The major part of the papers devoted to modeling the COVID-19 epidemic spread is based on the mathematical model named SIR-model. SIR − model (Susceptible-Infectious-Recovered model) is considered to be more realistic.
According to SIR − model the total population is divided into three parts: the proportion of the people susceptible to the disease -y (Susceptible), infected -x (Infectious) and recovered -z (Recovered). The presence of recovering people in the model reduces the number of infected people who are able to infect others.
The equations of the SIR model have the form:
where y -the part of people susceptible to the disease, x -the number of infected inhabitants, z -the number of recovered people, λ -the index of infected growth, µ -the index of recovered growth. This SIR-model was used earlier in the papers [1, 2] for modeling infection spread among people and animals. In this model epidemic spread is determined by the ratio of the parameters λ/µ. If this ratio is less than 1, the epidemic is fading; if it is more than 1, the epidemic is spreading. For modeling of COVID-19 epidemic spread SIR-model was used in the [3] [4] [5] .
There the SIR model development -the SEiR model. In this model, the group of infected people is divided into two subgroups. Group 1 is at the stage of the incubation period and is not contagious; Group 2 is made up of people who are contagious. Thus, the transition S → E → I → R occurs. SEiR-model as a software package is presented on an electronic resource [6] . The papers [8, 9] present the results of discrete logistic equation use to simulate the epidemic spread in Moscow (at the initial stages of the epidemic), in a number of European, Asian countries, and Israel. Modeling the epidemic spread not only at the initial stage, we noticed that the use of the only one discrete logistic equation is well applicable before the peak and on the peak of epidemic spread.
And in those countries where, after the peak of the increase in the number of infected people, the so-called "plateau" occurs and there is no drop in the increase in the number of infected people (as new sites of infection appear), we have to increase the capacity of the system all the time. So the model lost its predictive power, and mismatching between the calculated and actual data increased significantly. The parameters were adjusted to the right side (after the peak) of the curve of the increase of the infected people and we obtained the mismatching from the left side of the curve involved. And only the use of not a single but a system of discrete logistic equations led to the fact that the presence of waves superposition of reduced significantly the mismatching between the calculated and actual data both for the curve of infected people number and for the curve of the increase in the number of infected people.
Using the waves superposition approach we have achieved the smallest mismatching between actual and calculated data on the examples of modeling the epidemic spread in such countries as Spain and Italy (which are already almost completing the epidemic), in this work we consider the epidemic spread in the countries with the highest increase in the number of infected: the USA, Brazil, Russia, India as well as the epidemic spread all over the world.
For modeling with the use of superposition we have the system of discrete logistic equations: 
where variables ( ) and parameters ( ) are dimensionless. When values 0 < ( ) ≤ 1 regardless of the initial value of ( ) , the number of infected people during the i-th wave will tend to zero. When the values 1 < ( ) ≤ 3 regardless of the initial state, dimensionless number of the population infected during the i-th wave of epidemic spread will tend to the stationary steady state.
Therefore during the time the number of people infected in the i-th wave of epidemic spread (at the time of its completion) will be equal
Relations (4) and (5) 
From the relations (6) and (7) we can see that the capacity of the wave ( ) at the initial period of wave spread does not affect on the calculation of the number of infected people.
We will write the relation for determining the number of infected people at the local peak for the increase in infected people in the i-th wave of the epidemic 
We can see from the expression (9) that the system capacity ( ) can be defined in the vicinity of a local peak of the i-th wave of epidemic spread.
