Sample and data

The most well-known and reliable resources for storing global statistics on the spread of the COVID-19 coronavirus include the websites of Johns Hopkins University [22], the WHO [23], and Worldometers.info [24]. Here, information from numerous primary sources from different countries was accumulated, reflecting the current situation according to a variety of factors and conditions according to accepted medical reporting standards in each state.

The sources of primary information were thousands of official resources related to the current coronavirus situation. These included government websites, healthcare ministry databases, and regional organization data. All of these sources were verified and compared in order to ensure accuracy. Thus, the quality and quantity of information on statistical observations of cases of infection, recovery, mortality, etc., depend on a variety of conditions and factors characterizing the situation in each country, in particular, on the reporting standards adopted there. For example, in some countries, the number of infected people was represented only by the number of laboratory-detected cases. In other ones, it has taken into account the clinical picture of the diseases. It is noted that the gap between the actual incidence and reported cases depends on the number of coronavirus tests performed and the transparency of the country’s reporting [24]. There are expert estimates [24] showing that the number of undetected cases is a multiple of the number of identified cases, so it is better to use relative indicators for statistical comparison.

In Russia, information about the spread of the novel coronavirus infection was collected through the Ministry of Healthcare (Figure 1). Official statistical information on the number of new cases, recoveries, and mortality in the country and selected regions was published daily on the website of the Communication Centre of the Russian Federation Government [25]. Regional health ministries compiled daily statistics for municipalities and the region as a whole. Then the data were transferred to the federal database. Some data differences were recorded in the federal and regional databases. Firstly, a number of regions were late in submitting their data, and the information was not entered into the federal database until the next day. Secondly, the number of registered cases often differed: the most frequent discrepancies were noted between recovered and deceased patients. In a sample analysis of available statistics for 13 regions, such discrepancies were recorded in 39% of cases. The issue of failures and inconsistencies in the collection of coronavirus statistics has been raised several times in the Russian media.

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Figure 1. A diagram of the geodata database formation for the spread of a new coronavirus COVID-19 infection in the Russian Federation.

At the regional level, statistical information is received daily from municipal governments, where data is transmitted from health facilities (Figure 1). Since the WHO only provides recommendations for monitoring the spread of coronavirus, data collection and processing systems may vary at the national level, so direct cross-country comparison of raw data without pre-processing is difficult. To investigate the patterns of a new coronavirus infection spread within the borders of different countries, it is desirable to use databases from countries that use similar methods of recording the number of people who became infected, recovered, and deceased as a result of COVID-19 infection when compiling statistics.

Measures of parameters

For cross-country comparison, a relative indicator of the form F^*(t) = N(t)/N_k is used, where N(t) is the number of cases at the time t (the day from the beginning of the pandemic process on 12/31/2019), N_k is the same at the end of the study period t_k, which in our case includes time intervals (waves) of the first and the subsequent stages of the pandemic before the appearance of the omicron strain of the COVID-19 coronavirus (SARS-CoV-2) with unique features. That led to a sharp spike in the incidence and its rapid cessation, which requires further special quantitative analysis.

Figure 2 shows graphs of the proportion of daily increase in cases of COVID-19 coronavirus by countries of different continents (P(t) = [F^*(t + ∆t) – F^*(t)]/∆t, ∆t = 1 day) for the period from t₀ = 100 days (04/09/2020) up to t_k = 650 days (10/09/2021). Depending on the method of collecting and presenting pandemic information, relatively smooth and non-smooth with sharp daily changes in the value of P(t) differ. The graphs show a shift in the position of infection maxima by territories. During the period under review (2 years), four waves of morbidity were identified with a tendency to increase their amplitude.

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Figure 2. The chronogram of changes in the proportion P(t) of the daily increase in confirmed cases of COVID-19 during the period of 100–650 days of the pandemic by countries. 1—Russia; 2—USA; 3—Saudi Arabia; 4—South Africa; 5—Italy.

