SIR - Model Supported by a New Density 9783031052736

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Marcus Hellwig

SIR - Model Supported by a New Density Action Document for an Adapted COVID - Management

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Marcus Hellwig

SIR - Model Supported by a New Density Action Document for an Adapted COVID - Management

Marcus Hellwig Lautertal, Germany

ISSN 2197-6708 ISSN 2197-6716  (electronic) essentials ISSN 2731-3107 ISSN 2731-3115  (electronic) Springer essentials ISBN 978-3-031-05273-6  (eBook) https://doi.org/10.1007/978-3-031-05273-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Reinhard Dapper This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

What you can find in this essential

The SIR model and its derivatives receive a statistical data background from frequency distributions, from whose parameter values a​​ quality-oriented probability of the respective infection process can be inferred via the new density distribution. This gives the COVID management a functional basis for controlling the components of time planning, cost development, quality management and the use of personnel and materials. A precautionary consideration of the future of an infection process can be supported by the fact that the method now allows conclusions to be drawn about the near future of the infection process from the test data sets by means of statistical-probabilistic analyzes.

v

Preface

In contrast to regularly occurring infection processes, that of a COVID infection takes a different course. This is characterized by a dynamic that deviates from conventional, known processes in that the causative agents change their identity. As a result, in the infection figures, there are squeal in the population/the affected population, which, due to an apparent decay of the infection, i.e. the decrease in the number of infected persons, lead to the assumption of an imminent end—with fatal effects, as experience teaches. Obviously, the virus acts according to the slogan: “Seize the opportunity”. This opens up a completely new challenge for infectiology, namely the observation and analysis of dynamic infection processes, the course of which produces pathological pictures that did not occur before: • The death episode • The Long—Covid—Episode • The post—Covid—Episode. In contrast to a known virus infection, in Covid infections an end of a wave is not foreseeable, precisely because of the mutations that the virus produces according to its “ultiplication compulsion”. Marcus Hellwig

vii

Acknowledgment

I hereby thank Mr. Edward Brown, United States Department of Health and Human Services sincerely for his idea to supplement the SIR model with a probability density Eqb as a substitute for the component I (t) of the differential equation. This work was created with the excellent software Microsoft Office, the translations into the English language were mainly done by the Google translator with subsequent fine corrections.

ix

Contents

1 Occasion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

SIR Model as the Basis for a Probabilistic Model. . . . . . . . . . . . . . . . 5 3.1 Conditions of the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Discrete Values Versus ​​ Function Values. . . . . . . . . . . . . 7 3.1.2 From the 7-day Logarithm to the Exponential Increase . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.3 The Observation of the Gradient Coefficient. . . . . . . . . 10 3.2 Replacement of the Infection Rate I (t) of the SIR Model by the Eqb Function. . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Preliminary Clarification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.1 Consideration of a Sequence of Time Intervals. . . . . . . 15 3.4 Conclusion from Considering a Sequence of Time Intervals. . . . 24 3.5 Transition to Preventive Consideration. . . . . . . . . . . . . . . . . . . . . 26

4

Preventive Consideration Using Probabilistic SIR Modeling. . . . . . . 29

5

The “infection Curve” I (t) is Replaced by the Inclined, Steep Eqb Density Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1 The Power of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 In Earthly Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6

Events and Findings from the Recent Past. . . . . . . . . . . . . . . . . . . . . . 37 6.1 In Stochastic Systems (Hyperbolic Distribution Forms). . . . . . . . 37

xi

xii

Contents

7

Ways Out of Symmetry, Union with Asymmetry. . . . . . . . . . . . . . . . . 39 7.1 Right-Skewed and Left-Skewed Hypothetical Distributions . . . . 39

8

Random Scatter Areas of the NV and the Eqb . . . . . . . . . . . . . . . . . . 45

9

Presentation of the Equibalance Distribution, Eqb. . . . . . . . . . . . . . . 49 9.1 Presentation of Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 9.2 Adding the Kurtosis parameter to the density Eqb. . . . . . . . . . . . 54 9.3 The basis for the Eqb density, the test series results. . . . . . . . . . . 54 9.3.1 Density Eqb as a substitute for I (t), parameters, data sources. . . . . . . . . . . . . . . . . . . . . . . . . 54 9.3.2 Functions and parameter values. . . . . . . . . . . . . . . . . . . 54 9.4 Incidence from a probabilistic point of view . . . . . . . . . . . . . . . . 56

10 Infection Management in Relation to the Course of Incidence . . . . . 59 10.1 Phase Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.1.1 Phase scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1

Occasion

The course of infection can be described with the SIR model, a group of three differential equations. A method is to be presented which supports the model with a probabilistic component in such a way that part of the equation system is replaced by a probability density. This includes the application of sample analysis with the aim of being able to draw conclusions from the results obtained about the population - which reveals the character of the process. For this purpose, data collections are necessary that indicate the characteristic properties of a process when it is analyzed by obtaining statistical key figures from the sample numbers, which reveal their values in ​​ frequency tables or graphics. These key figures, also called parameters, are the basis for a theoretical consideration of the future of a process, which is called the density function. By far the most frequently used function is the Gaussian normal distribution when it comes to the spread of measured values, i.e. to describe those measured values ​​that provide infection surveys. The distribution used for the analysis of the test series of the infection process is the equibalanced distribution, which takes the dynamics into account by including skewness and kurtosis in the calculations. The procedure described below may serve to support the words of those quoted. President of the European Commission, quote from Ursula von der Leyen: “No virus should ever turn a local epidemic into a global pandemic”

United States Department of Health and Human Services, quoting Edward G. Brown:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_1

1

2

1 Occasion “Early treatment, prophylaxis and better testing are among the most important things to do when an outbreak begins. In addition, of course, the new virus needs to be identified and profiled so that all of these things have to happen as quickly as possible. From my point of view, vaccination should be the last thing, and nothing should be rushed there unless proper efficacy and safety studies have been carried out.”

2

Objectives

The following article deals with the use of a density function Eqb Fig. 2.1b, whose parameter values are generated from test data as a substitute for the course of I (t) of the number of infectious individuals in a SIR model, Fig. 2.1a. The following steps are carried out, which can be found in the following ­chapters: 1. The SIR model remains the basis for a future model. 2. The “infection curve” I (t) is replaced by the Eqb density function. 3. The basis for the Eqb density are the test series results as a data set of the frequency of the I (t) from which the parameters are derived which influence the “shaping” of the function. 4. The test data are significantly influenced by the day of the week on which the surveys took place. Therefore there is a “graphical difference” between the values of ​​ the density function and the frequency data, which can be adjusted by using the Gaussian method of the “least squares” in order to derive the most plausible parameters by choosing a weekday come into play in density. 5. Since the frequency distribution is linked to the quantities in connection with the time course, a common statement can be inferred, so that from the course of the density values from ​​ Eqb it is possible to infer the future of the development of the frequency distribution along the time scale. 6. Since the frequencies run “undulating” over time, depending on the behavior of the population, it is necessary to determine when the slope of the frequency distribution increases or decreases between time intervals, because then it is time to set the starting point of the observation and analysis anew.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_2

3

4

2 Objectives

a

b

Fig. 2.1   a https://upload.wikimedia.org/wikipedia/commons/b/bd/SIR-Modell.svg. b Frequency distribution and density Eqb. (Image rights: https://de.wikipedia.org/wiki/SIR-Modell)

7. From an interval logarithm, conclusions can be drawn about the exponent of the function and thus about the future development of the frequency distribution and thus about the course of future processes. 8. A shift in the summation of the Eqb before the start of the process start before the start of an infection process—may help to identify how a population could have been prepared before the influence of an infection event. 9. The use of the steps mentioned above means that measures can be planned at an early stage. The SIR models described so far and their derivatives are based on the ratio of a hypothetical relationship between differential functions. The replacement of the relationship between I (Infect), S (Infectious) and R (Recovered) by S´ = S(− β I), for b as an infection factor let S´ = S (− β p (I)), for p the percentage from the density.

3

SIR Model as the Basis for a Probabilistic Model

3.1 Conditions of the Model If we assume an infection, the following set terms (set term for populations) used: • Amount of infectious, susceptible (S) • Amount of infected people (I) • Amount of recovered, retained (R) The SIR model is derived from these populations. It is based on the context and assumption that the infection process occurs over a period of time such that: • The number of infected people (I) influences the number of infected people (S) and those who have recovered (R). This is shown in the following graphic, Fig. 3.1. The following chapter shows that test data collection and subsequent statistical-probabilistic calculations always come “very close” to a “true” course of an infection process when frequency density and function values ​​indicate agreement via a regression analysis. A future course can then be deduced from this and the basis can thus be laid for the planning of preventive measures. It is shown that there are fundamental differences between: • a mathematical proposition that is valid “forever” • a statistical-probabilistic statement that always needs to be checked for correspondence between theory and practice.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_3

5

6

3  SIR Model as the Basis for a Probabilistic Model

Fig. 3.1   SIR model. (Image rights: https://de.wikipedia.org/wiki/SIR-Modell)

