👶🏾 🏛️ 🖌️ Typical COVID-19 Epidemic Development Model 🤙🏼 🎷 🙋🏽

This publication is about the simplest mathematical model of the epidemic, with only two parameters, which is addressed to all those who want to know what will surround us and our neighbors tomorrow, in a week, in a month or two. I am grateful in advance to those who, having enjoyed the model, expose its vices and help make it better.

Introduction. A few words about half-naked numbers

Assessing the development of the epidemic, it should be remembered that its development is determined by the number of carriers of the virus, and manifests itself through the number of confirmed diagnoses. These quantities are not identical. We can only assume that the number of carriers of the virus in an infected society is directly proportional to the number of confirmed diagnoses. The proportionality coefficients can vary both in the autonomous regions of one country, and in different countries.

Sources of official and unofficial information on lung disease, which WHO named COVID-19 [5], consistently report confirmed cases where people have been tested positively for Sars-CoV-2. But these figures do not say anything about how many people in different countries actually carry or carried the virus, as well as how much sick or so far COVID-19 has been caused by the virus.

The truth is that the world community (so far) does not know how high the dark number of diseases is. The problem is that testing for Sars-CoV-2 is similar to testing for alcohol intoxication familiar to drivers. Testing takes place in two stages. At the first, a sample is taken, processed according to the standard procedure, and the concentration in the sample of a certain witness substance is determined, the presence of which is most likely associated with Sars-CoV-2 infection, or, as in the case of the driver, is associated with intoxication. Ensuring this stage lends itself to unification and even standardization, as in the case of "ppm." WHO provides this step by accrediting reference laboratories around the world [5]. The Russian Gosstandart also issues a certificate of verification of breathalyzers. But in the next step,At the stage of interpreting the result of measuring the concentration of the witness substance, the area of responsibility of the national authorities begins. The authorities, as well as with the ppm, establish national rules for confirming the diagnosis of COVID-19 on the basis of exceeding the threshold level of concentration of the witness substance in the sample set by the authorities. There is no agreement between countries regarding such a threshold level, and there has also been no international comparison of confirmation of diagnoses.There is no agreement between countries regarding such a threshold level, and there has also been no international comparison of confirmation of diagnoses.There is no agreement between countries regarding such a threshold level, and there has also been no international comparison of confirmation of diagnoses.

From this we can conclude that the number of carriers of the virus can be estimated rough at best. For the same reason, it makes no sense to compare the absolute numbers of confirmed diagnoses in different countries and draw conclusions from the results of dividing the number of diagnoses by the number of citizens.

Please always keep these uncertainties in mind when interpreting half-naked numbers.

Principles of the epidemic

According to the fundamental course of differential and integral calculus G.M. Fichtenholtz, trying to describe the development of the COVID-19 pandemic, we describe an ordinary variant - or a function of time, which reflects the sequence of daily values of the characteristics of the development of the epidemic (or pandemic - as you like). We get estimates of the value of this function in the form of a random sample - those numbers that daily bring reports on the number of new confirmed diagnoses, the number of dead or recovered. Our task is to find the law of function change (or Fichtenholtz variants) with time (day after day) according to the results of random sampling. The purpose of this search is to find a reliable basis for forecasting the development of the epidemic. And the scope of the results is the adoption of an informed decision, for example, aboutwhen it is already possible to go with a friend to Ples on the Volga, or when, finally, the authorities can remove quarantine measures in the country and invite foreign colleagues to the symposium. Unfortunately, so much depends on how the epidemic develops in our lives.

The function of interest to us is the dependence of the total number of Sars-CoV-2 carriers, or the number of confirmed COVID-19 diagnoses proportional to it over time. This is a non-decreasing function: the current value of the function must be greater than or equal to what it was yesterday. Such a function is a reflection of the most general law of the development of the epidemic [5, 6]. It is based on three principles.

The principle of proportionality. The daily increase in the number of carriers of the virus in an infected society is proportional to the total number of carriers of the virus that arose over the entire previous period of the development of the epidemic.

The principle of independence. The relative daily increase in the number of carriers of the virus does not depend on the total number of carriers of the virus that arose over the entire previous time of the development of the epidemic.

The mathematical expression of the law of the development of an epidemic that meets these principles is elementary:

N (n + 1) = N (n) + f (n) × N (n) = N (n) [1+ f (n)],

where N ( n) is the number of carriers of the virus on the n-th day of the development of the epidemic, for example, today, N (n + 1) is the number of carriers of the virus on n + 1-day of the development of the epidemic, that is, tomorrow, and f (n) is the relative daily an increase in the number of confirmed diagnoses, which is a "factor in the reproduction of the epidemic."