There are different types of geoinformational representation of spatial data on maps, reflecting different degrees of penetration into the content of the processes and phenomena under study: analytical, complex, assessment, synthetic, and systemic [26]. When studying and mapping the heterogeneity of population response to coronavirus spread, analytical maps display the spatial variability of initial indicators, which are taken into account in the classical SIR model (number of people who became diseased, deaths, recovered) [15], as well as their relative values (morbidity, mortality, lethality), e.g., the proportions of people who became diseased and died in relation to the total population of the country [1, 27]. Such primary information is usually displayed on the Internet using geographic infographics [22], often with elements of animation and zooming. Comprehensive maps show, with the help of diagrams, the ratio of population proportions belonging to different states of people in the epidemic process. Information for assessment maps is calculated by formulas based on initial indicators, for example, by calculating reliability, hazard, and public health risk values. Synthetic (integral) maps illustrate the circumstances and conditions of the epidemic by region. Synthesis is performed by solving direct and inverse modelling and mapping problems. System maps reflect the parameters of geoinformation models of epidemiological data linkage specific to different territories.

Models and data analysis procedure

The methods of fiber bundle in differential geometry (Supplementary material 1), as well as reliability theory (Supplementary material 2), are used as a basis for mathematical modelling and statistical data analysis.

The SIR model [15] describes the redistribution of the number of susceptible (S), infected (I), recovered (R), and differentially removed from the epidemic process parts of the whole susceptible population S₀, e.g., the number of recovered patients. Unfortunately, the resulting logistic curve of the solution function of the SIR equations does not fit well with statistical plots of the development of a new coronavirus pandemic by country [28], and new equations for modelling need to be sought. A promising direction of modelling in this area is the relations of probability and reliability theory [17, 18].

Processing of geodata by Equation S7 is provided by its reduction by the hierarchy of principal fibrations G{G[f(y)]} → G[f(y)] → f(y) to the linear form f(y) = K(τ). The probability equations F^*(z) for the occurrence of extreme values z of different types have the general form Equation S7 for maximum and minimum values τ = t and τ = θln(t/t_m) + t_m. For Equation S7, the integrated hazard E^*(τ) = –lnF^*(z) = exp[–α(τ – τ_m)]and the risk (failure rate) p(τ) = –dE^*(τ)/dτ = αexp[–α(τ – τ_m)], where p(τ_m) = p_m = α is the acceptable risk (AR). The value ln[–E^*(τ)] = –α(τ – τ_m) depends linearly on τ. The dependence K(τ) = –α(τ – τ_m) is a functional analogue of the universal Equation S2 when f(y) = K(τ), y = τ – τ_m, i.e., the value τ = τ_m corresponds to the point of tangency of the line K(τ) of the envelope manifold F(τ). The tangent K(τ, τ_m) to F(τ) at the new point τ = τ_m is the layer in which the distribution F^*(z) = exp{–exp[K(τ, τ_m)]} with individual parameters (τ_m, α) is formed. When τ_m changes, a path on F(τ) is constructed along which K(τ, τ_m) changes and the transition from one function of F^*(z) to the other takes place.

The constant α is the sensitivity of the change K(τ, τ_m) when τ changes, which determines the value and direction of the vector α individually in each layer of K(τ, τ_m) [21] (Figure 3). Content-wise, p_m = α = eP_m = eV_m/S₀ is determined by the value of the extremum (peak) of the distribution density function, increases with the maximum of the current incidence increase V_m and decreases with increasing infection potential S₀, varying across countries. The values of (τ_m, α) vary from place to place, preserving the general (true) form of the functional dependence P(τ, τ_m) with parameters (τ_m, α).

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Figure 3. Different types of density function curves for maximum and minimum values. 1–2—Gumbel’s function, 3–4—Fréchet function with the same parameters α = 0.3/day, τ₀ = 20 days, θ = 9 days.