The following chapter is listed for this purpose. Probability/Mathematical Theorem To determine the future occurrence of events that are statistically recorded, the probabilistic can make statements. Measurements are necessary for this, which on the one hand provide a basis for the starting conditions of the probabilistic and for a continuous prognosis. If scientists try to find the truth—in the sense of—100 percent certainty—they will fail. There will be a difference between mathematical and statistical truth at all times. A mathematical truth is defined as a mathematical proof. A statistical truth for which proof can never be found from a mathematical point of view is only valid as a comparison of samples of a series of test values ​​with a theoretical density function, which always depends on the amount of tests that have been carried out. The answer is ultimately: On the one hand, the strength of a deterministic algorithm is used in the mathematical sense, on the other hand, a quantity of data is evaluated in the statistical sense using a density distribution such as the density distribution to approximate a truth. In connection with the preceding explanations, it is pointed out that this work only makes statistical-probabilistic statements, influences from other specialist areas are not taken into account. An example of the development of a mathematical truth is the proof of the Pythagorean theorem (Fig. 3.2). The one that is often published is this graphic representation of a numerical proof in which the truth is brought about by a compelling logic case. In contrast, there is the statistical-probabilistic finding of truth, the approximation of a correspondence of relationships by means of a regression test, which is brought about by the method of least squares. This method is used in the further course of the comparison of the frequency values ​​of infection values and ​​ the

3.1  Conditions of the Model

7

Fig. 3.2   Proof of the Pythagorean theorem. (Image rights: [https://www.matheretter.de/ wiki/pythagoras-beweis])

a

b

Fig. 3.3   a Frequency values -​​ probability values. b Least squares method. (Image rights: [Copyright of the author])

probability values ​​(Fig.  3.3a, b) from the Equibalancedistribution. The percentage level of the determined coefficient of determination is therefore to be regarded as an approximate value for agreement.

3.1.1 Discrete Values ​​Versus Function Values It should be understood that the course—i.e. rise and fall—of the event results from the value of the respective test and its successor. The resulting picture, Fig. 3.4, then gives exactly one frequency value in a category 0 to 20 along the axis of discrete feature values. Rise and fall can therefore be represented as difference quotients on a gradient triangle delta x/delta y.

8

3  SIR Model as the Basis for a Probabilistic Model

Fig. 3.4   Frequency distribution of the characteristic values ​​along a time axis. (Image rights: [Copyright of the author])

y = f(x) 1.2

Werteberieich

Fig. 3.5   Density function. (Image rights: [Copyright of the author])

1 0.8

0.6 0.4 0.2 0

0 1 2 3 4 5 6 7 8 9 101112131415161718192021

x

Definionsbereich

It should also be understood that the course—i.e. rise and fall—of the function results from the sequence of the function values. The image from this, Fig. 3.5, then results in the function values ​​in a domain of definition t 0 to 20 along the axis of continuous function values. Rise and fall can therefore be used as differential quotients on a gradient triangle delta x/delta y whose limit values approach ​​ 0 and can thus be formulated as dy after dx (dy/dx).

3.1  Conditions of the Model

9

The superimposition of both graphs is therefore generally to be assumed as an approximation of two statements, especially since the parameter values that ​​ were determined from the random samples of test series arise from mathematical principles. Therefore the ideal case—that discrete test values ​​appear exactly congruent with continuous values from ​​ a density function—cannot be realized, Fig. 3.6. The categorization therefore plays a decisive role in the collective consideration and thus in the superimposition of frequency and function, since a frequency value from the random samples of the test measured values can ​​ sometimes not be assigned to each function value. It is therefore often necessary to assign test measured values averaged ​​ to the categories. It is found that, due to the continuity of the function Eqb, it is not possible to assign a function value to every frequency value in a category, Fig. 3.7a, b, this is due to the dynamics of the process, whose characteristic values fluctuate ​​ and therefore due to the different days of the week of the series of measurements can only be used as averaged values. In addition, the exponential course (increase/ decrease) can also be viewed as averaged values, since there is a range of fluctuation between the samples from day to day.

Fig. 3.6   Ideal frequency/density. (Image rights: [Copyright of the author])

10

a

3  SIR Model as the Basis for a Probabilistic Model

b

Fig. 3.7   Frequency value/function value—a, b different days of the week of the measurement series, test data. (Image rights: [Copyright of the author])

3.1.2 From the 7-day Logarithm to the Exponential Increase In this context, the exponential development must also be considered, since it shows to what extent, i.e. with what exponent, the Eqb process sequence is afflicted. To do this, it is necessary to know the exponent from a previous period of time; it is given in the information on the incidence values ​​with a previous week. A corresponding logarithm can be deduced from the interval averaged frequency distribution values, which in the inverse of its function allows conclusions to be drawn about the exponent of the interval. The following Fig. 3.8 reminds of the corresponding formulation from the textbook, an example: The reverse suggests that—if frequency values ​​exist—the exponent of the process, the exponential development, can be deduced from their logarithm. Figure 3.9 shows the corresponding development according to a data collection. This example shows that with a constant exponent > 1, the constancy is not accepted, but increases exponentially according to the decimal part after the decimal point.

3.1.3 The Observation of the Gradient Coefficient As can be seen from the past, infection processes sometimes run in waves. Since it cannot be seen directly from the observation of the measurement data from a time interval when a wave starts or when a wave changes into a subsequent wave,

3.1  Conditions of the Model

11 Logarithmus

1,50E+07 1,00E+07 5,00E+06 0,00E+00

x

0

10

20

30

40

50

y= hoch

Logarithmus

4

4

60

70

Exponent

Fig. 3.8   Logarithm value determination. (Image rights: [Copyright of the author])

positiv getestete Personen

Progdata basierend Exponent = Logarithmus

5,00E+03 5,00E+02

388

335

244

166 1,4

1,4

1.200,00

Folgetage

0

1.6.20

22.5.20

12.5.20

2.5.20

22.4.20

1.234,00 y= hoch

positiv getestete Personen

Tag Nummer

12.4.20

2.4.20

13.3.20

23.3.20

1,00E+00 3.3.20

5,00E+00

356 1,9

2,2

48

5,00E+01

Logarithmus = Exponent 1,00E+01

Progdata basierend Exponent = Logarithmus

Logarithmus = Exponent

Datum 0 1 2 3 4 5 6

10.03.2020 11.03.2020 12.03.2020 13.03.2020 14.03.2020 15.03.2020

48,00 60,00 92,00 80,00 60,00 128,00

Logarithm of 0 for basis 0 equals Logarithm of 48 for basis 1 equals Logarithm of 60 for basis 2 equals Logarithm of 92 for basis 3 equals Logarithm of 80 for basis 4 equals Logarithm of 60 for basis 5 equals Logarithm of 128 for basis 6 equals

because 2 to the power of because 3 to the power of because 4 to the power of because 5 to the power of because 6 to the power of

equals 60 equals 92 equals 80 equals 60 equals 128

Fig. 3.9   Exponential development from the mean logarithm of a measurement data interval. (Image rights: [Copyright of the author])

12

3  SIR Model as the Basis for a Probabilistic Model

it may be helpful to consider the course of the respective gradient for an upcoming increase. The first-time increase in a frequency distribution should therefore serve as the basis for achieving comparability. Figure 3.10 is shown for this, a 21-day interval with the measurement data from the 2nd day of the week in the 2nd interval. As can be seen from Fig. 3.11, the consideration of the slope of the following increase, interval with 21 days, turns out to be steeper than the previous interval. After the first frequency distribution reveals a decrease in the number of cases, the associated interval is followed by an interval with an increase, Fig. 3.12. Since this part of the process takes place in a period of time whose interval is exactly as long as its previous interval, the recognition of the extent of a renewed increase cannot be recognized prematurely. Therefore, the slope coefficient r should be taken into account, which indicates the turning point between the frequency distributions in such a way that it changes from decreasing to increasing on the diagram, Fig. 3.13.

Fig. 3.10   Gradient of the first rise, interval of 21 days. (Image rights: [Copyright of the author])

3.1  Conditions of the Model

13

Fig. 3.11   Gradient of the increase in the following interval. (Image rights: [Copyright of the author])

Fig. 3.12   Gradient of the following rise. (Image rights: [Copyright of the author])

It is also helpful to check the deviation of the probability density and frequency distribution of the test data with regression analysis, since the test data also ran different frequency data on different days of the week and thus influence the parameter values that ​​ are transferred to the function, Fig. 3.14.

14

3  SIR Model as the Basis for a Probabilistic Model

Fig. 3.13   Gradient of the following rise. (Image rights: [Copyright of the author])

Fig. 3.14   Regression. (Image rights: [Copyright of the author])

3.2 Replacement of the Infection Rate I (t) of the SIR Model by the Eqb Function The infection rate of group I (t) of the SIR model described in SIR-Modell— Wikipedia is to be replaced by the parameter values ​​from a frequency distribution that originate from a temporal survey, a sample. These are the parameters

3.3  Preliminary Clarification

15

Fig. 3.15   Frequency/density/SIR model. (Image rights: [Copyright of the author])

­maximum, scatter, skewness and kurtosis, Fig. 3.15. The functions develop from this:

I(t) = Eqb(t = x; σ , max, r, κ) ∗ N(total) S(t) = N − Eqb(t = x; σ , max, r, κ) A function for R (t) develops separately from the parameter values from ​​ the frequency of the genesis numbers:

R(t) = Eqb(t = x; σ , max, r, κ) ∗ N(Recovered)

3.3 Preliminary Clarification The function values for ​​ S and R are determined accordingly, which then form the following overview in the relationship between SIR and Eqb density, Fig. 3.16a–c.

3.3.1 Consideration of a Sequence of Time Intervals In the following, the consideration of a series of interval considerations should provide information on the extent to which they can be useful in pointing out the rise and fall of infection processes. Consideration initial interval 1, Fig. 3.17. In Fig. 3.17 are shown.

