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Analyzing Response Efficiency to COVID-19 and Underlying Factors of the Outbreak With Deep Assessment Methodology and Fractional Calculus

This study focuses on modeling the daily deaths per new case of COVID-19 by using the Fractional Calculus and the Least Squares Method. Based on our prior work, we proposed a new modeling approach, assessed the strength of outbreak response, and analyzed possible underlying factors of the outbreak for 8 countries including China, France, Germany, Italy, Russia, Spain, the UK, and the US. First, we modeled weekly deaths per new case of COVID-19 using our new modeling method Deep Assessment Methodology - Second Derivative (DAM-SD). Later, we defined a performance indicator to understand how well each country copes with the pandemic. Lastly, Pearson correlations between the performance indicator and several economic, social, and environmental indices, such as Human Development Index, Human Freedom, Democracy, Competitiveness, and Trust Index are computed, and p-values are reported. Results showed that DAM-SD successfully models the daily new cases of COVID-19 with 3.7390% Mean Average Precision Error and outperforms the DAM by 1.5678% MAPE. China is the best-modeled country with 4.0975e-08% MAPE whereas the model produced the highest error rate for France as 8.8317% MAPE. According to the analysis with the performance indicators, China is the most successful country against the pandemic while the United States and France fail to confine COVID-19 outbreak compared to the others. Indicators such as Human Development Index, Human Freedom, Human Democracy, GINI Index, Workers Rights, the Trust Index, and Air Pollution are found significant for COVID-19 response according to the p-values. In the correlation analysis, Average Class Size, Government’s Long Term Vision, Responsiveness to Change, Better Life Index, and Population Density were the least significant indicators. Long Term Care Beds, Social Capital, and Global Social Mobility indicators are found correlated with the COVID-19 response. Household Spending and Student Skills are found insignificant.