The appearance of equations of the form (Equation S7) is associated with the description of the self-regulation mechanism of the infection spread in indicators of integrated hazard:

i.e., the values of differential p(τ) and integrated hazard E^*(τ) are proportional to the AR factor p_m = α. The steady decrease of E^*(τ) with the time course τ is governed by the current infection hazard E^*(τ):

When E^*(τ) = –lnF^*(τ) ≈ 1 – F^*(τ) = P^*(τ) and F^*(τ) = I(τ)/S₀, P^*(τ) = S(τ)/S₀; A = α/S₀ = eV_m/S₀² is the frequency of population contacts. In addition, Equation 1 is a variant of the universal Equation S2 presented in the form of

Statistical graphs of the daily increase in the number of confirmed cases of COVID-19 coronavirus infection grow rapidly at the beginning, reach a plateau, and then slowly decline (Figure 2). Of the curves of the considered functions (Figure 3), the Gumbel and Fréchet distribution functions (Equation S7) of maximum values qualitatively satisfy these features. The last one is convenient because by varying θ in the calculation of the variables of proper time τ = θln(t/t_m) + t_m, it is possible to increase the size of the plateau of the curve of the epidemic development process. When comparing statistical and theoretical curves [19], the Fréchet function is favoured as more general and adequate according to the similarity criteria, and also because of the fact that in the vicinity of the epidemic peak t = t_m the value τ = θln(t/t_m) + t_m ≈ θ[t/t_m – 1] + t_m = θ(t – t_m)/t_m + t_m = (θ/t_m)(t – t_m) + t_m depends linearly on time and at θ = t_m coincides completely with the time course τ = t, and the Fréchet function with the Gumbel distribution. The relation η = θ/t_m indicates the degree to which the situation under consideration corresponds to this ideal case.

For the first wave of the 2020 pandemic in Russia in MS Excel spreadsheets according to the algorithm of description of the statistical graph of daily increase in the number of infected people, the moment of the beginning t₀ = 67 day of 2020, when S(t₀) ≥ 10 established cases of infection, and the end t_k = 240 day of the first (spring) wave of the epidemic [local minimum on the graph S(t)] were distinguished. The value S(t_k) = S₀ = 975,576 people is considered the potential for infection. The unreliability index is calculated—the probability of health loss of the sensitive population S₀: F^*(t) = S(t)/S₀. We calculate the density of distribution of infection moments—the share of daily increase of sick people P(t) = F^*(t + 1) – F^*(t) and other reliability indicators (Equation S1): integrated hazard E^*(t) = –lnF^*(t), risk p(t) = E^*(t + 1) – E^*(t). By graphing P(t), the mode of the distribution is approximately determined—the moment of epidemic peak t_m = 149 days, the maximum value of P_m = P(t_m) = 0.0097/day and the AR α = eP_m = 0.0264/day. The proper time course is calculated τ = θln(t/t_m) + t_m at θ = t_m. Using Equation S7, approximating curves of the distributions P(t) and P(τ) over time t and τ are calculated by varying the parameters t_m, P_m and to θ ensure similarity in terms of correlation coefficient R. The Fréchet function of the marginal distribution at η = θ/t_m = 130/149 = 0.87 gives a better approximation (R = 0.95) of the data than the Gumbel function (R = 0.89).

To calculate the proportion of confirmed cases of COVID-19 in Russia P_k(t) (Figure 4) for each epidemic wave k, Equation S6 is applied in real time τ = t with K_k correction for the wave amplitude: P_k(t) = K_kP(t). The coefficients of the equation are α = 0.0217 and P_m = 0.008. The location of the seasonal peaks τ_m = t_mk of the proportion of infected people differs by about 179 days, i.e., half a year. For forecasting, it is necessary to know in advance the value of K_k and the location of t_mk (modal values) of infection peaks, which is approximately determined by their growth trends: t_mk = 170k, K_k = 0.139k. As a result, it is possible to calculate epidemic curves (Figure 4) based on Equation S6 P(t) = P_k(t)/K_k with corrections for the amplitude of K_k and the location of t_mk peaks of epidemic waves.

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Figure 4. Daily change in the proportion of COVID-19 confirmed cases in Russia. 1—initial data; 2—curves of data approximation according to Equation S6 when τ = t; 3—forecast of changes while maintaining the pandemic situation.