The function f (n), as the “epidemic reproduction factor” in an infected society, which I described in [1,2], is the main indicator of the spread of the virus during an epidemic. The reproduction factor completely determines the development of the epidemic and is a complete analogue of the bank deposit rate, which in turn is a factor in the reproduction of money on a fixed deposit:

f(n) , , , , ;
f(n) , , , , ;
f(n) , , , , , ;
f(n) , . .

Like money on a fixed-term bank deposit, the total number of carriers of the virus in an infected society, which we estimate as the number of carriers of the virus, grows as

N (n + 1) = N (n) [1+ f (n)],
N (n + 2 ) = N (n) [1+ f (n)] × [1+ f (n + 1)],
N (n + 3) = N (n) [1+ f (n)] × [1+ f (n + 1)] × [1+ f (n + 2)],

and so on.

In terms of managing the development of the epidemic, the type of function of the reproduction of the epidemic is determined by the action of several factors [1-7]:

Random changes in the virus due to mutations accompanying the reproduction of the virus in the body of the virus carrier;
By the actions of doctors who, through testing, determine the carriers of the virus in an infected society and do not allow further transmission of the virus, limiting the contact of the established carrier with others;
By the actions of authorities, which, through quarantine measures, limit the possibility of contact between an unidentified carrier and others.

Bearing in mind the randomness and independence of the processes of mutation formation, virus transmission during contacts, it should be expected that under the conditions of constant measures to control the epidemic on the part of physicians and authorities, a typical epidemic reproduction function will look like a decreasing exponential function of time with constant parameters:

f (n) = B × exp (-Ax)

The development of any infectious disease is characterized by a latent (latent) period of its development from the transmission of the virus by its carrier to a healthy person until the symptoms of the disease manifest and subsequent confirmation of the diagnosis in the new virus carrier. Therefore, a change in the authorities' measures to control the epidemic or sabotage of such measures by members of an infected society will not manifest itself in a change in the number of carriers of the virus instantly. These actions and the visible changes caused by them, for example, the number of diagnoses or deaths, will be separated by a period of time approximately equal to the latent period of the development of the disease. This implies the third principle for the development of a controlled epidemic.

The principle of aftereffect. The relative daily increase in the number of carriers of the virus with delay reflects measures to control the epidemic or their sabotage.

A good example of this principle is the remote control of the rover. The operator on Earth presses the brake pedal, and the device on Mars stops only after 3 minutes due to the finite speed of propagation of radio waves.

Typical epidemic

A typical epidemic is an epidemic whose development is subject to the principle of proportionality, the principle of independence and the principle of aftereffect, and the reproduction function has the form of a decreasing exponential function of time with constant parameters.

In the future, we will consider the number of confirmed diagnoses as a characteristic of the COVID-19 epidemic, proportional to the number of carriers of the Sars-CoV-2 virus. WHO statistical reports [8] on the development of the COVID-19 pandemic show that in all countries without exception there are quite long periods of typical development in the history of the COVID-19 epidemic. With a typical development of the epidemic, in order to estimate the number of confirmed diagnoses in a month, it is enough to know how many have already been registered to date and the parameters of the function f (n) for the next 30 days.

The growth in the number of confirmed diagnoses of ND (n) in the Chinese province of Hubei, the capital of which is the famous Wuhan, as the epidemic develops, is shown in Fig. 1. The dependence f (n), referred to as the relative daily increase in the number of confirmed diagnoses, is also presented there. At the first stage of the development of the epidemic, the rate of “reproduction” of diagnoses in Hubei grew rapidly, but from 10 days f (n) begins to behave like a falling exponential function with constant parameters, and the development of the epidemic becomes typical.

In the period of 10-40 days, the parameters of the exponential function f (n) were equal to: A = 0.14604 1 / day. and B = 1.234. Moreover, on February 13 there was a crisis in the system for confirming diagnoses and their number in one day increased by almost 17 thousand [1,2]. At the same time, such an injection of new diagnoses did not lead to a change in the parameters of the reproduction function. After 40 days, at the third, final stage of the development of the epidemic, A = 0.324 1 / day. and B = 3.273.

Changing the parameters of the reproduction function f (n) was a consequence of the actions of the authorities aimed at strengthening quarantine measures that impede the reproduction and transmission of the virus. Moreover, according to the principle of aftereffect, these actions and subsequent changes in the function of reproduction did not occur simultaneously. They are separated by a time interval of approximately 5-15 days.