16

a

3  SIR Model as the Basis for a Probabilistic Model

b

c sum of death 14.863 sum of death that could have been avoided if Infecon prevenon measures or vaccinaons had started on 27.2.2020 =8.148

Fig. 3.16   a Compilation of the courses S, best R and I x Eqb, b cases, I x Eqb, c total deaths, best R course of avoidance of deaths. (Image rights: [Copyright of the author])

• a: the frequency distribution (test data) in connection with the density function (I > Eqb for the state with the sum of the density = 1) as well as the expected positive cases of infection (prognosis data logscaled 7 days in history for future) and the accumulated prognosis cases of infection (CumulPrognosedata logscaled). • b: the course of the positive tested persons, the forecast data from the determination of the 7-day logarithm to determine the exponent for the further exponential development (progdata based on logarithm, the course of the exponent (logarithm = exponent) and the incidence (Incidence). • c: the slope of the initial consideration of the density function, the slope of the current density function, the profile of the initial density function, the profile of the current density function, the slope of the density function. • d: the frequency distribution, the density function. • e: the regression line between the sums of the frequency distribution and the density function. Consideration of the following interval, Fig. 3.18.

1

518

Apr-20

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

518

Aug-20

intervals

Oct-21

251 27

0

1

2

3

4

5

6

7

1000 900 800 700 600 500 400 300 200 100 0

d

Sep-20

518

21.11.2020

518

518

518

147.097 167.817 104.103122.233139.845

I > Eqb Prog Texas 1,00

May-21 Testdata

Jul-21 0

0,005

0,01

0,015

0,02

0,025

17.09.2021

R² = 0,9738

1.800,00 1.600,00 1.400,00 1.200,00 1.000,00 800,00 600,00 400,00 200,00 -

e

Aug-21

Dec-19

1.000.000 0,015 100.000

Linear (Sum Eqb / Sum frequency)

Sum Eqb / Sum frequency

10

100 0,005 10

10.000 0,01 1.000

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.17   a Frequency density and density function, b positive tested, forecast data from a 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,010

0,020

0,030

0,040

0,050

c

Jan-20

Mar-20

Oct-20 Mar-20

26.01.2020

Feb-20

Jun-20

b

P

Mar-20

Sep-20

Nov-20 Mar-20

May-20 Jan-21

Dec-20 Apr-20

Jun-20 Apr-21

Jan-21 Apr-20

Jul-20 Jul-21

Mar-21 May-20

79.757

0,00

56.447

0,01

Feb-21 Apr-20

Nov-21 0,01

20.187

Apr-21 May-20

Oct-21

0,025 1.000.000.000

0,02

Jun-21

May-20

Dec-21 0,02

Sep-21

100.000.000 0,02 10.000.000

0,03

09.04.2020

0,03

1.000 900 800 700 600 500 400 300 200 100 0

0,04

a

3.3  Preliminary Clarification 17

0,030

0,040

0,050

1.272

1.272

d

intervals

0

1200 1000 800 1 600 400 0,5 200 0 1,5

1.272

21.11.2020

1.272

1.272

I > Eqb Prog Texas 1,00

1.272

350.873 401.753 245.297289.817333.065

Jun-21

Testdata

Jul-21 0

0,005

0,01

0,015

0,02

0,025

17.09.2021

R² = 0,9814

12.000,00 10.000,00 8.000,00 6.000,00 4.000,00 2.000,00 -

e

Aug-21

Dec-19

Linear (Sum Eqb / Sum frequency)

Sum Eqb / Sum frequency

10

100 0,005 10

10.000 0,01 1.000

1.000.000 0,015 100.000

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.18   a Frequency density and density function, b positive tested, forecast data from 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,010

0,020

1

251 27

May-20

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

Mar-20

c

Jan-20

26.01.2020

0

Feb-20

Jun-20

Sep-20

128.273 185.513

Aug-20 Oct-21

200

Mar-20

Sep-20

Oct-20 Mar-20

400

Apr-20 Jan-21

b

P

Jun-20 Apr-21

Nov-20 Mar-20

Jul-20 Jul-21

Dec-20 Apr-20

39.233

Mar-21 May-20

600

0,00

Jan-21 Apr-20

Oct-21 0,01

Feb-21 Apr-20

Nov-21 0,02

800

0,03

May-21 May-20

Dec-21 0,04

Apr-21 May-20

0,025 1.000.000.000

0,05

Sep-21

100.000.000 0,02 10.000.000

0,06

23.04.2020

0,07

1.200

0,08

1.000

0,10

a

0,09

18 3  SIR Model as the Basis for a Probabilistic Model

3.3  Preliminary Clarification

19

Comparison of the consideration of the initial interval 1 and the consideration of the subsequent interval 2. It is—essentially—observed: • b: the exponent drops from 1.7 to 1.6 • c: with almost the same gradient of the straight line (gradient), the skewness of the function decreases • e: The regression line in that coefficient of determination remains close to 1.0 Consideration of the following interval 3, Fig. 3.19. Comparison of the consideration of the initial intervals 1 and 2 and the consideration of the following interval 3: It is—essentially—observed: • b: the exponent remains at 1.6 • c: with a decreasing gradient of the straight line (gradient), the skewness of the function decreases—very strongly • e: The regression line in that coefficient of determination remains close to 1.0 Consideration of the following interval 4, Fig. 3.20. Comparison of the consideration of the initial intervals 1, 2 and 3 and the consideration of the following interval 4. It is—essentially—observed: • b: the exponent drops to 1.5 • c: with a decreasing gradient of the straight line (gradient), the skewness of the function decreases—very strongly • e: The regression line in that coefficient of determination remains close to 1.0 Consideration of the following interval 5, Fig. 3.21. Comparison of the consideration of the initial intervals 1, 2, 3 and 4 and the consideration of the following interval 5. It is—essentially—observed: • a: the position parameters of the function shift significantly • b: the exponent increases to 1.6 • c: as the gradient of the straight line (gradient) decreases, the skewness of the function increases again • e: The regression line in that coefficient of determination remains close to 1.0

0,030

0,040

1

Aug-20 0

0,2

0,4

0,6

0,8

1

1,2

1400 1200 1000 800 600 400 200 0

d

Sep-20 1,4

intervals

Oct-21

1.131

1.131

I > Eqb Prog Texas 1,01

21.11.2020

1.131

Oct-20 Mar-20

1.131

1.131

Jun-21

Testdata

Jul-21 0

0,005

0,01

0,015

0,02

17.09.2021

Sep-21

R² = 0,9661

30.000,00 25.000,00 20.000,00 15.000,00 10.000,00 5.000,00 -

e

Aug-21

Dec-19

Sum Eqb / Sum frequency

Linear (Sum Eqb / Sum frequency)

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.19   a Frequency density and density function, b positive tested, forecast data from a 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,010

0,020

Jan-20

Mar-20

Mar-21 May-20

1.065

May-20

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

Feb-20

Jun-20

c

26.01.2020

Mar-20

Sep-20

b

P

Apr-20

Jan-21

Nov-20

Mar-20

Jun-20 Apr-21

Dec-20 Apr-20

Jul-20 Jul-21

Jan-21 Apr-20

Oct-21 0,00

251 27

0,02

Feb-21 Apr-20

Nov-21 0,04

May-21 May-20

Dec-21 0,06

313.150 358.390 219.277258.862297.316

0,08

115.225 166.120

0,10

36.055

0,12

07.05.2020

Apr-21 May-20

0,02 1.000.000.000 0,018 100.000.000 0,016 10.000.000 0,014 1.000.000 0,012 100.000 0,01 10.000 0,008 1.000 0,006 100 0,004 0,002 10 0 1

0,14

1.400 1.200 1.000 800 600 400 200 0

0,18

a

0,16

20 3  SIR Model as the Basis for a Probabilistic Model

0,030

0,040

1

1.460

d

intervals

-0,5

1600 1400 1200 1 1000 800 0,5 600 400 200 0 0 1,5

Sep-20

1.460

I > Eqb Prog Texas 1,03

21.11.2020

1.460

Nov-20

1.460

1.460

393.227 451.627 272.047323.147372.787

May-21

Testdata

Jul-21

0,016 0,014 0,012 0,01 0,008 0,006 0,004 0,002 0

e

Aug-21 R² = 0,978

-

5.000,00

10.000,00

15.000,00

20.000,00

25.000,00

17.09.2021

Sep-21

Dec-19

0,016 1.000.000.000

Linear (Sum Eqb / Sum frequency)

Sum Eqb / Sum frequency

0 1

1.000.000 0,01 100.000 0,008 10.000 0,006 1.000 0,004 100 0,002 10

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.20   a Frequency density and density function, b positive tested, forecast data from 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,010