In order to generate the basic parameter maps of the model (Equation S7), a table of coefficients of the Fréchet equation for the distribution of identified cases by country is generated (Table 1). The values α = eP_m are obtained to be the same across countries. This means that the pandemic across countries belongs to the same type of phenomenon, and there is a line P(t_m) = P_m = 0.0947/day enveloping the territorial P(t) functions from above. The curves of the P(t) functions differ in the timing t_m of the peak of the P(t) = P(t_m) = P_m wave of the epidemic and its duration θ. The morbidity by country (layers, precedents) on the map forms peculiar time-shifted fields of infection spread. Increasing the value θ of the relative rates of proper time τ leads to a reduction in the duration of the epidemic period Δt₀ = t₀ – t_k in a country, which also individualizes the process, and on the other hand, allows meta-analyses to bring the dependencies P(t) to one typical model with constant coefficients.

Figure 4 shows the first four waves of the COVID-19 epidemic in Russia over two years: observed data and forecasted calculations of the population’s response using Equation S6. The calculations (dotted curve of Figure 4) were performed at the time of 10/25/2021 (666 days of observation) and were confirmed by the current statistics on COVID-19 cases until the beginning of January. However, from 01/14/2022 (day 747), the trend was broken due to the emergence of a new omicron strain of COVID-19, whose epidemic is radically different in terms of both the time of onset and the scale of manifestation, and requires a special study.

Meta-analysis

In order to combine the results of studies from different territories, to support and test scientific hypotheses, we propose using meta-analysis methods [20]. They are applied to the assessment of the population response of different countries to the spread of the COVID-19 disease. It allows us to identify the general regularities of the epidemic process [18] and to present them in the form of a mathematical model of its gradual development in order to solve direct and inverse problems of modelling and forecasting the emergence of crisis situations.

Meta-analysis is a set of procedures through scientific methodology, which consists of combining the results of a number of studies using statistical methods to test one or several interrelated scientific hypotheses. Meta-knowledge is pure knowledge abstracted from local circumstances, territorial peculiarities. The high generality of the approach, its interdisciplinary character, and consideration of environmental conditions allow us to consider the methodology of meta-analysis at the meta-theoretical level, which implies the use of mathematical formulas for knowledge representation with fundamental substantive constraints [20]. Meta-analysis is most widely applied in the social, medical, and biological sciences [29–31], where it is used to identify common patterns, to explore and explain observed differences due to heterogeneity of study results, and thus to increase the accuracy of the assessment of intervention effects through statistical processing of the available information. This allows a more precise definition of the categories of objects and types of environments for which the results obtained are applicable [20].

The methodology of meta-analysis is based on a meta-theoretical framework—the hypothesis of systemic stratification of Earth reality on the diversity of geohistorical environment. Locally, processes and phenomena are described by homogeneous Equations S1 and S2 of data integration, so each situation is reduced to the properties of the type layer and universal equations f(y) of the variables’ relationship [20]. This allows understanding and investigating different objects as the same object, and matching them by comparing them with a chosen etalon.

Reliability functions are analytically related to P^*(t) and to each other according to Equation S3, therefore, specifying one of them, we obtain all the others. For example, F^*(t) = exp[–E^*(t)] is the unreliability, which exponentially increases with decreasing hazard. In particular, it reflects the relative increase in the number of infected with decreasing risk of getting infected at the last stages of the pandemic. The meta-dependence p(τ) = αE^*(τ) + ε is reliably manifested in the Russian Federation population response and satisfies Equation 1 p(τ) = 0.0217E^*(τ) – 0.0169 with a correlation coefficient R = 0.919.

For the Fréchet function, the following relation is correct

Equation 4 in E^*(τ) coordinates does not depend on the environmental parameters (τ_m, α), so it can be used in the process of meta-analysis to highlight general patterns.

According to Equation 4, the normalized value ρ(τ) = P(τ)/P_m should ideally be determined only by the integrated hazard E^*(τ). For the individual epidemic wave curves, this value is transformed E^*(τ) → e^*(τ) for each country so that the plots pass through the origin (0, 0) when the normalised value e^*(τ) = 0 and the maximum point (1, 1) when P(τ)/P_m = 1 and e^*(τ) = 1. The transformation e^*(τ) = [1.61E^*(τ) – 0.78]^λ, λ = 1.45 common for all countries shows that the dependence (Equation 4) linear at λ = 1 is generally non-linear λ = 1.45, i.e., there is an interaction of hazard indicators E^*(τ), which is usually associated with a positive effect of self-organization.