The system capacity will be equal
where M -the number of epidemic waves for the particular country. The number of all the infected people is determined according to the relation
We have developed the algorithm for searching the parameters ( ( ) , ( ) ) on the base of actual data on the number of infected people as well as on the daily increase in the number of infected that provides the minimum mismatching between calculated and actual data on the number of infected and on the increase of the number of infected. Table 1 shows the values for the indices of the number of infected and the capacities of the waves in Moscow. Table 3 shows the values for indices of the number of infected people and the waves capacities in India. Table 3 The values of mathematical model parameters for COVID-19 epidemic spread in 4. Discussion of the results Figure 1 shows the calculation results on the daily increase of COVID-19
infected people, which we presented in the paper [9] . We examined 3 scenarios with different capacities: 150,000 inhabitants (as for the 5th wave), 300,000 inhabitants and 500,000 inhabitants. The peak of the 6th wave was in the vicinity of July 9, and the ending is expected in mid-August. The sixth wave will add to the number of infected people according to the scenarios, respectively: 15,000, 30,000, 50,000 inhabitants, and will bring the total number of infected to the values, respectively: 237,000, 252,000, 272,000 Moscow residents.
Which of scenarios to be implemented in Moscow will be clear in mid-July.
It should be noted that the indices of the number of infected people growth and waves capacity were determined from actual data [12], the largest error in the number of infected people did not exceed 10% (relative error) and in most cases it was less then l%. Table 7 shows the number of COVID-19 infected people that have brought these 6 waves. it was predicted in our work [9] . It should be mentioned that in the calculations according to the number of infected in Russia the error did not exceed 5%, in most cases it was less than 0.5 %. Table 3 shows the values of indices for the number of infected people growth and the system capacity. We can see that with the introduction of restrictions in the country, the index of infected people growth was consistently decreasing. But such a huge capacity indicates about very strong mixing in India, apparently such high capacity value is likely to be associated with large population crowding.
The system capacity is not directly related to the country population. So, the wave capacity for China is only 750 thousand people, for Russia ~ 6.3 million people (which is also not rather small). The wave peak of the epidemic spread in India is expected on June 30 with the increase of infected people not less than 15 thousand inhabitants.
The epidemic beginning in India was on February 14, the end of the epidemic will be in late November -early December. The size of a single wave will give 1,103,000 infected India residents. But we believe that in the future there may be other waves that can increase the number of infected people. The error in calculating the number of infected in India did not exceed 25% in some cases. The first two waves arose one after another and appeared in early March in Table 9 .
Тable 9
Waves characteristics in the whole world [9] was 760 thousand inhabitants and the capacity of the first two waves in the world was 750 thousand inhabitants (see Table 9 ). The error in the calculations for the number of infected people in comparison with the actual data (for the first 2 waves) does not exceed 2%, and when calculating the remaining waves it does not exceed 6%.
The third wave occurred on 10.02 and ended in late May and brought the world a little less than a million infected people (948,586 see table 9). We believe that the third wave covers the epidemic spread in Europe. The fourth wave of the epidemic spread was determined on March 14 and gave a local peak on April 27. The end of the fourth wave is mid-September (see Table 9 ) and it will bring 3,875,593 infected people. The 4th wave is likely to capture the epidemic spread in the countries such as Russia, some states in the USA, etc.
The fifth wave was confirmed on April 21, its local peak occurred on June 16
with an increase of infected being 146,170 people (see Table 9 ), the end date is The fifth wave is slightly less than 7 million infected residents, see table 9 (6,802,485 inhabitants). With the advent of the fifth wave the epidemic spread in the world has passed into a "new phase", the daily increase in the number of infected people in the world has exceeded 100,000 people daily. Thus, the mathematical model for COVID-19 epidemic in the world predicts (calculated on June 22) that the superposition of five waves will give the world 11,709,311 infected people.
We believe that the world expects the following waves which have not been observed in actual data yet, the next waves will increase the number of infected people in the world.
On the base of a system of discrete logistic equations we developed an approach by superposition of waves for modeling of the COVID-19 epidemic both in the whole world and in particular countries. In countries that strictly enforce restrictive measures and have small "capacities", the epidemic spread fits on one single wave (case study is the epidemic spread in China).
The COVID-19 epidemic spread in the whole world fits into the superposition of 5 waves (calculated on June 22). In further calculations to correct the model only the fifth wave will be corrected (the remaining waves will remain unchanged) and the subsequent waves will take their place corresponding to the epidemic spread in the world. 