0,020

Jan-20

1.065

Jun-20

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

Mar-20

c

Feb-20

Jun-20

Aug-20 Oct-21

26.01.2020

Mar-20

Sep-20

Oct-20 Mar-20

251 27

Apr-20 Jan-21

Jan-21

137.727 203.427

Mar-21

36.396

May-20 Apr-21

b

P

Jul-20 Jul-21

Oct-21 0,00

Dec-20 Apr-20

Nov-21 0,02

Feb-21 Jun-20

Dec-21 0,04

Apr-21 Jul-20

0,014 100.000.000

0,06

Jun-21 Aug-20

10.000.000 0,012

0,08

21.05.2020

0,10

1.600 1.400 1.200 1.000 800 600 400 200 0

0,12

a

3.3  Preliminary Clarification 21

0,040

0,030

d

Sep-20

intervals

0

0,2

0,4

0,6

0,8

1

0

500

1000

1500

2000 1,2

21.11.2020

1.819

I > Eqb Prog Texas 1,06

Jan-21

1.819

1.819

Mar-21

1.819

May-21

Testdata

Jul-21 0,012 0,01 0,008 0,006 0,004 0,002 0

17.09.2021

R² = 0,961

-

5.000,00

10.000,00

15.000,00

20.000,00

25.000,00

e

Aug-21

Dec-19

0,008 1.000.000

Linear (Sum Eqb / Sum frequency)

Sum Eqb / Sum frequency

10

100 0,002 10

0,004 1.000

100.000 0,006 10.000

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.21   a Frequency density and density function, b positive tested, forecast data from a 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,010

0,020

Mar-20

0,050

May-20

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

Feb-20

Jun-20

c

Jan-20

1.819

Aug-20 Oct-21

26.01.2020

Mar-20

Sep-20

Nov-20

1

Apr-20 Jan-21

Oct-20 Mar-20

1.065

Jul-20

251 27

Jun-20 Apr-21

b

P

Jul-20 Jul-21

Apr-21

475.587 548.347 324.610388.275450.121

0,00

157.262 239.117

0,01

Dec-20 Apr-20

Nov-21 0,02

36.396

0,04

Feb-21 Jun-20

Dec-21 0,03

Jun-21 Aug-20

Oct-21

0,012 1.000.000.000

0,05

Sep-21

100.000.000 0,01 10.000.000

0,06

04.06.2020

0,07

2.000 1.800 1.600 1.400 1.200 1.000 800 600 400 200 0

0,09

a

0,08

22 3  SIR Model as the Basis for a Probabilistic Model

0,150

Jan-20

26.01.2020

Mar-20

2.207

intervals

d 3000

4000

5000

Sep-20 -2

0

0

1000

2 2000

4

6

2.207

2.207

2.207

Jan-21

Nov-20 I > Eqb Prog Texas 0,99

2.207

May-21

Testdata

Jul-21 0,04 0,035 0,03 0,025 0,02 0,015 0,01 0,005 0

-

0,20

0,40

0,60

0,80

1,00

e

Aug-21

Mar-21

Dec-19

Linear (Sum Eqb / Sum frequency)

Sum Eqb / Sum frequency

01

0,005 10

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.22   a Frequency density and density function, b positive tested, forecast data from 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,050

0,100

Feb-20

Jun-20

0,200

May-20

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

Sep-20

c

Mar-20

1.065

Aug-20 Oct-21

1

Apr-20 Jan-21

Oct-20 Mar-20

251 27

Jun-20 Apr-21

b

P

Jul-20 Jul-21

Oct-21 0,10

559.172 647.452 375.991453.236528.274

0,20

Dec-20 Apr-20

Nov-21 0,30

Feb-21 Jun-20

Dec-21 0,40

172.947 272.262

0,50

Apr-21 Jul-20

10.000.000 0,03 1.000.000 0,025 100.000 0,02 10.000 0,015 1.000 0,01 100

0,60

36.396

Jun-21 Aug-20

100.000.000 0,035

0,70

Sep-21 0,00

0,04 1.000.000.000

0,80

18.06.2020

0,90

5.000 4.500 4.000 3.500 3.000 2.500 2.000 1.500 1.000 500 0

1,00

a

3.3  Preliminary Clarification 23

24

3  SIR Model as the Basis for a Probabilistic Model

This is the first indication that it is to be expected that the frequency distribution will experience a change. Consideration of the following interval 6; Fig. 3.22. Comparison of the consideration of the initial intervals 1, 2, 3 and 4 and the consideration of the following interval 5. It is—essentially—observed: • a: the position parameters of the function shift significantly • b: the exponent increases to 1.7 • c: if the gradient of the straight line (gradient) increases, the skewness of the function increases again • e: The regression line in that coefficient of determination is no longer comprehensible As a result of the obviously large deviations, it is concluded that the frequency distribution is subject to a new wave and thus has an effect on the parameters that flow into the density function. Therefore, the consideration of the test data is readjusted under a corresponding starting time—consideration of the initial interval 1—and the method is repeated. Consideration of the following interval 7; Fig. 3.23. Comparison of the consideration of the initial intervals 1, 2, 3, 4, 5 and 6 and the consideration of the subsequent interval 7. It is—essentially—observed: • a: the position parameters of the function shift significantly • b: the exponent increases to 3.0 and decreases to 2.5 • c: as the gradient of the straight line (gradient) decreases, the skewness of the function increases again • e: The regression line in that coefficient of determination drops to 0.84

3.4 Conclusion from Considering a Sequence of Time Intervals The interval consideration begins with the definition of a reference graphic which results from the frequency distribution of an initially considered initial interval. The four parameters maximum, scatter, skewness and kurtosis are transferred to the density function. It refers to the expected numerical development of the

Mar-20

1

9

9

intervals

gradient of density after reaching inlection point 1 previous Equibalancedistribution 4 Parameters 1 Equibalancedistribution 4 Parameters new gradient of density after reaching inlection point 2 actual r = n (%)

26.01.2020

528

0

1

2

3

4

Sep-20 8000 7000 6000 5000 4000 3000 2000 1000 0

d

9

1.158

1.563 9

1.986 9

I > Eqb Prog Texas 1,01

9

2.733 2.301 2.607 9

3.093

Testdata

Jul-21 0

0,005

0,01

0,015

0,02

Sep-21

R² = 0,8431

350.000,00 300.000,00 250.000,00 200.000,00 150.000,00 100.000,00 50.000,00 -

e

Aug-21

Mar-20

Linear (Sum Eqb / Sum frequency)

Sum Eqb / Sum frequency

01

10.000 0,015 1.000 0,01 100 0,005 10

1.000.000.000 100.000.000 10.000.000 1.000.000 100.000 10.000 1.000 100 10 1 0 0 0 0 0 0

Fig. 3.23   a Frequency density and density function, b positive tested, forecast data from a 7-day logarithm, c slope of the densities. d frequency distribution, the density function, e regression line. (Image rights: [Copyright of the author])

0,000

0,010

0,020

0,030

0,040

c

0

Jun-20

132

Nov-20

21.06.2020

Jul-20 Oct-21

1.000

Jan-20

Jun-20

Oct-20 Apr-20

2.000

Feb-20

Sep-20

Jan-21

3.000

Apr-20

Jan-21

b

P

May-20 Apr-21

Apr-21

4.000

Mar-21 Aug-20

Aug-20

Jul-21

Oct-21 0,00

Dec-20 Jun-20

Nov-21 0,05

Feb-21 Jul-20

Dec-21 0,10

May-21 Oct-20

100.000.000 0,03 10.000.000 0,025 1.000.000 0,02 100.000

0,15

Jun-21

0,035 1.000.000.000

0,20

5.000

0,25

6.000

0,30

a

3.4  Conclusion from Considering a Sequence of Time Intervals 25

26

3  SIR Model as the Basis for a Probabilistic Model

p­ rocess—in this case the COVID—19 infection process starting with the month of February 2020. Exponential increase By determining a 7-day logarithm, the exponent of the development can be inferred, which in turn influences all further process-dependent factors in the future. This includes the incidence as well as an increase or decrease in the number of cases. Regression, coefficient of determination If the process is observed over time in further subsequent intervals, it soon turns out that, due to the changeable numerical ratio in the determined frequencies, fluctuations occur in the test data, which may be influenced by the day of the week on which the data situation was announced the best relationship between the frequency distribution and the probability density must be checked. A use of the regression analysis according to Gaussian version therefore provides information about the agreement if the choice of the day of the week leads to the coefficient of determination being close to 1. Skew and pitch However, it also shows that it makes sense to observe the rate of increase from interval to subsequent interval, because there may be a short-term increase in the number of cases. It is not possible to foresee this obviously via the frequency distribution, but possibly by tracking the slope and the skewness of the density from the initial interval over subsequent intervals. It turns out that the start time of the initial interval has to be adjusted if the positions of the inclination and slope measure as shown between the following interval 5 and following interval 6. Location parameters It is obviously clear that all position parameters, i.e. maximum, scatter, skewness and kurtosis must be included in the observation, because their values ​​ultimately determine the decision as to how the further course of a process event will be.

3.5 Transition to Preventive Consideration Decision-makers would like to confront an infection process early and preventively—and before an infection event can ever achieve pathological effects. To do this, it is necessary to have the resources available in advance—i.e. before a

3.5  Transition to Preventive Consideration

27

process has a start. A preventive consideration using a probabilistic SIR model should provide information on how preventive measures or healing measures are timed. This will ask the following questions: How many cases of illness or death could have been avoided if. • premature preventive or healing measures • Preventive or curative measures accompanying the time could be hit? The preventive measures or curative measures are listed in accordance with Coronavirus—infektionsschutz.de. Keep your distance, observe hygiene, wear a mask in everyday life, ventilate regularly and use the Corona warning app: The AntiCOVID—formula is part of our everyday life in times of Corona. We will inform you what still needs to be observed. Testing for an acute infection with the SARS-CoV-2 coronavirus helps identify infected people and break chains of transmission. PCR tests, rapid antigen tests and antigen self-tests are available as tests. Vaccination against the SARS-CoV-2 coronavirus is the most effective protection against COVID-19 and the spread of the virus. As a demonstration, the following graphics show how many deaths could have been avoided if the preceding preventive or healing measures could have worked if they were used. For this purpose, the probabilistic SIR model was equipped with a back calculation, which corresponds to the determination of the values of ​​ the cumulative curve – from an intensity of preventive measures or curative measures – from a period of time before an initial interval Permits modeling that can help to better plan the necessary resources with regard to – staff – Time – Costs – quality The following chapter lists cases that may give an insight into probabilistic SIR modeling.