The graphical dependence ρ(τ) = P(τ)/P_m on e^*(Figure 5a–b) of the observed values for individual countries corresponds to the calculated curve ρ[e^*] of the Equation 4. The similarity of P_n(e^*) plots for the population of different countries (Figure 5b) is revealed, which allows us to consider various dependencies (Figure 5a–b) as a single meta-law, which satisfies Equation 4 for normalized values of E^*(τ) → e^*(τ) and accounts for 94% of the points on the graph. For Russia, the similarity between the empirical and theoretical indicators of the dependence ρ[e^*] is very high (Figure 5a, R = 0.97).

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Figure 5. The dot plots of the dependence of the normalized proportion ρ = P(τ)/P_m of confirmed cases of the second wave of COVID-19 on the value of e^*(τ) of standardized integrated hazard E^*(τ): (a) in Russia, (b) as well as with overlapping graphs of different countries (Russia, USA, South Africa, Mexico, India, Kazakhstan, Canada): 1—initial data; 2—the curve of data approximation according to Equation S7. Calculated based on data from [22].

In the course of meta-analysis, the information is compressed. Constant coefficients and general regularities of the relationship between the characteristics of systems are identified in a ‘pure’ form in the central informative part of the data series of one wave. The minimum scatter (homogeneity) of data around the graphs of theoretical formulas determines the reliability of connections and allows their use in different conditions for processing data series and solving forecasting problems. The graphs in Figure 5 confirm the applicability of Equation 4, and thus of Equations S5 and S6 in statistical studies, but do not necessarily justify them definitively, because the recommended form of the Equation E^*(τ) may be different H: F^*(τ) → E^*(τ) = H[ln(F^*(τ))] → K(τ) = H[ln(E^*(τ))], which requires clarification.

Population response

The spread of COVID-19 disease is considered an emergency pandemic situation, dangerous for the population and economy of states. Ensuring the safety of global and regional life systems requires prompt analysis of incoming information about the epidemic process of infection and the creation of adequate models and methods of risk management [17]. The complexity of the proposed hierarchical models is expressed in the multiplicity of the embedding (superposition) of exponential functions G{G[f(y)]}. In the epidemic model, this is managed by reducing the value of the AR α of infection through organizational pressure on the value of this risk α = f(y). The effectiveness of the pressure is assessed by the values of sustainable controllability indicators, individualized for each area.

To describe the reliability function P^*(t) of the first wave, Equation 5 in the z = α(t – t_m) variant of the Gompertz Equation [17] was used:

where P(t_m) = P_m is the largest value of the function P(t). AR α defines the safety boundaries of existence of the considered system at t = t_m, when the distribution curve P(t) reaches its maximum P(t_m). According to Equation S1, the failure rate is equal to p(t) = αE(t), from which the coefficient α = p(t)/E(t) is calculated, even if it has a variable value α(t).

At the first hierarchical level, the exponential hazard function E(t) = αexp[α(t – t_m)] is manifested, and at the second level, the twice-exponential reliability function P^*(t) (Equation 5). By analogy, at the zero level of complexity, we should expect an exponential dependence of AR α(t) (the riskiness of the situation) on time:

where α_m is the limiting value of AR (t → ∞); β is a constant control coefficient of AR reduction. In Equation 6, the value of g = α₀β is the strength of the control pressure on the development of the epidemic process (controllability); χ = α₀ + gt_m + α_m is the AR value at the very beginning (t = 0) of the global pandemic manifestation in a given country. The coefficient χ is an indicator of the unpreparedness of the state and society for the epidemic, which is positively determined by the value of gt_m, starting α₀ and final α_m level of AR. Obviously, preparedness increases with increasing manageability β, later moment of epidemic peak t_m and a higher value of AR allowed in the country at the beginning α₀ and at the end α_m of the process.