Around March 10, the relative daily increase in the number of confirmed diagnoses fell below the level of 0.0001 1 / day. The Chinese authorities began to lift quarantine restrictions and on March 15 officially announced that the COVID-19 epidemic in Hubei was over.

Fig. 1. The development of the COVID-19 epidemic in Hubei Province (China)

Switzerland is another example of a typical epidemic development. In Switzerland (see Fig. 2), the dependence of the reproduction function f (n) on time remains unchanged from March 16, on the tenth day after the number of confirmed diagnoses exceeded the critical level of 100 [1,2]. The development of the COVID-19 epidemic is typical. The parameters of the epidemic reproduction function: A = 0.0910 and B = 0.6443.

Fig. 2. The development of the COVID-19 epidemic in Switzerland The

steady decline in the reproduction rate of the epidemic in Switzerland reflects the efforts of the authorities and residents to combat the epidemic. Tough measures of intervention by the Chinese authorities, with full submission to these measures by the residents, made it possible to ensure a quick end to the epidemic. The liberal quarantine measures of the authorities, together with the full support of these measures by the residents, allowed the authorities and residents of Switzerland to stop the epidemic, but only by the 80th day of its development can the results be achieved that Hubei Province reached 30 days earlier, by the 50th day.

Atypical epidemic

There are countries in which the actions of the authorities were ineffective, and their sabotage by those whom the authorities should protect, completely reduced these measures to nothing. Such countries provide examples of the atypical development of the COVID-19 epidemic.

Fig. 3. The development of the COVID-19 epidemic in Iran

In fig. Figure 3 presents a picture of the development of the COVID-19 epidemic in Iran. Strong secular Iranian authorities in the first month of the development of the epidemic have made great strides in managing the epidemic. But then two events occurred that led to disastrous consequences. The consequence of the first event is a surge in f (n) on March 24-31. If there wasn’t a surge, in Iran the number of confirmed diagnoses for today (May 25) would be about 30 thousand instead of 135 thousand. What was the impetus that caused the surge in the reproduction of the epidemic? Such a push is an act of sabotage - a gross violation of quarantine measures, for example, public festivities in the middle of a period of self-isolation. As shown earlier, the time between the act of sabotage and the surge in the number of requests for medical care should be approximately 5-15 days. What are the events of the mass violation of quarantine measures,associated with crowding occurred in Iran 5-15 days before March 24-31? There are several such events in the calendar of secular and religious holidays in Iran: March 8, Father's Day in Muslim countries and the birthday of Imam Ali; March 10 - Day of traditional Kurdish clothes; March 21 - Navruz and March 22 - Israv al-Miraj. Father's Day and Navruz are worthy occasions to get together, chat (kisses are an indispensable attribute of a cordial meeting in the East) and contribute to the reproduction of carriers of the virus. Following these events, the secular authorities of Iran again established tight control over the development of the epidemic, but on May 1, it became clear that control of the epidemic became ineffective. On April 18, Iran celebrated Army Day, which was preceded on April 11 by the lifting of many quarantine restrictions [9].The result is a partial loss of control of the epidemic and an uncontrolled increase in the number of confirmed diagnoses in May.
Another example of the loss of control over the development of the epidemic is the situation in the Austrian capital, the beautiful Vienna. In fig. 4 presents a picture of the development of the COVID-19 epidemic in Vienna.