I. INTRODUCTION
C OVID-19 disease was first identified in Wuhan, capital of Hubei province of China, in December 2019 and declared as a global pandemic by the World Health Organization on March 11, 2020. As of October 2021, the disease has infected 244 million people and caused more than 4 million deaths worldwide. Governments, institutions, and policymakers are trying to design the optimal outbreak response to curb the spread of the disease and ultimately reduce the number of people becoming infected. Therefore, understanding the dynamics of the outbreak and response are crucial for effective filtering and evaluation of candidate strategies.
Computational, statistical, and mathematical methods have attracted researchers for decades in analyzing the characteristics of any natural phenomena, including biological epidemics. Because of the ongoing COVID-19 crisis, scientific institutions, universities, and companies are actively investigating various mathematical methodologies for predicting the trend, estimating the peaks, the end, and modeling the course of the pandemic. For instance, in [1], three types of mathematical models, the logistic model, Bertalanffy model, and Gompertz model, were adopted for fitting COVID-19 data. A compartmental model that takes super-spreaders into account is proposed, and the proposed model is evaluated with the data from Wuhan in [2]. Reference [3] combines ARIMA and wavelet-based predicting techniques for short-term future prediction of Canada, France, India, South Korea, and the UK. In [4], a method based on a dictionary learning and online non-negative matrix factorization is proposed to predict the COVID-19 outbreak, and the near future is extrapolated by one step recursive predictions. Reference [5] investigates the early transmission dynamics of COVID-19 in Wuhan by fitting a stochastic transmission dynamic model using sequential Monte Carlo simulation and obtains the basic Reproduction number R t . Schüttler et al. [6] proposed a Gauss Model to map the time to the Gauss function, modeled the daily deaths per day, and forecasted the further progression of the fatalities per country. In [7], authors use a mean-field model to understand the temporal dynamics of the epidemic spread and predict the time of the peak of confirmed cases for Mainland China, Italy, and France. In [8], the mortality rate is estimated with the ARIMA and the regression models. Further, [9] focuses on uncovering persistent dynamics of COVID-19 cases by using the Switching Kalman Filter algorithm to predict the future of the infection. The family of compartmental models is being investigated for mathematical modeling of COVID-19 [10]- [12], as well. For instance, the SIR model that uses three variables (susceptible (S), infected (I), and recovered (R)) is employed to predict the epidemic characteristics in mainland China and estimate the final size of the COVID-19 pandemic [13], [14]. In the original SIR model, transmission rate, β and recovery rate, γ are constant variables and do not change by time. In [15], the authors formulate the transmission and recovery rates as time-dependent variables to track and control the disease spread better. Further, a version of the SIR model, "SEIR" which has an additional element E for exposed, is introduced in [16], and the evolution of the epidemic in Hubei Province is shown. Another compartmental mathematical model with six variables including asymptotic (A), isolated infected (I) q , and quarantine susceptible (S q ) is proposed and sensitivity analysis of parameters is carried out in [17]. Also, the mathematical models are used to identify the effect of co-morbidities on COVID-19 mortality rates too. In [18], the authors have estimated the link of demographic and co-morbidities with COVID-19 test positivity alone, or with mortality, with logistic regression. An Elastic Net regularized binary classification is performed to select for the most important elements such as co-morbidities and demographics in [19].
Fractional Calculus (FC) is a branch of mathematics that allows operations over arbitrary positive order of integrals and derivatives and extends the concept of calculus to the entire real and complex axis. Fractional operators are superior to their integer-order counterparts for two reasons. First, fractional operators provide flexibility and are capable of modeling systems with complex dynamics. Also, conventional integer-order operators have no memory because their locality properties and solutions of integer order differential equations do not depend on past values. On the other hand, fractional operators have non-locality properties effective for modeling systems that possess historical dependence. Hence, fractional operators have attracted various disciplines of science and engineering including mechanics, biology, biomedical devices, nanotechnology, diffusion, diffraction, economics, and control theory [20]- [39].
Fractional Calculus (FC) has been widely utilized for analyzing the characteristics of various epidemics such as Measles, Tuberculosis, Malaria, Dengue, and Ebola [20]- [24] in the literature. Recently, [25] presented a mathematical model based on Fractional Calculus for the transmission rate of the COVID-19 pandemic. Also, [26] illustrated that the fractional operators are superior to their integerorder counterparts for exploring the hidden dynamics of the infection by using the data of India.
Several studies are published recently on the containment of COVID-19 and optimal COVID-19 response. [27] argues that game-theoretic modeling should be used for obtaining the best policy response to control the COVID-19 pandemic. Another study [28] proposes an epidemiological model on a social network to study how the COVID-19 epidemics evolve and how it is contained by different vaccination strategies.
Earlier, we introduced various mathematical modeling and prediction methods that employ Fractional Calculus where we focused on analyzing time series such as children's physical growth, subscriber numbers of operators, GDP per capita, and compared the results to other modeling approaches such as Fractional Model-1, and Polynomial Models [30], [31]. Results showed that Fractional models had superior results when compared to Polynomial Models and Fractional Model-1 [31], [30]. Most recently, we used Deep Assessment Methodology (DAM), our previously proposed modeling and prediction method that exploits Fractional Calculus, for analyzing COVID-19 [33]. In this paper, the non-locality property of fractional operators, existing research on employing FC for infections, and our previous modeling results with DAM and FC are the foundations of our motivation for analyzing COVID-19 by using FC.
Previously, we proposed Deep Assessment Methodology (DAM), a modeling and prediction method, which exploits Fractional Calculus and Least Squares Method. In [33], we showed that the DAM performs better than Polynomial Method and vanilla Long Short-Term Memory (LSTM) for modeling and prediction, respectively. DAM represents any function by its previous values and its derivative. In this paper, we propose a new approach Deep Assessment Methodology -Second Derivative (DAM-SD) which we extend the function representation of DAM with secondderivative.
The main objectives of this study are assessing the performance of DAM-SD, modeling daily deaths per new COVID-19 case, and analyzing the correlation between various social and economic indicators with the combat against COVID-19 outbreak. To achieve these goals, we modeled the daily deaths per new case trend of COVID-19 for China, France, Germany, Italy, Spain, Russia, the UK, and the US using our new modeling method, DAM-SD. Later, we defined a performance indicator using the modeled functions to understand how well each country fights against the COVID-19 spread. The performance indicator is formulated based on the assumption that the daily new case curve is a wave with symmetrical left and right-hand sides around the peak day. This indicator is referred to as β p throughout the paper. Lastly, we investigated possible underlying factors behind the success against COVID-19 outbreak by computing the Pearson correlation between β p and various indicators related to the capacity of coping with a pandemic, social-economical status, educational and behavioral status, and vulnerabilities.
The organization of this paper is as follows. Section II introduces our new methodology DAM-SD and performance indicator β p . Section III reports the modeling results and values of the performance indicators and analyzes the correlation between β p and the picked subset of indicators. Lastly, in Section IV, the conclusion is given.