4

Preventive Consideration Using Probabilistic SIR Modeling

Case 1, Fig. 4.1, Avoided deaths if the date of recovery had started 0 days before the start of the infection process. It is—essentially—observed: • a: the density function Eqb in connection with an S curve for the susceptible and R ideal for those who have recovered in the event that healing occurs with the onset of an infection • b: the frequency distribution of the infected, the frequency distribution of the vaccinated • c the frequency distribution of the infected, the ideal R—sum curve, if the date of recovery had started 0 days before the start of the infection process • d: the total number of deaths recorded, the total number of deaths if the recovery date had started 0 days before the start of the infection process Case 2, Fig. 4.2, Avoided Dead, if the date of recovery had started 30 days before the start of the infection process and the density of the preventive action rate per day was 1.5 It is—essentially—observed: • a: the density function Eqb in connection with an S curve for the susceptible and R ideal for those who have recovered in the event that healing occurs with the onset of an infection • b: the frequency distribution of the infected, the frequency distribution of the vaccinated

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_4

29

Apr-20

Feb-20

dS*Eqb

b=

Total deaths 487

S

25.145.561

SIR-Modell

Apr-20

frequency of death

R (number of vaccinated related to avoided death)

I = Eqbl x Habitants

0,000

ideal R

Feb-21

Jun-20

number of vaccinated

R (number of vaccinated related to avoided death)

1,294%

dR

0

0,01

50000

100000

150000

200000

250000

frequency of death

0,00

100,00

200,00

300,00

400,00

500,00

Sum of frequency of death

25.01.2020

Start of contemplation

Density of vaccination rate per day

21.04.2020

end of contemplation

1,0

0,0019%

percent Sum of No of Death

0

0,013

Sum of avoided death if date of real recovering had started 0 days before start of infection process 600,00

preventioned vaacinations

Avoided dead 6 Avoided deaths if the date of recovery had started 0 days before the start of the infection process

dI

g=

0

5.000.000

10.000.000

Jan-20

15.000.000

sum of I death avoided

Sum of frequency of death

Apr-20

20.000.000

c

Jul-20

25.000.000

Oct-20

ideal R

Density of vaccination I death rate per day avoided minus 0 days

Feb-21

30.000.000

May-21

Sum of avoided Sum of death if avoided date of death if date real of I sum of I recoverin recovering death g had had started avoided started 0 0 days days before start before of infection start of process infection process

487

Sum of death

avoided death if date of recovering had started 0 days before start of infection process

6

avoided death

0

5

10

15

20

25

30

35

40

45

50

Fig. 4.1   a Probabilistic SIR model with density function. b past frequency of infections in the start interval, late current vaccinations. c past frequency, course of the R curve with the beginning of the infection curve. d total deaths, avoided deaths if the recovery date is 0 days would have started before the infection process began. (Image rights: [Copyright of the author])

d

date of sequence

Habitants

Texas

0,0

10,0

20,0

30,0

40,0

50,0

Jan-20

Jan-20

b

Jul-20

Jul-20

May-21

S

Oct-20

Sep-20 Aug-21

= Eqbl x Habitants

Aug-21

CommandBI

Dec-21

a

Oct-20 Dec-21

450.000 400.000 350.000 300.000 250.000 200.000 150.000 100.000 50.000 0

30 4  Preventive Consideration Using Probabilistic SIR Modeling

Jan-20

Jan-20

Feb-20

Apr-20 frequency of death

dI

R (number of vaccinated related to avoided death)

46,931%

dR

0

0,47

50000

100000

150000

200000

250000

0,00

100,00

200,00

300,00

400,00

500,00

600,00

700,00

21.04.2020

Density of Sum of preventive frequency measuresrat of death e per day

25.01.2020

1,5

0,0019%

percent end of Start of contemplati contemplati Sum of No of Death on on

30

0,469

Sum of avoided death if date of real recovering had started 30 days before start of infection process 800,00

preventio ned frequency vaacinati of death ons

229 Avoided deaths if the date of recovery had started 30 days before the start of the infection process and the density of the preventive action rate per day was 1.5

ideal R

g=

number of vaccinated

0

5.000.000

10.000.000

sum of I death avoided

Jan-20

15.000.000

Sum of frequency of death

Apr-20

20.000.000

c

Jul-20

ideal R

Oct-20

25.000.000

Density of preventiv sum of e I death I death measures avoide avoide rate per d d day minus 30 days

Feb-21

30.000.000

Avoided dead

I = Eqbl x Habitants

0,000

Jun-20

487

dS*Eqb

b=

Feb-21

Total deaths

S

25.145.561

SIR-Modell

Apr-20

R (number of vaccinated related to avoided death)

May-21

Sum of avoided death if date of I recoverin g had started 30 days before start of infection process

30

avoided death

0

5

10

15

20

25

487 229 Sum of avoide avoided d death if death date of if date recovering of real had started recove 30 days ring before had start of started infection 30 process days before

Sum of death

40 35

Fig. 4.2   a Probabilistic SIR model with density function. b past frequency of infections in the start interval, late current vaccinations. c past frequency, course of the R curve with the beginning of the infection curve. d total deaths, avoided deaths if the recovery date is 0 days would have started before the infection process began. (Image rights: [Copyright of the author])

d

date of sequence

Habitants

Texas

0,0

10,0

20,0

30,0

40,0

50,0

b

Jul-20

Jul-20 May-21

S

Oct-20

Sep-20

Aug-21

I = Eqbl x Habitants

Aug-21

CommandB

Dec-21

a

Oct-20 Dec-21

450.000 400.000 350.000 300.000 250.000 200.000 150.000 100.000 50.000 0

4  Preventive Consideration Using Probabilistic SIR Modeling 31

32

4  Preventive Consideration Using Probabilistic SIR Modeling

• c the frequency distribution of the infected, the ideal R-sum curve, if the date of recovery had started 30 days before the start of the infection process and the density of the preventive measures rate per day were 1.5 • d: the sum of the recorded deaths, the sum of the deaths if the date of recovery had started 30 days before the start of the infection process and the density of the preventive action rate per day would be 1.5 Case 3, Fig. 4.3, Avoided deaths if the date of recovery had started 30 days before the start of the infection process and the density of the preventive action rate per day was 1.5 It is—essentially—observed: • a: the density function Eqb in connection with an S curve for the susceptible and R ideal for those who have recovered in the event that healing occurs with the onset of an infection • b: the frequency distribution of the infected, the frequency distribution of the vaccinated • c the frequency distribution of the infected, the ideal R-sum curve if the date of recovery had started 30 days before the start of the infection process and the density of the preventive measures rate per day were 3.0 • d: the sum of deaths recorded, the sum of deaths if the date of recovery had started 30 days before the start of the infection process and the density of the preventive action rate per day was 3.0 A final consideration may show how a precaution can work, if. • premature preventive or healing measures. • Preventive or curative measures accompanying the time. could have an early effect. Case 4, Fig. 4.4, Avoided deaths if the date of recovery had started 33 days before the start of the infection process and the density of the preventive action rate per day was 1.5

Jan-20

Jan-20

dI

R (number of vaccinated related to avoided death)

93,863%

dR

0

0,94

50000

100000

150000

200000

250000

Sum of avoided death if date of real recovering had started 30 days before start of infection process

preventio ned frequency vaacinati of death ons

0,00

200,00

400,00

600,00

800,00

1.000,00

1.200,00

1.400,00

1.600,00

457 Avoided deaths if the date of recovery had started 30 days before the start of the infection process and the density of the preventive action rate per day was 3.0

ideal R

g=

number of vaccinated

0

5.000.000

10.000.000

sum of I death avoided

Jan-20

20.000.000

21.04.2020

Density of Sum of preventive frequency measuresrat of death e per day

25.01.2020

3,0

0,0019%

Start of end of percent contemplati contemplati Sum of No on on of Death

Apr-20

15.000.000

Sum of frequency of death

Jul-20

25.000.000

c

Probabilistic SIR -Model page 3

30

0,939

Oct-20

ideal R

Density of preventiv e sum of I death measures I death avoide rate per avoide d day d minus 30 days

Feb-21

30.000.000

Avoided dead

I = Eqbl x Habitants

0,000

Jun-20

487

dS*Eqb

b=

Feb-21

Total deaths

S

25.145.561

SIR-Modell

R (number of vaccinated related to avoided death)

May-21

Sum of avoided death if date of I recoverin g had started 30 days before start of infection process

avoided death 487 457 Sum of avoided avoided death if death if date of date of real recovering recoverin had started g had 30 days started 30 before days start of before infection start of process infection process