Fig. 4. The development of the COVID-19 epidemic in Vienna

According to Viennese residents, quarantine measures in the city boiled down to the termination of offices and enterprises, the closure of city parks, museums and everything where people could communicate voluntarily or involuntarily, including grocery stores. Those that continued to work were crowded. On the streets, the police did not allow the appearance of only elderly residents of the city. Everyone else had to take care of their own security themselves. And people went for walks in parks and forests that do not have gates and fences, which are many in Vienna and the surrounding area. As a result, there is a very large variation of f (n) indicating a poor epidemic management. After April 14 (Catholic Easter), something similar to the events in Iran after May 1 began in Vienna. The function of reproduction of the epidemic has practically ceased to change. Her sample is throwing around 0.012 1 / day.As a result, there is a continuous increase in the number of confirmed diagnoses over the past 50 days, similar to Iranian. In addition, there was a crisis in the medical care system: after April 4, the number of patients fell at around 1,500, apparently this is the maximum for Vienna, and in two days April 16-17, there were less than 1,000 patients in Vienna. This thousand was recognized as recovered. As a result, over the past 40 days the number of patients in Vienna has not changed and remains at the level of 450-500 people. People are cured, but they are immediately replaced by new ones and nothing changes. Nothing like this happened in the remaining 8 federal Austrian lands [6]. If the authorities retained control of the epidemic in late April - early May, the situation could develop as the red dotted line shows.after April 4, the number of patients fell at around 1,500, apparently this is the maximum for Vienna, and in two days April 16-17, there were less than 1,000 patients in Vienna. This thousand was recognized as recovered. As a result, over the past 40 days the number of patients in Vienna has not changed and remains at the level of 450-500 people. People are cured, but they are immediately replaced by new ones and nothing changes. Nothing like this happened in the remaining 8 federal Austrian lands [6]. If the authorities retained control of the epidemic in late April - early May, the situation could develop as the red dotted line shows.after April 4, the number of patients fell at around 1,500, apparently this is the maximum for Vienna, and in two days April 16-17, there were less than 1,000 patients in Vienna. This thousand was recognized as recovered. As a result, over the past 40 days the number of patients in Vienna has not changed and remains at the level of 450-500 people. People are cured, but they are immediately replaced by new ones and nothing changes. Nothing like this happened in the remaining 8 federal Austrian lands [6]. If the authorities retained control of the epidemic in late April - early May, the situation could develop as the red dotted line shows.As a result, over the past 40 days the number of patients in Vienna has not changed and remains at the level of 450-500 people. People are cured, but they are immediately replaced by new ones and nothing changes. Nothing like this happened in the remaining 8 federal Austrian lands [6]. If the authorities retained control of the epidemic in late April - early May, the situation could develop as the red dotted line shows.As a result, over the past 40 days the number of patients in Vienna has not changed and remains at the level of 450-500 people. People are cured, but they are immediately replaced by new ones and nothing changes. Nothing like this happened in the remaining 8 federal Austrian lands [6]. If the authorities retained control of the epidemic in late April - early May, the situation could develop as the red dotted line shows.

In the history of the development of the COVID-19 epidemic in Moscow, as in Russia as a whole, sabotage also occurred. In fig. 5. The development of the COVID-19 epidemic in Moscow is shown.

Fig. 5. The development of the COVID-19 epidemic in Moscow

The impetus that led to an upsurge in the function of reproducing the epidemic on April 30 - May 3 was most likely Easter events, together with congestion at the entrance of the Moscow Metro stations on Monday, April 15. The second event was the result of a restrictive regime for the use of public transport introduced by the Moscow government. Moscow authorities imposed restrictions on the use of public transport in two stages. On March 25, reduced fare in transport for seniors and students was canceled. On April 11, it was decided to introduce access control for transport from April 15, immediately after the Easter events. The congestion of people in churches and at the entrance to the metro was an impetus (sabotage?), Which led to a surge in the function of reproduction of the epidemic in early May. The fact that the same surge occurred in all of Russia,the version about the involvement of Easter events in it is most likely.

Parameters of a typical epidemic development model

Since the development of a typical epidemic is predictable, I consider it necessary to make available to the reader the parameters of the epidemic reproduction function in a number of countries and cities. They are shown in the table.

The existing experience of using the typical epidemic development model COVID-19 gives a forecast of the number of confirmed diagnoses with an uncertainty of about 10-15%. According to existing practice, with a decrease in f (n) below 0.001 1 / day, the authorities can decide to weaken or completely withdraw quarantine measures. In this area, the typical conditions of the epidemic will not be respected and predictive estimates of the number of confirmed diagnoses will be underestimated. This should be taken into account when evaluating the forecast for countries in which the epidemic is nearing an end.

Table. Parameters of the function of reproduction of the number of confirmed diagnoses during the development of a typical COVID-19 epidemic.

Conclusion

In mid-April, Moscow and Russia stepped on the first rake that Iran had stepped on a month earlier - in mid-March. Vienna stepped on the second Iranian rake. If, unlike Vienna, we do not step on the second Iranian rake, then, as the estimates in fig. 5, holding the Victory Parade, appointed by the President on June 24, can be safe.

External links

[1] COVID-19: What and when to expect?
[2] Antilapsha No. 2 COVID-19: Why do we need quarantined science?
[3] Anti-noodle No. 3 COVID-19 in a big city
[4] Anti-noodle No. 4 COVID-19: Avalanche that covered Europe
[5] Anti-noodle No. 5 COVID-19: Who is now?
[6] Antilapsha No. 6 COVID-19: End of a pandemic?
[7] Antilapsha No. 7 COVID-19: Half-naked numbers. Human casualties

[8] WHO COVID-19 reports

[9] –In Iran, they said they expect the second and third waves of coronavirus in the country / TASS.ru (05/02/2020; 14:27)

[10] Digital passes: how it will work access system in the city. MOS.ru. (04/11/2020; 19:35).

Typical COVID-19 Epidemic Development Model