A. DEEP ASSESSMENT METHODOLOGY WITH SECOND DERIVATIVE
As aforementioned, we previously proposed a mathematical method for modeling and prediction, DAM, which exploits the first-order derivative of functions and their Taylor series expansion in [33]. DAM represents any time instance of a function g(x) as a weighted sum of its previous values and their first-order derivatives. Here, we propose including the second-derivative in the function representation as well as the first-order derivative. We refer to this model as DAM-Second Derivative (DAM-SD). Let g(x) be continuous and differentiable. With DAM-SD, the function g(x) is represented as shown in (1).
We can express g(x) as follows using Taylor Series expansion: Likewise, any previous time instance of g(x) is written as: By substituting the Taylor expansion into (1), g(x) and its derivative g ′ (x) are expressed as shown in (4) and (6), respectively.
We can re-write the equation above as: Substituting Taylor expansion into derivative g ′ (x) yields, The next step is including the heritability property with fractional calculus. DAM-SD employs Caputo's description of the fractional derivative as shown in (7).
Here, Γ is the Gamma function and γ suggests the order of the fractional derivative. The j th order derivative with respect to x is referred to as g (j) . This way, the derivative in (5) is generalized by changing the first-order derivative into fractional-order γ as in [29]- [31]. In DAM-SD, the order of derivative, j, is set to 1, and the fractional-order γ changes in the range of [0,1]. This study consists of two parts of numerical analysis as explained in Section III. For the first part, it is assumed that function f (x) represents the daily deaths per daily new cases of COVID-19 pandemic over time which is equal to (8). For the second part, it is assumed that function f (x) represents the daily new cases VOLUME 4, 2016 of the pandemic.
The function f (x) that satisfies (9) models any time instant of the discrete daily new COVID-19 dataset similar to the arbitrary continuous g(x) function is given in (6).
Equation (9) is the fractional-order derivative of function f (x) where x denotes the time. Exploiting the fractionalorder γ parameter in (9) provides more general and flexible modeling compared to (6) [32]. Compared to the integer counterparts, the fractional derivative has one additional parameter γ to optimize that helps to achieve better optimality. Next, (9) needs to be converted to an algebraic form. Here, we use the Laplace Transform. After taking the inverse Laplace Transform of the transformed algebraic equation, we get the final form of f (x) as: where, Now, the representation for function f (x) can be obtained by finding the unknown coefficients a kn , b kn , c kn and f (0), and parameters M , l, and γ by minimizing the squared total error between the real data. The squared total error referred to as ϵ 2 T , is found by summing the error of each instance in the dataset and calculated as shown in (12): where m is the total number of instances in the dataset and P i and f (i) denote the real and approximated data instances, respectively. The square of total error, ϵ T 2 , is minimized with a gradient-based approach and the unknown coefficients are found by solving a linear system of equations. Readers can find detailed explanation in [33].

B. MEASURING THE SUCCESS OF THE FIGHT AGAINST COVID-19
Understanding the performance against the COVID-19 pandemic is crucial for policymakers to pick optimal measurements and take necessary actions to stop further progression. In this section, we define a performance indicator β p for understanding how well each country copes with the pandemic. We introduced this indicator by assuming the daily new cases curve of a pandemic is a curve with a single symmetrical wave that decays after the peak point with an inverse trend of its rise. By this assumption, the pandemic is restrained after the first wave of the daily new cases curve. Fig. 1 illustrates both the expected daily new cases curve we assumed to be ideal and the real daily new cases curve, represented by f cm and f c , respectively. Horizontal axis refers to time where x 0 , x p and x t represent the initial day of the pandemic, the peak, and today, respectively. The expected curve, f cm , is a symmetric curve around the day of the peak, x t . Here, f cm is obtained by modeling the real data starting in the range [x 0 , x p ] and mirroring the modeled curve around x = x p .
We calculate the β p indicator as shown in (13) as the ratio of area under the expected daily new cases curve to the area under the actual daily new cases curve. We used a similar notation as Fig. 1. The magnitude of β p is a measure of the success of a country's fight against the pandemic. A value close to 0 is a sign of an ineffective response.
Multiple peaks on the daily new case curve produce a larger area under the curve when compared to the ideal expected state. The multiple peaks also produce a β p closer to 0. On the other hand, the β p indicator is close to 1 when the actual data curve has symmetrical left and right-hand sides around x = x p . The magnitude of β p hints at the overall situation of the pandemic. The greater values of β p indicate the pandemic is relatively under control. Similarly, smaller values of β p imply poor performance against COVID-19.