Sum of death

0

5

10

15

20

25

Fig. 4.3   a Probabilistic SIR model with density function. b past frequency of infections in the start interval, late current vaccinations. c past frequency, course of the R curve with the beginning of the infection curve. d total deaths, avoided deaths if the recovery date is 0 days would have started before the infection process began. (Image rights: [Copyright of the author])

d

date of sequence

Habitants

Texas

0,0

10,0

20,0

30,0

40,0

50,0

b

Feb-20

Apr-20

Apr-20

frequency of death

Jul-20

Jul-20 May-21

S

Oct-20

Sep-20

Aug-21

I = Eqbl x Habitants

Aug-21

CommandB

Oct-20 Dec-21

450.000 400.000 350.000 300.000 250.000 200.000 150.000 100.000 50.000 0

Dec-21

a

4  Preventive Consideration Using Probabilistic SIR Modeling 33

dI

100,000%

R (number of vaccinated related to avoided death)

Avoided deaths if the date of recovery had started 33 days before the start of the infection process and the density of the preventive action rate per day was 3.0

14384

ideal R

g=

number of vaccinated

Avoided dead

I = Eqbl x Habitants

0,000

Jun-20

14384

dS*Eqb

b=

Feb-21

Total deaths

S

25.145.561

SIR-Modell

Apr-20

frequency of death

0

dR

0

1,00

50000

100000

150000

200000

250000

5.000.000

0,00

5.000,00

10.000,00

15.000,00

20.000,00

25.000,00

14.09.2020

Density of Sum of preventive frequency measuresrat of death e per day

05.02.2020

1,5

0,0572%

Start of end of percent contemplati contemplati Sum of No on on of Death

33

1,000

Density of preventiv e sum of I death measures I death avoide rate per avoide d day d minus 33 days

Sum of avoided death if date of I recoverin g had started 33 days before start of infection process

avoided death

0

200

400

600

800

1000

1200

14384 14384 Sum of avoided avoided death if death if date of date of real recovering recoverin had started g had 33 days started 33 before days start of before infection start of process infection process

Sum of death

Sum of avoided death if date of real recovering had started 33 days before start of infection process

sum of I death avoided

Sum of frequency of death

preventione frequency d of death vaacinations

10.000.000

15.000.000

20.000.000

c

Dec-21

Fig. 4.4   a Probabilistic SIR model with density function. b past frequency of infections in the start interval, late current vaccinations. c past frequency, course of the R curve with the beginning of the infection curve. d total deaths, avoided deaths if the recovery date is 0 days would have started before the infection process began. (Image rights: [Copyright of the author])

d

date of sequence

Habitants

Texas

1.400,0 1.200,0 1.000,0 800,0 600,0 400,0 200,0 0,0

b

Jan-20

Jan-20

0

50.000

100.000

150.000

Jul-20

Jul-20 May-21

ideal R

Jan-20

25.000.000

Apr-20

30.000.000

Jul-20

200.000

Feb-20

Apr-20

R (number of vaccinated related to avoided death)

Oct-20

Sep-20 Aug-21

S

Oct-20

Dec-21

I = Eqbl x Habitants

Oct-20

CommandB

Feb-21

250.000

May-21

300.000

Aug-21

a

34 4  Preventive Consideration Using Probabilistic SIR Modeling

5

The “infection Curve” I (t) is Replaced by the Inclined, Steep Eqb Density Function

Asymmetry dominates nature and its laws. Nothing to be observed and measured appears to have completely symmetrical properties.

5.1 The Power of Symmetry The power of symmetry is present. It even influences our behavior. We are always inclined to give priority to symmetry. It has a profound effect on how we think and act. We are happy to be tempted by the ideal image of symmetry—the scales, Fig. 5.1—to first examine the point of view of the objects for their symmetry properties. Even small deviations from a state of equilibrium can lead to blatant consequences, especially when it comes to moving, dynamic things (> exponential instability). At an upper point, the dead center, small deviations decide whether the next step is backwards or forwards. The cyclist receives support from the pedal opposite, which drives the decision-making process in the right direction—an asymmetrical step. It is a polarizing principle, which forces the one or the other (binomial principle,—coefficient, Fig. 5.2), it offers no further options for decision-making, it offers no intermediate states, it is two-valued as prescribed by the probability function for a binomial distribution.   n k B(k|p, n) = p (1 − p)n−k (5.1) k

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_5

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36

5  The “infection Curve” I (t) is Replaced by the Inclined …

Fig. 5.1   Balance in a state of equilibrium (proof by author). (Image rights: [Copyright of the author])

Fig. 5.2   Symmetrical binomial distribution. (Image rights: [Copyright of the author])

5.2 In Earthly Physics Everything earthly and physical is shaped by the omnipresent force of gravity. It has the same effect on the left pan of a scale as it does on the right. That is why it is in balance. The system is always in equilibrium as long as the weightings on the right and left are the same. We also find the principle of equilibrium in mathematical equations. What is asymmetrical? How do you deal with asymmetry?

6

Events and Findings from the Recent Past

6.1 In Stochastic Systems (Hyperbolic Distribution Forms) Not all forms of distribution lose their validity if the quantity formation for measurements comes from observation areas such as the financial world. However, as the recent past has shown (> stock market crash, “wild coincidence”), the normal distribution cannot be used there. Risks are underestimated when the applied stochastics fail. Practitioners are often “spoiled for choice” when it comes to finding a suitable form of distribution for the subject of the study. Often measurement data show up in a normally distributed form, possibly in a slightly left or right oblique shape. Financial data from stock prices, Fig. 5.3, often show very steep climbs and very wide ends that seem to have nothing in common with those from other sciences. However, these often have clearly symmetrical shapes.

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6  Events and Findings from the Recent Past

Fig. 6.1   Discount certificates. (Proof: https://www.geldanlage-brief.de/images/newsletter/20150909-DiscountZertifikate.PNG) Image rights: [Copyright of the author]

7

Ways Out of Symmetry, Union with Asymmetry

The development of the normal distribution was developed during the lifetime of the author Gauss. A further differentiation with regard to the skew would have lengthened the computation and checking effort for plausibility (checking that the sum of the density distribution converges to 1) by a multiple of the time. Therefore, at different times, different authors have dealt with the problems of the skewed distributions Fig. 7.1.

7.1 Right-Skewed and Left-Skewed Hypothetical Distributions Of course, it is also a matter of detecting those events that are observed beyond the limit values. But where are the limit values to ​​ be set if distributions are not symmetrical, or, even more fatally, the imbalances from sample to sample move from the left around the mean to the right? Most processes are subject to influences that prevent a constant spread of events from being observed. In this respect, the normal distribution must not be used at all. Other areas of science are also struggling with the existing statistical analysis tools. Julia Prahm reports in her diploma thesis “An application via peak over threshold” about the investigation that a combined Pareto distribution is more suitable for adjusting the values ​​than the Gaussian normal distribution. Approaches such as those developed by mathematicians who saw a specific reason to supplement or replace the normal distribution can help.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_7

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7  Ways Out of Symmetry, Union with Asymmetry

Fig. 7.1   Skewed distributions. (Proof of author). [Author’s copyright]

Decisive for the replacement of the normal distribution by another probability density function is the appearance of so-called “thick tails”, a “heavy tail” distribution Fig. 7.2, as it is revealed when series of events tend to deliver measured values ​​that exceed the permitted number of the target exceed. The answer to this lies in developing a thought experiment by the author: • corresponds to the symmetrical scattering, Fig. 7.3, to the distribution of spheres on a symmetrical Galton board • so the inclined scattering corresponds to the principle of an inclined Galton board, Fig. 7.4.

Fig. 7.2   Normal distribution, skewed distributions, heavy tail distributions. (Proof of author). [Author’s copyright]

7.1  Right-Skewed and Left-Skewed Hypothetical Distributions

41

Fig. 7.3   Normal distribution. (Proof http://www.biosensor-physik.de/biosensor/imk_1747_m. jpg). [Author's copyright]

Fig. 7.4   Galton board. (Proof https://upload. wikimedia.org/wikipedia/ commons/1/1a/Galton_ box_3.jpg). [Author's copyright]

42

7  Ways Out of Symmetry, Union with Asymmetry

Considered analogously to this: The maximum of any symmetrical expression of a function lies at an expected value, which should correspond to the mean value of a frequency distribution in a large approximation, as is the case with the Gaussian normal distribution. A skew is recognizable if you notice different spreads on the left and right. In the same way, the maximum moves to the left or right and is then called the mode value. Exactly this migration, it can assume extreme values, leads to an imbalance that should give rise to the formulation and naming of the new probability density function. The Gaussian normal distribution is determined by two parameters, as shown above - there is now another, the equilibrium parameter equiweight ρ. It results from the thought that the skewness of a distribution is based on the weighting point G and the proportions Gl / Gr, which takes into account the variable properties inherent in the events Fig. 7.5. This additional parameter ρ

r: = (1 − (ρ%(x − µ)))

Fig. 7.5   Equilibrium parameter equiweight ρ. (Proof of author). [Author’s copyright]

(7.1)

7.1  Right-Skewed and Left-Skewed Hypothetical Distributions

43

opens up the new probability function to science. (x−m2 )

1 2s2 Eqb(x; m, s, r) =  e− 1−r(x−m) 2 2π s (1 − r(x − m))

(7.2 (Proof author))

beziehungsweise

(x − µ2 ) 1  exp − 2 2 2σ (1 − ρ(x − µ)) 2π σ (1 − ρ(x − µ))

(7.3 (Proof of Andrej Depperschmidt))

