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The major medical and social challenge of the 21st century is COVID19, caused by the novel coronavirus SARSCoV2. Critical issues include the rate at which the coronavirus spreads and the effect of quarantine measures and population vaccination on this rate. Knowledge of the laws of the spread of COVID19 will enable assessment of the effectiveness and reasonableness of the quarantine measures used, as well as determination of the necessary level of vaccination needed to overcome this crisis.
This study aims to establish the laws of the spread of COVID19 and to use them to develop a mathematical model to predict changes in the number of active cases over time, possible human losses, and the rate of recovery of patients, to make informed decisions about the number of necessary beds in hospitals, the introduction and type of quarantine measures, and the required threshold of vaccination of the population.
This study analyzed the onset of COVID19 spread in countries such as China, Italy, Spain, the United States, the United Kingdom, Japan, France, and Germany based on publicly available statistical data. The change in the number of COVID19 cases, deaths, and recovered persons over time was examined, considering the possible introduction of quarantine measures and isolation of infected people in these countries. Based on the data, the virus transmissibility and the average duration of the disease at different stages were evaluated, and a model based on the principle of recursion was developed. Its key features are the separation of active (nonisolated) infected persons into a distinct category and the prediction of their number based on the average duration of the disease in the inactive phase and the concentration of these persons in the population in the preceding days.
Specific values for SARSCoV2 transmissibility and COVID19 duration were estimated for different countries. In China, the viral transmissibility was 3.12 before quarantine measures were implemented and 0.36 after these measures were lifted. For the other countries, the viral transmissibility was 2.282.76 initially, and it then decreased to 0.871.29 as a result of quarantine measures. Therefore, it can be expected that the spread of SARSCoV2 will be suppressed if 56%64% of the total population becomes vaccinated or survives COVID19.
The quarantine measures adopted in most countries are too weak compared to those previously used in China. Therefore, it is not expected that the spread of COVID19 will stop and the disease will cease to exist naturally or owing to quarantine measures. Active vaccination of the population is needed to prevent the spread of COVID19. Furthermore, the required specific percentage of vaccinated individuals depends on the magnitude of viral transmissibility, which can be evaluated using the proposed model and statistical data for the country of interest.
The first mathematical models to predict the development of infectious diseases were used in the early 20th century [
When a new infection appears, neither the set of population categories to be considered in the model nor the rate of transition of people from one category to another is known. Current information about the features of the COVID19 infection caused by the novel coronavirus (SARSCoV2) and the manner in which people perceive it and act should serve as a basis for building a model to describe the spread of this virus. These features can be described as follows: first, the presence of a long incubation period, during which the infected persons are contagious to others, and second, the isolation of discovered infected persons, which as a result become conditionally noncontagious. The combination of these two factors makes this novel coronavirus infection unique. In general, the opposite is true—infected people are not dangerous to others during the incubation period and become contagious after its expiry. For this reason, a new model that considers these circumstances is needed to predict the spread of COVID19. However, the duration of the immunity produced after recovery from COVID19 is currently unknown. In addition, there is also very little information available to accurately calculate the rate of recovery among patients with COVID19: a small percentage of the population recovers within just a week after contracting infection, whereas the majority of people experience the illness for a long time. Therefore, the proposed model cannot be final, but it is necessary for forecasting and management decisions.
The model for COVID19 spread is based on a set of parameters whose values are unique for each country due to differences in population density, human behavior, date of virus penetration, and government actions. The set includes the following parameters:
The evaluation of the spread of the virus is based on the calculation of the following data:
where
In the case of vaccination of the population and considering the temporary nature of the immunity received due to SARSCoV2 infection or vaccination, the above expression will be as follows:
where
At the start of the epidemic (date
Thus, in order to calculate the virus spread dynamics, it is necessary to know the values of only two parameters—
However, it is more difficult to model human losses correctly. Two more parameters need to be considered:
These two parameters depend on the efficacy of treatment and may vary as physicians gain experience and as hospitals overflow. The number of deaths on date
Because of the presence of two parameters (
The situation with predicting the number of recovered persons is even worse due to the appearance of an even greater number of independent parameters:
where
The model equations are presented in the discrete form (instead of differential one), so that the model can be easily reproduced for calculations in any spreadsheet editor. At first glance, it seems that the model does not take into account the existence of asymptomatic carriers of infection, but this is not true: since the share of asymptomatic carriers in the population does not change over time, their presence is taken into account implicitly by the value of the transmissibility. This model can be denoted by the abbreviation SILRD, which means that it takes into account Susceptible, Infected, Isolated, Recovered, and Dead persons.
Based on historical data on disease development in eight countries (China, Italy, Spain, the United States, the United Kingdom, Japan, France, and Germany [
Time dependences of the total number of COVID19 cases, deaths, and recovered cases. Dots show actual data, whereas lines represent the result of calculations using the model.
Parameters identified from the models used in different countries.
Parameter  China  Italy  Spain  USA  Japan  UK  Germany  France 