III. NUMERICAL RESULTS
In this section, we provide the modeling results of the daily deaths per case ratio using plain DAM and DAM-SD, compute the performance indicator β p , and discuss the correlation between β p various social, economic, governmental, educational, and behavioral indicators possibly linked to the outbreak of the pandemic. We acquired the data of daily new cases of COVID-19 from [40]. The proposed approach DAM-SD is implemented on MATLAB   and publicly available at [41]. The countries are selected considering two main factors: virus transmission risks and economic activity. The selected countries have large tourist rates, carry a large number of air passengers, and have a high amount of population. The countries are members of the G-20, except Spain. We report the results using the Mean Absolute Percentage Error (MAPE) metric computed as follows: where k is the total number of samples, P (i) is the real value and f (i) is the predicted value for i th sample.

A. MODELING OF DAILY DEATHS/NEW CASE RATIO
In Section II, briefly, we introduced our new modeling method Deep Assessment Methodology Second Derivative (DAM-SD). In this section, we report the modeling results of the COVID-19 daily deaths per case ratio and assess the performance of DAM-SD by comparing it with DAM. To eliminate the variables of the day of the week, we averaged the weekly data and reported the modeling results of the weekly deaths/new case ratio. We denote the weekly deaths per new case ratio as r. For week i, the ratio r is computed as: Daily new cases and daily deaths are obtained from [40]. In [33], we showed the superiority of DAM to vanilla LSTM and Polynomial method. Here, we compare our new approach DAM-SD to DAM [33]. Table I reports the modeling errors, optimized M , l and, γ values of both DAM and DAM-SD methods for modeling weekly deaths per new COVID-19 case. As shown in the table, the DAM-SD models the deaths per case ratio with 3.7390% average MAPE. For all countries, DAM-SD yields a smaller MAPE and is superior to DAM. Fig. 2 illustrates the DAM-SD modeling curves of the deaths per case ratios. We employed grid search for finding the optimal fractional-order γ values.
Optimal γ values are reported in row 4 and row 8. For both models, α's different from 1, demonstrating the advantage of fractional-order derivatives compared to integer-order counterparts. Both DAM and DAM-SD methods model China with the minimum error. Modeling France yields the largest error for both models with 8.8317% and 12.6112% error rates for DAM-SD and DAM, respectively. For DAM-SD, the largest γ is found for Russia as 0.9938 and the smallest value of γ is found for Spain as 0.0972. As mentioned in VOLUME   Section II, when the fractional-order is 1, the method is equivalent to employing a classical integer-order derivative. The search area of the optimal γ includes 1. As seen in the last column, optimized γ values for all countries are other than 1, indicating the superiority of the fractional-order methodology over integer order differentiation. As seen in Fig. 2, countries such as Germany and France consist of relatively large variance. These countries produce higher error rates.

B. ANALYSIS OF COVID-19 RESPONSE
In this section, we assess the effectiveness of the COVID-19 response using the performance indicator β p . For computing β p , we need to determine the expected curve of the daily new cases of each country. We assumed that the expected daily new cases curve is a single peak curve with symmetric left and right-hand sides. Our purpose is to understand the relative performance benefit of the analyzed countries among themselves. The performance indicator β p measures how close the real data curve is to the ideal daily new cases curve. Fig. 3 illustrates the real data and the ideal expected curves of daily new cases for China, France, Germany, Italy, Spain, Russia, the UK, and the US. Red dashed vertical lines denote the day with the maximum daily new cases during the first wave, except for China. For China, we treat the second-largest number of daily new cases as the peak. To find the expected curves, we modeled the data from the initial day until the peak day using DAM-SD. Later, we mirror the curve found by modeling around the peak day and obtain a symmetric expected curve. Lastly, we measured the COVID-19 response performance at 11 th of March by computing the performance indicator β p as described in section II. Table II reports the Performance indicators for China, France, Germany, Italy, Russia, Spain, the UK, and the US. The largest β p is produced by China as 0.3206 whereas the lowest is produced as 0.0082 by the US. As mentioned earlier, β p is a measure of how effectively a country combats with COVID-19. A larger β p indicates a better COVID-19 VOLUME 4, 2016 performance. According to the results, the most successful country against the pandemic is China, whereas France, the US, Spain, and the UK are fighting inadequately. The aforementioned indicators are grouped into four main categories: indicators of coping with the pandemic, socialeconomical indicators, educational and behavioral indicators, and indicators of vulnerability against the pandemic. Table III, IV, V, and VI reports the indicators and corresponding p-values. Indicator data are provided in terms of either score or rank. The values are converted to scores by taking the multiplicative inverse of the data when only ranks are provided. Lower p-values indicate strong correlations between the indicator and the performance coefficient. Regarding Long Term Care Beds, Population Health Coverage, Better Life Index, Average Class Size, and Lack of Social Support indicators, values for China or Russia were not available. Therefore we computed correlations by excluding the missing country/countries for these indicators.