The statistical materials available at the end of May 2020 were processed by numerical methods using the given equations. The inverse problem of modelling—determination of informative model coefficients using time series data—was solved. The epidemic curves were compared across countries to find out the functional complexity of the process under study in each area.

The exponential increase in AR (Equation 6) corresponds to a linear dependence of lnα(t) on time t with an individual offset t_m at the beginning of the epidemic process (R > 0.98) (Figure 6). Calculations for Italy give the ratio ln[α(t) – α_m] = –0.0355(t – 18.0) (R = –0.97), α_m = 0. Then we get α(t) = exp[–0.0355(t – 18.0)] (R = 0.97), where the controllability index β = 0.0393. The initial AR was highest in China (α₀ = 0.267) and lowest in Russia (0.194). In Western countries α had a close value of 0.243. The value of 1/α₀ indicates the degree of preparedness of a country’s health care system for a pandemic.

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Figure 6. Changes over time in the AR lnα(t) of the COVID-19 coronavirus epidemic in some countries. 1—Russia; 2—China; 3—Italy; 4—Germany; 5—Spain.

On the basis of the models, the peculiarities of the population response of different countries to the actions of state authorities in managing the risk of infection through their influence β on the value of AR α. Sustainable indicators (parameters) of controllability β, individual for each territory, become characteristics of the impact efficiency. The authorities of China demonstrated high manageability of the population in the conditions of the beginning of the epidemic, the average level is typical for Western countries, and the Russian population showed low manageability with high state readiness for the pandemic. The territorially close countries—Italy, France, and Spain—showed similar trends of pandemic development with slightly different coefficients of Equation 6 (Figure 6).

Statistical analysis and mapping

The comparative analysis of reliability functions [19] makes it possible to identify the Fréchet distribution Equation S7 as the best approximation of F^*(t) curves. This equation uses multilevel independent variables t, τ, z, and their functions F^*(t), F^*(τ), F^*(z), with the same values F^*(t) = F^*(τ). Equation S7 reflects the probability F^*(z) of occurrence of extreme (maximum) values of z[τ(t)], associated with moments t of detection of failures from normal (healthy) functioning (facts of disease)—the proportion of those who became infected and re-infected out of the number of those susceptible to infection during time t from the beginning of the epidemic. The rate of change of F^*(z) in time P(t) = dF^*(z)/dt = (dF^*(z)/dz)(dz/dτ)(dτ/dt) corresponds to the daily increment of disease cases—the density function of the failure distribution (disease incidence) (Figure 2), i.e., it shows the excess incidence of new coronavirus infection. The value of the AR coefficient α is found by the position τ = τ_m = θln(t_m/t_m) + t_m = t_m, z = 1 of the maximum P(t_m) = P_m of the function P(t): α = eP_m, τ(t_m) = t_m. The value calculated from the data is α = eP_m = 0.0264/day [19].

The integrated hazard (threat, possibility of getting infected by time t) is calculated by the Equation E^*(τ) = –lnF^*(z) = exp[–α(τ – τ_m)] and the risk of disease p(τ) = –dE^*(τ)/dτ = αexp[–α(τ – τ_m)]. In model Equation S7, the indicator K(τ, τ_m) = ln[E^*(τ)] = –α(τ – τ_m) depends linearly on τ and is the functional analogue of the universal Equation S2 with f(y) = K(τ, τ_m), a = –α, y = τ – τ_m. At the same time, the modelling problem can be set in the general form f(y), taking into account the influence of various environmental factors y = x – x₀ on the hazard indicator K(x, x₀) = f(y). An example is the Cox model [32] of risk assessment p(x) of an event occurrence taking into account the influence of independent variables (predictors) x = {x_i}.

The calculated values of K(τ, τ_m) are well approximated by a linear dependence on time t (Figure 7):

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Figure 7. The variation in time t of hazard indicator K(t, t_m) = lnE^*(t) of population infection with coronavirus in different countries. 1—Russia; 2—USA; 3—Brazil; 4—Peru; 5—Mexico; 6—Chile; 7—Bangladesh; 8—South Africa; 9—Italy; 10—Sweden.

