key: cord-0623198-8g8xpazu authors: Koltsova, E. M.; Kurkina, E. S.; Vasetsky, A. M. title: Mathematical Modeling of the Spread of COVID-19 in Moscow and Russian Regions date: 2020-04-21 journal: nan DOI: nan sha: 7994c453261120138f8fb00432331bf1f4aafe02 doc_id: 623198 cord_uid: 8g8xpazu To model the spread of COVID-19 coronavirus in Russian regions and in Moscow, a discrete logistic equation describing the increase in the number of cases is used. To check the adequacy of the mathematical model, the simulation results were compared with the spread of coronavirus in China, in a number of European and Asian countries, and the United States. The parameters of the logistics equation for Russia, Moscow and other large regions were determined in the interval (01.03 - 08.04). A comparative analysis of growth rates of COVID-19 infected population for different countries and regions is presented. Various scenarios of the spread of COVID-19 coronavirus in Moscow and in the regions of Russia are considered. For each scenario, curves for the daily new cases and graphs for the increase in the total number of cases were obtained, and the dynamics of infection spread by day was studied. Peak times, epidemic periods, the number of infected people at the peak and their growth were determined. The discrete logistic equation is use to describe the spread of the epidemic coronavirus COVID -19 in Russia and its regions. For the first time, the logistic equation in differential form was suggested by the Belgian mathematician Pierre Verhulst in 1845 [1] to model population growth. The essential difference from the mathematical model of Thomas Malthus (presented in the famous work "Experience of the population law" [2] ), which describes exponential population growth, is that the Verhulst model took into account competition for resources, * Corresponding author. Email: kolts@muctr.ru which leads to limited population growth. In 1920 logistic equation in differential form = (1 − ), (1) where y(t) is the population size at time t, the parameter λ characterizes the population growth rate, and the parameter N determines the maximum possible population size in conditions of limited resources. The logistic equation began to be widely used, starting from the 20s of the last century, when it was rediscovered by R. Pearl [3] and confirmed its adequacy in experiments with the reproduction of Drosophila flies. Currently, the use of this equation has found wide application in mathematical Biophysics, which is well reflected in the monographs of Russian biophysicists G. Yu. Riznichenko and A. B. Rubin [4] . The Verhulst equation was also used to describe the spread of epidemics. In this case, the entire population that can become infected is divided into two parts: susceptible to the disease (Susceptible) and infected (-Infectious). It is believed that the disease is transmitted through contacts of healthy people with patients with a probability of λ, and under conditions of good mixing, the increase in patients is described by the equation: Here = / is the part of infected people, (1 − / ) is the part of people susceptible to the disease. N is the maximum number of cases. This model, written in the form of two equations, is called the SI model (Susceptible-Infectious model). China [5] and Sweden [6] . It has two fixed (stationary) points: .: = 0, = . The second point is the only attractor. Thus, over time, no matter what value the growth indicator λ in equation (2) takes, the population size will tend to N only with different periods of time to reach this stationary state. where μ − is the rate of recovery. There are even more complex models [7] that describe the spread of infections more accurately. However, the more complex the model, the more unknown parameters it contains, the value of which cannot be estimated with good accuracy. Note, that there is a large difference between the maximum number of residents, who are potentially ill (N max ) and the maximum number of residents, who are actually ill (N in ). The value of N max is much greater than the number of N in . The main parameters that most strongly affect on the spread of COVID-19 coronavirus and on the characteristics of the peak are two parameters: the growth rate of the number of cases and the maximum value of the number of residents who can potentially be infected N in . The number of recovered in the case of COVID-19 almost up to the peak can be ignored. The main parameters that most strongly affect the rate of spread of COVID-19 coronavirus infection and the characteristics of the peak are two parameters: the growth rate of the number of cases and the maximum value of the number of residents who can potentially be infected with N in . We used a discrete logistic equation that contains exactly these two parameters, and we believe that in this case it describes the spread of the epidemic better than any other model. The discrete logistic equation is widely known thanks to the work of the American scientist M. Feigenbaum [8] We used a discrete logistic equation to describe the spread of COVID-19 coronavirus in different countries and cities. It has the form: we reduce equation (3) to the form: where the variables x n and the parameter α are dimensionless. For values 0 < α ≤ 1, regardless of the choice x 0 , the population size tends to zero. That is, the number of cases will tend to zero, no matter how many cases there were at the beginning. At values 1 < α ≤ 3, the dimensionless number of the diseased population tends to