8

Random Scatter Areas of the NV and the Eqb

The differences between the probability density distributions NV and Eqb and their random scatter ranges are considerable. This results from the application of the third parameter r or ρ. The value range of the standard normal distribution (see Fig. 8.1) is in the negative as well as in the positive range with respect to an expected value μ in open intervals in the negative as well as in the positive area. The integral above this is 1. The respective density sections are represented by a multiple of σ. Here, μ is derived from the 2nd derivative (intersection with the x -axis) of the NV function. The range between ±3μ gives a probability density of 99.73% The graphic shows a symmetrical picture with the intersections of the turning points and the 2nd derivative of the function (see Fig. 8.2). A differentiated picture shows the probability density distribution of the Eqb. The 2nd derivative of the function (see Fig. 8.3) defines the intersection points with the x-axis and thus the different σ-positions with respect to an expected value. This means that the random scatter ranges of the Eqb are also different from those of the NV (see Fig. 8.4). Obviously, they are closely related to the described inclination of the control r or with effects on the “tails”, the ends of the random scatter ranges. The new Eqb function extends the perspective to include the skewness of the distributions with the following consequences for the random scatter ranges and thus the exceeding of the previously valid limit values. This case is shown in a table using an example, since the total density of the NV and the multivariate Eqb can turn out differently depending on the different values for ​​ the parameters μ, σ, ρ (see Fig. 8.5). The parameter values shown ​​ in the graphic below apply. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_8

45

46

8  Random Scatter Areas of the NV and the Eqb

Fig. 8.1   Normal distribution blue, 1st derivative purple, 2nd derivative light blue. (Proof of author). [Author's copyright]

Fig. 8.2   Derivation normal distribution blue, mean normal distribution green, position μ left red, position μ right black. (Proof of author). [Author's copyright]

8  Random Scatter Areas of the NV and the Eqb

47

Fig. 8.3   2nd derivative Eqb blue, mean normal distribution green, position σ left red, position σ right black. (Proof of author). [Author's copyright]

Fig. 8.4   2. Average normal distribution green, position 3 σ left red, position 3 σ right black, inclination ρ light blue, different “tail” circle. (Proof of author). [Author's copyright]

48

8  Random Scatter Areas of the NV and the Eqb

Fig. 8.5   2. Average normal distribution green, position 3 σ left red, position 3 σ right black, slope ρ light blue, parameter values. (Proof of author). [Author's copyright]

9

Presentation of the Equibalance Distribution, Eqb

Its probability density Fig. 9.1 remains at 1 in “misalignments” and includes the normal distribution in the symmetrical case. However, it also opens up extreme imbalances with a permanent probability density 1, Fig. 9.2.

r = 0,5

r = −1

9.1 Presentation of Density Symmetrical forms of appearance, as they are evident in almost all specialist areas, influence the objective recording of facts in such a way that they are often used as a basis for judgment. The process world also likes to use simple, memorable graphic representations. The symmetrical normal distribution density developed by Gauss is a good example of this. On the other hand, there are numerous asymmetrical process positions for which specially adapted density functions were developed. The newly developed equibalance distribution Eqb is intended to remedy the situation by replacing as many of the specially adapted density functions as possible via a skew parameter. The newly developed formula of a right or left skewed distribution, the “Equibalancedistribution Eqb” for the analysis of measured values, is trend-setting for the quality-effective monitoring and action management. The symmetrical normal distribution used to describe this is still contained in the Eqb as a simplified special case.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_9

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9  Presentation of the Equibalance Distribution, Eqb

Fig. 9.1   Eqb density distribution (proof author). (Author’s copyright)

Fig. 9.2   2. Eqb density distributions, normal distribution. (Proof of author) [Author’s copyright]

However, due to the mutual influence of the parameters on the values ​​that the Eqb delivers, it will not be possible to estimate individual parameters with conventional statistics, because they all already occur in the expected value. The mathematical function Equibalancedistribution Eqb is examined:

9.1  Presentation of Density

51

(x − µ)2 1  exp − 2 2 2σ (1 − ρ(x − µ)) 2π σ (1 − ρ(x − µ))

(9.1)

A family of distributions on ℝ Andrej Depperschmidt und Marcus Hellwig August 2016

Summary We consider a parametric of functions on ℝ that contain the densities of normal distributions. We show that all the densities in this family are themselves densities of distributions. 1 Family of Densities Für r ∈ R, µ ∈ R und σ 2 > 0 definiert durch

fρ;µ,σ 2 (x) =



betrachten wir die Funktionen fρ;µ,σ 2 : R → R

1

√

2π σ 2 (1−ρ(x−µ))

0

  (x−µ)2 exp − 2σ 2 (1−ρ(x−µ)) : x < 1/ρ + µ

: x ≥ 1/ρ + µ.

In case ρ = 0 f ρ; µ, σ matches with the normal distribution with the parameters μ und σ2. (We reach for constancy 1/0 as ∞. In analyzing the function, we will initially limit ourselves to the case μ = 0 und σ2 = 1 and put f (ρ) = f (ρ; 0, 1).



 2

Theorem 1 the family {f ρ : ρ ε R} is a family of densities based on probability distributions R. (i) For ρ = 0 the distribution is the standard normal distribution (ii) For ρ > 0 the distribution is given by      2 x 2 x + e2/ρ � √1−ρx : x < 1/ρ, − ρ √1−ρx � √1−ρx + Fρ (x) = 1 : x ≥ 1/ρ. (iii) For ρ  1/ρ.

(9.2)

(9.3)

52

9  Presentation of the Equibalance Distribution, Eqb

Proof: For each ρ ∈ ℝ is f (ρ) not negativ. For ρ > 0 f (ρ) on the interval is defined as (-∞,1/ρ). Fot ρ  0. It applies:      2 x x 2 lim Fρ+ (x) = lim � √ + e2/ρ � √ − √ xր1/ρ xր1/ρ 1 − ρx 1 − ρx ρ 1 − ρx 2

=�(∞) + e2/ρ �(−∞) = 1 + 0. So F + ρ is continuous in 1/ρ and thus to the whole ℝ. Furthermore applies lim F + ρ (x) = 0. xր−∞

By deriving on x it is easy to convince oneself that F + ρ is the distribution function of a probability distribution whose density is given by (ρ). It is true     d + ρx x 1 √ ϕ Fρ (x) = + dx 2(1 − ρx)3/2 (1 − ρx)1/2 1 − ρx   ρx x 2 2 − e2/r ϕ √ − √ 3/2 2(1 − ρx) 1 − ρx ρ 1 − ρx      x x ρx 2 2/ρ 2 √ √ √ − e ϕ =fρ (x) + ϕ − 2(1 − ρx)3/2 1 − ρx 1 − ρx ρ 1 − ρx =fρ (x).

Here we used that follows because of

f ρ(x) =

√ 1 ϕ 1−ρx



√ x 1−ρx

 . The last equation in the display

9.1  Presentation of Density

53

    x 2 x 2 ϕ √ −e2/ρ ϕ √ − √ 1 − ρx 1 − ρx ρ 1 − ρx       2 x 2 x2 1 1 4x 4 exp − − exp − =√ − + 2(1 − ρx) ρ2 2 1 − ρx ρ(1 − ρx) ρ 2 (1 − ρx) 2π     2 2 2x 1 2 x 1 − exp + =√ − 2 exp − 2(1 − ρx) ρ2 ρ(1 − ρx) ρ (1 − ρx) 2π     x2 1 exp − =√ 1 − e0 = 0. 2(1 − ρx) 2π

(iii)

Let us now consider the case ρ  0. F− ρ Lemma 1.1 For every zero sequence applies (ρn ): n+∞  3 2 e1/(ρn ) � −1/|ρn | 2 −−−→ 0. Proof: We use the following estimate

1 − �(x) ≤

φ(x) x

f¨ur alle x > 0

With that we get:     1 2 2 exp e1/ρn � −1/|ρn |3/2 = e1/ρn 1 − � 1/|ρn |3/2 ≤ |ρn |3/2 √ 2π



1 1 − ρn2 2|ρn |3

n+∞ −−−→

0.

− Lemma 1.2 For ρ → 0 F + ρ und F ρ converge weakly against Φ. In other words:

lim Fρ+ (x) = lim Fρ− (x) = �(x) f¨ur alle x ∈ R.

ρր0

ρր0

Proof: Let x ∈ ℝ and let (ρn ) a sequence with (ρn ) > 0 for all n and (ρn ) → 0 für n → ∞. For sufficiently large then n is x  0.00109009 for 10 years Per year: 0.00109009 Per 100,000 people: 109 This means. Every year 121 new cases arise per 100,000.

The cumulative incidence can be used to estimate the likelihood that one person in the considered group will develop the disease. The above work can show that the development of an incidence can be completed through a probabilistic view-

Fig. 9.3   2. Course of the incidence in relation to and as a function of the exponential development. (Proof of author) [Author’s copyright]

9.4  Incidence from a probabilistic point of view

57

point when the number of cases increases exponentially. The solitary exponential view can also be completed by a density function view, which provides a more detailed view by taking into account the number of cases in a few days in history that give a logarithm value as an exponential forecast base. Based on the development of the data corresponding to the probability calculation, the following expected value for the course of the incidence in relation to and as a function of the exponential development, Fig. 9.3, can be derived from a week for a prognosis. For this purpose, all values ​​of the density equation are determined from the parameter values modal ​​ value, standard deviation, skewness and kurtosis of the previous interval for the future one.