Here, the ratio of the coefficients γ = αθ/t_m individually parameterises the epidemic process in different countries. The linear Equation 7 reflects the general trend of increasing K > 0 and decreasing K < 0 morbidity without taking into account periodic fluctuations (epidemic waves). The values τ_m, γ vary from country to country are unique to each country.

The linear regression coefficients a and b are statistically estimated:

According to the difference of a and b by countries, it is possible to estimate the peculiarities of epidemic processes and to display them differently on cartograms, in particular, according to the value of a = –γ—about AR a = –β = –aθ/t_m, and according to the values of b = βt_m = αθ—about the optimality of the management of the infection process: the smaller b—the more effective the preventive and anti-epidemic measures, the results of which are revealed during the period T = 100 ÷ 650 days.

Territorially, the values θ = b/α and t_m = –αθ/α vary, indicating the optimality and environmental conditions of the epidemic process realisation. Coefficients a and b of Equation 8 vary by country, for example, for Russia K = –0.00488t + 2.06 = –0.00488(t – 422.9) (correlation coefficient R = –0.98), for the USA K = –0.00625t + 2.26 = –0.00625(t – 362.1) (R = –0.98), for Brazil K = –0.00929t + 3.09 = –0.00929(t – 332.6) (R = –0.96). In Russia, γ = 0.00488/days, t_m = b/γ = 422.9 days, the proper time value θ = γt_m/α = b/α = 78.1 days. The value of t_m indicates the moment of culmination of the epidemic process, when K = 0 and the hazard has a critical value E^*(t_m) = 1. Later t_m dates characterize the best pandemic situation in terms of disease spread: Russia—423 days, South Africa—400 days, Germany—247 days, etc.

The value θ = b/α is proportional to the free term b of the regression Equation 8, so its increase indicates a worse quality of management of the epidemic process, which lasts longer. Of the countries studied, the best by this criterion are Saudi Arabia and Russia (θ < 80 days), while the worst (θ > 110 days) are Brazil and Sweden, which have been noted in the press as countries with low intensity of government pandemic management. The time indicators t_m and θ are not correlated (R = 0.02), so they can be used as independent indicators for mapping (Figure 8).

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Figure 8. Global change of the mathematical model parameters of the population response of different countries to COVID-19 infection: (a) the moment t_m (number of days) of the epidemic process peak; (b) the index θ (number of days) in the proper time.

In geoinformation mapping, the indicator method is implemented [33] when the unknown system status is identified by a set of indicators using a formulaic integrating estimation, which gives a graphical or cartographic representation informing the user about the required state. Graphs and mappings are used to compare indicators (signals)—individual parameters of a mathematical model of the system’s behavior—with each other and with limit values. In this case, the indicator method is implemented by estimating the parameters of the dependence K(t, t_m) = –(αθ/t_m)(t – t_m) (Equation 7).

The presented cartograms are filled with the results of calculations for countries for which statistical information is available. On the basis of this information, it is possible to estimate the values of t_m and θ, and to calculate the coefficients of Equation 8: α = –γ = –αθ/t_m, b = –αt_m = αθ, and then calculate K = K(τ(t), τ_m) = at + b, E^*(τ) = exp[K(τ, τ_m)], F^*(z) = exp[–E^*(τ)] in order. Thus, the set of cartographic geo-images-cartograms (Figure 8) reflects integral (synthetic) properties of the population response of different countries to COVID-19 infection, which is described by the equation of the corresponding mathematical model Equation S7, according to which derived indicators of the reliability of public health preservation can be calculated and then analytically mapped.

Of the processed spatial data on coronavirus epidemics in 86 countries, in 95% of cases, they satisfy the linear Equation 8 with a coefficient of determination R² > 0.90. The t_m values reflecting the moment of maximum coronavirus morbidity trend are concentrated in the interval of 300–500 days. They are shown on the cartogram in steps of 100 days, dividing this interval into two parts with different lags of pandemic deployment by country. The values of θ of the proper time course indicators are mainly located in the interval 60–100 days and are shown on the cartogram with a color scale interval of 20 days. The best indicators of a country’s healthcare system for managing the pandemic process are shown in green on the cartograms, while the worst are shown in red.