a stationary stable state ̅ , equal to Therefore, over time, the number of people who become ill at the end of the epidemic will be equal to The relations (6) and (7) are correct if the α index remains constant throughout the epidemic. Note that there is an important difference between the differential equation describing logistic growth and the discrete one. In the differential equation at 1<α, the number tends to the value N, and in the discrete equation-to the value (7), which depends on the growth rate indicator, and may differ greatly from N. The current quarantine measures reduce the probability of infection and the indicator α decreases, which leads to a decrease in the total number of cases . To test the feasibility of using the logistic equation The normalizing factor N was also determined for a number of European and Asian countries. It was found from a comparison of estimated and actual data. The actual data was taken from the site [12]. After analyzing the ratio of the normalizing factor to the number of residents The values of growth rates of the diseased population for fourth scenarios are presented in Table 3 . Infected cases in the regional part of Russia Fig. 5 . Curves of the number of infected people in the regional part of Russia by epidemic days in accordance with the four scenarios (1 -4),  -reported cases. Figure 6 shows the actual and estimated daily growth of people infected with COVID-19 coronavirus for four scenarios during the epidemic. Daily new cases in the regional part of Russia Fig. 6 . Curves of the daily new cases in the regional part of Russia by epidemic days in accordance with the four scenarios (1 -4),  -reported cases. Table 5 shows the calculated data under four scenarios: the time of peak population growth, the number of infected people and the value of new case at the peak, the total number of cases, and the end time. The growth rates of the number of people infected with coronavirus in the Russian regions were calculated in such a way as to best describe the actual data from March 1 to April 7. We see that they changed twice. Since we use only recorded data, the change in the indicator for the first time is probably due to an increase in the number of tests and an improvement in their quality. The decrease in the indicator since April 1 st is due to the reaction of the virus spread to quarantine measures Then, starting from April 8 th , a forecast was made using equation (3), which used the value of the last growth indicator. April 15 7127 7456 7425 7305 7185 April 16 8738 8774 8681 8477 8277 April 17 10358 10218 10131 9807 9481 April 18 12111 11985 11739 11303 10829 April 19 13941 13915 13323 12975 12287 April 20 15647 16190 15324 14825 13857 April 21 17402 18793 17554 16853 14768 Analyzing the data in the table 6, you can see that the best agreement with the actual data is given by calculations for the first and second scenarios with the parameter N = 2×10 6 and N = 1×10 6 . But the final choice of the normalizing factor N can be made closer to the peak value of the daily new cases. The simplest model of infection spread chosen by us, which is based on a discrete logistic equation (3) Note, that the rate of increase in the number of cases is evaluated in the first stage of the epidemic, when the linear term prevails in equation (5) , and the nonlinear term can be ignored: Then the disease increases exponentially, which in the logarithmic scale is a straight line. The growth indicator determines the slope of the straight line. If the growth rate changes, then the slope of the straight line also changes. It may change with the introduction of various quarantine measures or their cancellation. Table 1 shows the average estimated growth rates of the number of cases before the peak of the epidemic in a number of European and Asian countries and the United States. The best agreement with actual data was obtained for these values of growth rates. These values determined both the time of the peak and the number at the peak. Table 1 shows that Sweden and Japan have the lowest rates of growth in the number of cases. Perhaps because of these indicators, these countries did not impose strict quarantine measures. As mentioned above, the parameter N begins to affect strongly on the development of the epidemic closer to its peak. Although this parameter represents the maximum number of residents who can potentially be infected, it does not mean that all of them will get sick. In this model, as much as corresponds to the growth indicator α will get sick in contrast to SI, SIR, and other models. The normalizing factor N depends on the nation's immunity to the virus, on the living conditions of the inhabitants (crowding, etc.), on the mentality of the nation, etc. It is calculated based on the agreement of calculated and actual data. Table 2 The parameter values have not been re-selected or changed since April 7. All scenarios up to April 7 described the actual data well, but by April 14 it became clear that the 3 rd and 4 th scenarios were lagging behind the actual data, especially the 4-th scenario with N = 3×10 5 . Splitting of forecast data under four scenarios begins as the number of cases increases and approaches the peak (see Fig. 3, 4) . Table 5 suggests optimism that the second scenario of the epidemic will be held. But there is a chance that Moscow will follow the worst-case first scenario. Table 5 shows that scenarios 3 and 4 produce results that lag more and more behind the actual data every day. So, it was not