid-19 an exact mathematical theory is needed. There will be an endless discussion of the efficiency of different measures and different management approaches until verifiable trustworthy mathematical models and the corresponding software are developed and implemented practically. The verification does require algorithms that can be used by anyone, not only by their creators, the ultimate test of their validity. This is one of the reasons we wrote our own programs; they are posted in [Ch1, Ch3] and can be used by anyone for any countries and regions, though only for the late stages of Covid-19 so far (mode (B)).

The new theory we suggest seems a solid basis for reaching the next level, which is forecasting. It already describes the curves of total numbers of detected infections with high accuracy and with surprisingly high level of stability of the auto-projections, but forecasting is always a challenge. The small number of the parameter we employ explains very well the uniformity of the curves of total numbers of detected infections of Covid-19 in many countries, as well as mathematical similarity of the first and the second waves.

These parameters are: (1) the initial transmission rate c, which can be seen at relatively early stages of the spread, (2) the intensity of hard measures a, which become sufficiently stable near the turning point, and (3) the intensity b of the measures (mostly soft) during the second phase, toward the saturation. The scaling coefficient of u(t) is adjusted to match the real numbers of cases. The coefficient of u 2 is actually of importance, but it is mostly used to capture some effects of the second order and does not seem really necessary for forecasting; the dominant Bessel-type solution u 1 is expected to be sufficient for this.

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The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10. 1101 The fact that we were able to describe such complex stochastic processes as epidemics "just" with 3 parameters seems a real discovery. The confirmations are solid in many countries, but it will take time to understand the scope of this development and to begin using it practically.

The author thanks very much David Kazhdan for valuable comments and suggestions; a good portion of this paper presents author's attempts to answer his questions. Many thanks to Eric Opdam and Alexei Borodin for their kind interest and ETH-ITS for outstanding hospitality; special thanks are to Giovanni Felder and Rahul Pandharipande.

Author Affiliations. University of North Carolina at Chapel Hill, Mathematics Department, Phillips Hall, Chapel Hill, NC 27599-3250. .

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