3.12  2.55  2.76  2.46  2.46  2.28  2.34  2.34 

31.12.19  01.02.20  12.02.20  11.02.20  19.01.20  11.02.20  08.02.20  07.02.20 

1.68  1.74  1.56  1.20  1.29  1.74  1.32  1.56 

23.01.20  26.02.20  10.03.20  21.03.20  27.01.20  10.03.20  14.03.20  07.03.20 

1.14  1.38  0.90  0.90  N/A^{a}  1.32  0.87  0.87 

29.01.20  08.03.20  23.03.20  06.04.20  N/A  24.03.20  24.03.20  26.03.20 

0.36  0.92  N/A  N/A  N/A  1.05  N/A  N/A 

10.02.20  18.03.20  N/A  N/A  N/A  01.04.20  N/A  N/A 

0.04  0.05–0.14  0.03–0.105  0.03–0.06  0.027  0.03–0.135  0.004–0.04  0.03–0.14 

0.24  0.16  0.5  0.14  0.973  N/A  0.75  0.3 

14  9  14  14  31  N/A  17  12 

29  39  unknown  unknown  31  N/A  unknown  unknown 

N/A  12.06.21  24.03.21  30.05.21  21.03.21  >31.12.21  31.12.20  03.01.21 

N/A  310000  330000  1690000  52800000  2270000  210000  220000 
^{a}Not applicable.
^{b}If an interval is indicated, it means gradual growth.
On the date on which the analyzed data set ends, all European countries (except the United Kingdom) only managed to reduce the transmissibility slightly below 1.0. From a practical point of view, this means that the number of people falling ill on a daily basis in these countries was gradually decreasing, but it was at such a slow pace that the end date of the epidemic in these countries could not have been before, at best, the end of 2020. In reality, these countries have partially canceled quarantine measures, causing an increase in viral transmissibility and, consequently, a new rise in the number of infected persons and a shift in the date of a possible end of the epidemic to the future. It should be understood that any alleviation of quarantine measures would lead to increased transmissibility and resumption of an accelerated spread of the virus. To prevent this from happening after the quarantine restrictions have been removed, the viral transmissibility must remain below 1.0. By way of example, the original transmissibility was 2.55 in the case of Italy; therefore, it is necessary that 61% of the Italian population be either infected and then recovered (provided the immunity produced is durable and strong) or vaccinated against SARSCoV2 so that when quarantine measures are lifted, the transmissibility remains less than 1.0. At the time of writing this manuscript, 0.33% of the Italian population had been infected according to official statistics [
Thus, the model allows forecasting of the situation development and concluding about the effectiveness of quarantine measures. By way of example, it helps determine the current number of active infected persons (
This work was carried out within the State Program of A.V. Topchiev Institute of Petrochemical Synthesis.
None declared.