possible to keep the milder scenarios of the epidemic development with a smaller number of cases. In recent days, the results of scenarios 1 and 2 have also started to lag behind the real data. It seems that the easing of already soft restrictive measures has led to an outbreak in some regions. This means that the growth rate parameter α has changed, and it will need to be slightly increased and selected in the model. Which of the scenarios, the harder 1 st scenario or the less hard 2 nd scenario, is implemented will be clear in a few days. If you compare the growth in the number of infected people in Russia with the growth in the number of infected people in Moscow, we can see that the indicators for the Russian regions is generally lower than for Moscow. The average population growth rate for Moscow, calculated before the peak, is 1.112, which is lower than in European countries, but higher than in Japan (see Table 1 ). The average number of COVID-19 infected people (before the peak of the epidemic) in It should be noted that the distribution of the epidemic in China fell well within the scenario with the choice of a single normalizing factor. To model the epidemic in a number of European countries, such as Italy and Spain, we needed two logistic equations with different rationing factors. This is due to fluctuations with a large amplitude in the peak area and on a downtrend. The results of the logistics equation with a smaller multiplier described the actual data better before the peak. The model with a larger multiplier described the real data better after the peak. In other words, the actual data went along the corridor between these two curves. What is the scenario for the spread of the epidemic in Russia after the "Chinese" or "European" peak will become clear when the peak will be passed. Mathematical modeling of the dynamics of the spread of COVID-19 coronovirus infection in Moscow and in regional Russia was carried out and forecasts were made. The simplest nonlinear discrete equation describing logistic growth was chosen as the model. This model contains only two parameters that are chosen based on statistical data at the initial stage of the epidemic. The first parameter is a rate of the increase in the number of cases. It can change with the implementation of quarantine measures and affect on the total number of cases at the end of the epidemic. The second parameter is a normalization multiplier that only estimates the maximum potential number of residents who may become ill. The predicted total number of cases is calculated in the model and may differ from the N value by more than an order of magnitude. Its exact value can only be determined closer to the peak of the epidemic, so based on the experience of modeling the spread of the epidemic in different countries that have already passed the peak, different N values are selected and different scenarios are considered. Four different scenarios of the epidemic development are proposed for Moscow and regional Russia. For each scenario, the growth rates of the population infected with COVID-19 coronavirus were founded, the peak time was calculated, the maximum increase in cases at the peak and the total number of cases during the epidemic were found. While the work was being prepared, it became clear that Moscow did not manage to isolate citizens well from each other, and two easier scenarios are no longer being implemented. The tightening of quarantine measures in the last week inspires optimism that the development of the epidemic in Moscow will follow the second, not the worst scenario, which currently best describes the actual data. Mathematical modeling of the dynamics of the spread of coronavirus infection in the regions of Russia showed that it is delayed by 6 days compared to Moscow, and the growth rates of the number of cases is slightly less than the growth rate in Moscow. Four scenarios for the development of the epidemic in regional Russia were considered too. It is calculated that for the" light "scenario for the Russian regions, the number of cases will be ~ 40,000 people, for the "heavy" scenario -160,000 people. Note that we based on official data on the number of cases, which determine the burden on health care. And unrecorded cases, which according to some estimates may be 20% of those accounted for, ensure the spread of infection, and the model can be used. Mathematical researches into the law of population growth increase // Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles. 1845 An essay on the principle of population as it affects the future improvement of society On the rate of growth of the population of the United States since 1790 and its mathematical representation // Proceedings of the National Academy of Sciences of the United States of America Mathematical methods in biology and ecology. Biophysical dynamics of production processes. Part 2: textbook for undergraduate and graduate studies -3-rd ed A mathematical model for the coronavirus COVID-19 outbreak // arXiv preprint Model studies on the COVID-19 pandemic in Sweden // arXiv preprint Сonceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action Universal behavior in nonlinear systems // Physica D: Nonlinear Phenomena Methods of synergetics in chemistry and chemical technology Nonlinear dynamics and thermodynamics of irreversible processes in chemistry and chemical technology The authors are grateful to the postgraduate student A.S. Shaneva of D. Mendeleev University for help in editing the article.