C. ANALYZING POSSIBLE UNDERLYING FACTORS OF
Indicators of Coping with the pandemic show strength of governments' effort and existing structure. Infrastructure indicator measures the quality of transportation and communications infrastructure. Institutions indicate the level of basic security, enforcing property rights, transparency, and efficiency. The government's Long-Term Vision indicates the capability of carrying out long-term projects. Respond to Change indicator measures to what extent the government responds effectively to the technological, economic, societal, and security changes. Efficiency enhancers is the indicator of goods market and labor market efficiency, higher education and training, technological readiness. Basic requirements indicator considers features related to the macroeconomic environment, health and primary education, institution, and infrastructure status. The better life index allows comparing well-being across countries based on topics including health, education, safety, access to services.
(h) The real and the expected daily new cases curves for the US. The day of the peak is April-25. this category are larger than 0.1. The smaller the p-value, the stronger the evidence that one should reject the null hypothesis. By implication, the least significant indicators for the performance against the pandemic are Government's Respond to Change and Basic Requirements. Better Life Index, Basic Requirements, and Efficiency Enhancers did not find significant in this analysis. The most significant indicator of this category is Infrastructure. Long Term Care Beds is the second most significant indicator under the category. Table IV  Results indicate Human Development Index, Human Freedom, Human Democracy, Worker's Rights, and GINI Index are significant for coping with the pandemic. The least significant indicator of this category is Household Spending.
Next, Table V reports the correlation between the performance indicator β p and various educational and behavioral indicators such as Mean Years of Education, Critical Thinking, Average Class Size, Student Skills, Educational Attainment, and The trust index. The most significant indicator is computed for the Trust Index, producing a pvalue smaller than 0.05. Critical Thinking, Average Class Size, and Student Skills are found insignificant for the fight against the pandemic. Lastly, Table VI reports indicators related to the vulnerability of a country against a pandemic. The p-value for Air Pollution is found smaller than 0.05 meaning Air Pollution is significant for the fight against the pandemic. The least significant indicator of this category is found Population Density. Table III, IV, V, and VI showed that Human Development Index, Human Freedom, Human Democracy, Worker's Rights, GINI Index, Trust Index, and Air Pollution indicators are significant for the COVID-19 battle. The global impacts of COVID-19 are highly unpredictable, and decisions concerning the outbreak are made in a rapidly evolving environment. Policymakers may embrace these findings for designing rational strategies against COVID-19 or a further pandemic.

IV. CONCLUSIONS
In this paper, we proposed a new modeling approach Deep Assessment Methodology SD (DAM-SD) and, modeled the weekly deaths per case data of COVID-19 for China, France, Germany, Italy, Russia, Spain, the UK, and the US. We assessed the performance of our new method by comparing it with DAM [33]. Later, we assumed the daily new cases of a pandemic would be a single wave curve, symmetric around its peak, and defined a performance indicator to measure how well each of the 8 countries copes with the ongoing COVID-19 pandemic. Lastly, we investigated the possible underlying factors of the outbreak control by Educational and p behavioral Indicators Mean Years of Education [42] .1033 Critical Thinking [42] .8130 Average Class Size [57] .9444 Student Skills [42] .7069 Educational Attainment* [58] .2799 The Trust Index [60] ** < .05 .6266