Infection Management in Relation to the Course of Incidence

10

Infection management as such can be viewed as a system that—apart from the infection system—COVID—can map some other systems. Infections are part of a pathological process that is generally regarded as “harmful”. The term “infection” translates “inward”, as an act of surrender. In connection with the present work, this is the transfer of pathogens. In a general context, it is a fundamental formulation that applies to many contexts. In this respect, infection management is the “handling of handovers”. That under respect of views of: • temporal • cost-effective • qualitative • personnel • material There are differences between the types of process, e.g. infectious diseases of living beings, computer virus infestation, deliberate dissemination of false reports on damage, neglect of education systems …, i.e. a number of processes that the probabilistic SIR model can provide information on, if the statistical basis for it is given. This applies to processes whose start conditions and the subsequent sequence are essentially based on experience that can be used as a basis for a comparison. This was not possible with the COVID infection, since no comparable process can serve as a basis from the past, which is meant from the fundamental difference that different qualities were revealed to the COVID process, which only

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6_10

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10  Infection Management in Relation to the Course of Incidence

appeared in the course of time the occurrence of variants that cannot be qualified as long as their behavior over time and measurements is not known. Everything in the course of the experience with infection management, the “handling of handovers”, be it health-damaging viruses or data-destroying viruses or personality-damaging fake news or inadequate support and equipment of educational institutions is based on this. In all cases, the processes can be divided into phases, which can be outlined as follows, in the special case for the infectious disease of living beings.

10.1 Phase Structure The phases are based on the probabilistic SIR model. Phase S Phase S depending on Phase I = Eqb × Habitants based on the frequency distribution of the test series. Quote: (http://www.medizinfo.de/infktionen/allgemeines/phasen.shtml) Beginning of the infection process An infectious disease runs in several phases from the beginning of the infection. Invasion phase: This phase describes the actual contagion. The pathogen enters the body. However, it initially remains inactive and does not multiply. Incubation phase: Depending on the pathogen, it will multiply after hours or days in which the pathogen has got used to its new environment. But there are no complaints yet. In many infectious diseases, at the end of this phase there is a massive, often almost explosive, multiplication of the pathogens. The incubation period describes the period from infection to the onset of the disease—precisely considered as the first two phases of an infection. It is of different length depending on the infectious disease, e.g. B. Diphtheria 1 to 7 days, with rubella 14 to 21 days, but with AIDS the incubation period can last more than 10 years. Disease phase: The first symptoms are now showing. Depending on the severity of the infection, there may be slight complaints, e.g. B. mild headache or hoarseness or local redness. But I can also develop serious symptoms such as B. high fever, dizziness and weakness. Overcoming phase: After an infectious disease has been overcome, all pathogens are destroyed in this phase.

10.1  Phase Structure

61

Phase I. Phase I develops over the entire duration of the process. It is determined by the test series, which provide information about a predicted future of the process depending on the phases S and R via their results = frequency distribution and probability density. Phase R. Experience shows that a recovery process (R) only takes place after time intervals. • the identification of the cause, the pathogen – based on empirical values – based on research results • the development of prevention and treatment measures – Behaviors – Protection – medication • a test phase of avoidance or treatment measures – Medication by means of sample tests on test subjects – Protection through personal protective equipment – Behaviors about contact restrictions • the production of prevention and treatment measures – accompanying the determination of suitability from the results of the test phase Determination of the quantity structures Infrastructure (test center, information center, hospitals) Personnel requirements (ambulance, emergency services, nursing staff, medical staff, pathology, aftercare staff, psychologists) Material requirements (medication, protective equipment) • the provision of the elements of the quantity structures according to the checklists: Quote: State Health Office Baden-Württemberg in the regional council of Stuttgart; Manual for operational pandemic planning, second expanded and updated edition, December 2010 • Pre-pandemic action – V1 Operational and personnel planning – V2 Procurement of medical and hygiene products – V3 information and communication – V4 Preparatory Medical Planning • Measures during the pandemic – P1 Maintaining minimum operation – P2 Organizational measures for the staff

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10  Infection Management in Relation to the Course of Incidence

– P3 External information – P4 Medical measures – P5 measures for relatives and foreign employees • Post-pandemic measures • N1 return to normalder Beobachtung der Wirkung und Rückkoppelung der Ergebnisse aus den Phasen auf Prozesskorrekturen im Verlauf. At the beginning of the “COVID process, 1st wave”, the planning of the phases was not possible because none. Experience in the phases mentioned was available, but success was achieved in terms of positive development as soon as the following were used: • Behaviors • Protection • medication The ideal, but not feasible recovery process (R) starts right at the beginning of the disease process, Fig. 10.1).

10.1.1 Phase scheduling On the basis of a time sequence, a phase schedule Fig. 10.2 can be created, which can be used to plan and monitor the process. It can be seen that the first few weeks have a decisive influence on the effects of the activities undertaken on the development of an infection process.

Fig. 10.1   Probabilistic SIR model, but unrealizable recovery process (R) for the COVID process. ((Proof of author)[Author’s copyright])

y y y y

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Fig. 10.2   Phase planning as a sample for the first 20 weeks after identification. ((Proof of author)[Author’s copyright])

Phase S Beginning of the infecon process Invasion phase, infecon, pathogens do not mulply. Incubaon phase: explosive mulplicaon of the pathogens. Disease phase: first symptoms, complaints Phase I. Provision of infrastructure (test center, informaon center, hospitals) Carry out test series Prevenve planning for personnel requirements (paent transport, emergency services, nursing staff, medical staff, pathology, a‚ercare staff, psychologists) Informaon events for the populaon Phase R. Idenficaon of the cause, the pathogen Carry out test series Collecon of empirical values Collecon of research results Connuaon of informaon events for the populaon Development of prevenon and treatment measures Behaviors protecon medicaon Test phase of avoidance and treatment measures Medicaon by means of sample tests on test subjects Protecon through personal protecve equipment Behaviors about contact restricons Producon of prevenve and treatment measures, procurement of personnel Determinaon of the quanty structures Infrastructure (test center, informaon center, hospitals) Personnel requirements (ambulance, emergency services, nursing staff, medical staff, pathology, a‚ercare staff, psychologists) Material requirements (medicaon, protecve equipment) Pre-pandemic acon V1 Operaonal and personnel planning V2 Procurement of medical and hygiene products V3 informaon and communicaon V4 Preparatory Medical Planning Measures during the pandemic Carry out test series P1 Maintaining minimum operaon P2 Organizaonal measures for the staff P3 External informaon P4 Medical measures P5 measures for relaves and foreign employees Post-pandemic measures N1 return to normal Observaon of the effect and feedback of the results from the phases on process correcons in the course.

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10  Infection Management in Relation to the Course of Incidence

Even if the infection process cannot be fully recognized in a timely context, activities carried out quickly can result in the number of deaths and post-infection cases being lower than delaying the execution of the activities. A probabilistic SIR model from a real development provides information on how a phase R process is changed if a subsequent population can be protected by identifying an outbreak in a starting population: Timeline of the COVID-19 pandemic in the United States (2020)—Wikipedia “On January 3, CDC Director Robert Redfield was notified by a counterpart in China that a “mysterious respiratory illness was spreading in Wuhan [China]”; The superimposition of the frequency distribution by the R curve in Fig. 10.3 shows that—even if an infection process has already started—a fatal course for many people could have been avoided if the activities were carried out according to the phase schedule.

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Fig. 10.3   Sum of the dead, sum of the dead that could have been avoided. ((Proof of author)[Author’s copyright])

Summary

In the previous chapters it is explained that infection processes can be systematically analyzed using the known methods from statistics, stochastics and probability theory. The case of a biological infection was taken into account in the present study. The same way can be applied to other areas, this means any infection process, including the digital one. The matter to be considered equally in all areas is the earliest possible detection of an “attack” on the organism to be protected, be it a biological, digital or organizational structure, of any kind, the frequency of which can increase rapidly at the beginning if this is not possible early and moderately recognized or even prevented. Infection risks remain active as long as they are described, observed and measured, as long as they are not recognized and without resistance. The graphics show the multi-exponential multiplication in which infections develop, decrease and flare up again if the emission pathways are not continuously interrupted. The modeling of an infection process using the S.I.R. methods described can benefit from the fact that the parameters for skewness and kurtosis taken into account in the equi-balance distribution take into account the actual skewness and steepness of an exponential expression of the frequency distribution, a good approximation of the asymmetrical course over time.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6

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References

„Phasen einer Infektionskrankheit“, http://www.medizinfo.de/infektionen/allgemeines/ phasen.shtml Landesgesundheitsamt Baden-Württemberg im Regierungspräsidium Stuttgart; „Handbuch Betriebliche Pandemieplanung“, zweite erweiterte und aktualisierte Auflage, Dezember 2010 Hellwig, Marcus, Springer Verlag 2017, „Der dritte Parameter und die asymmetrische Varianz, Philosophie und mathematisches Konstrukt der Equibalancedistribution“, Hellwig, Marcus, Springer Verlag 2020 „Partikelemissionskonzept und probabilistische Betrachtung der Entwicklung von Infektionen in Systemen Dynamik von Logarithmus und Exponent im Infektionsprozess, Perkolationseffekte“ Datenerhebungen Fallzahlen aus https://github.com/nytimes/covid-19-data/blob/master/ us-states.csv Datenerhebungen Genesene aus https://raw.githubusercontent.com/owid/covid-19-data/ master/public/data/vaccinations/us_state_vaccinations.csv

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 M. Hellwig, SIR - Model Supported by a New Density, essentials, https://doi.org/10.1007/978-3-031-05273-6

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