Modeling of COVID-19 by scientists, epidemiologists, and health experts was considered early on in the pandemic as it began to ravage the world. The socio-economic determinants of this pandemic taken into account in this modeling are important because they condition the severity of an affected country and the way in which it is controlled, which leads to consider the corresponding variables alongside the daily reproduction rates during the period of contagiousness of individuals infected by the pandemic. The aim of this article is to show the joint variations of socio-economic determinants and epidemiological parameters, which can be observed between developed and developing states and between successive epidemic waves.

Some researchers have already worked on socio-economic analysis of the COVID-19 pandemic: in [1], the authors examined the geoclimatic, demographic, and socio-economic determinants of COVID-19 prevalence and have shown that the influence of these determinants varies by comparing the first and second wave of the pandemic. The socio-economic impact of the COVID-19 pandemic in United States of America (United States) was studied by Barlow and Vodenska [2], where the authors investigate the systematic risk posted by sector-level industries within the United States. Ahmed et al. [3] modeled daily confirmed cases of COVID-19 in different countries across the globe using regression models with predictions for upcoming scenarios. Kong et al. [4] worked on the socio-economic and environmental factors influencing the basic reproduction number of the COVID-19 pandemic by fitting a logistic growth curve to the reported daily cases up to the first peak of the pandemic while Qiu et al. [5] studied the impact of socio-economic factors on the transmission of COVID-19 disease with China as a case study using an empirical model, and the authors conclude that these determinants have rich implications for ongoing efforts in containing the pandemic. The work in this present article is an extension of [6], which was based on the analysis of the reproduction numbers of COVID-19 based on the Current Health Expenditure as Gross Domestic Product Percentage (CHE/GDP) across several countries using some machine learning tools. The results of this study show that some countries with a high CHE/GDP improved their public health strategy against the virus during the second wave of the pandemic, fighting it all the more effectively against it the more effectively they were. the most affected during the first wave. The difference with the present study lies in the fact that the latter takes into account data from twice as many public sites [7–17] and that it is more focused on social inequality, quantified for example by the Social Fracture coefficient (SF equal to the ratio between the incomes of the richest 10% and the poorest 10% of a given population), and the Theil and Gini indices. It was shown previously in [18] that the Gini index was highly correlated (r = 0.93) to another Demo-economic index denoted DI and equal to the quotient (CHE/GDP)/SF, proving that all these indices are closely related and carry part of the causality of inter-country variations in epidemiological parameters.

The main objectives of this article are to establish a relationship between Theil and Gini index, analyze critically some of the socio-economic determinants of the pandemic, correlate them, predict three of the socio-economic variables, and perform some regression analyses. We have also clustered countries according to these parameters with the help of the lasso (least absolute shrinkage and selection operator) method, and we were able to select the best variables for the COVID-19 modeling.

The paper is divided into seven sections: after an introductive section, we explain in Section 2 the methodology used in this research, Section 3 deals with the variables used, Section 4 establishes a mathematical relationship between Theil and Gini index, Section 5 is dedicated to the visualization of the results obtained, while we finally give the discussion and conclusion in Sections 6 and 7, respectively.

The use of machine learning methods to analyze data has been helpful over the years to get a proper view on how a model behaves. In this research, we used some supervised and unsupervised machine learning methods and we also tried to use one deep learning method for the identification and visualization of clusters. To jointly interpret the socio-economic and epidemiological data, we have chosen these three main classes of the descriptive statistics, which allow us to compare these socio-economic and epidemiological data. Supervised learning is used in its regression function (prediction of a quantitative variable from annotated examples) and unsupervised learning (in which the data is not labeled) in its classification function. As for deep learning, it makes it possible to create a model from large-scale unlabeled data.

The supervised machine learning methods we used are first univariate polynomial regression, linear regression, lasso regression, and ridge regression. We also use some of these methods to make prediction by training the model and testing some percentage of the values. Lasso regression helped us to know the best variables to be used in the modeling. After the univariate regressions, we introduced multivariate least square methods, allowing us to test much more complex relations between variables. It can be represented as follows: y = β 1 x 1 + β 2 x 2 + ⋯ + ∈ , (1) Where β 1 , β 2 , ⋯ are coefficients or weights, ∈ is the residual noise, y is the dependent variable, and x 1 , x 2 , ⋯ are the independent variables. Ridge and lasso regression are simple methods to reduce the model complexity and prevent over-fitting, which may result from linear regression. The cost function for ridge regression is given below: ∑ i = 1 n ( y i − y ^ i ) 2 = ∑ i = 1 n ( y i − ∑ j = 0 m β j × x ij ) 2 + α ∑ j = 0 m β j 2 (2) with for some c > 0, ∑ j = 0 m β j 2 < c , while α is the penalty term that regularizes the coefficients such that if the coefficients take large values, the optimization function is penalized. Ridge regression puts constraint on the coefficients β . We define the cost function for lasso regression in the same way, but by replacing the L2 penalty term by an L1 one as: ∑ i = 1 n ( y i − y ^ i ) 2 = ∑ i = 1 n ( y i − ∑ j = 0 m β j × x ij ) 2 + α ∑ j = 0 m | β j | (3) with for some c > 0, ∑ j = 0 m | β j | < c .

After the supervised learning methods, we used unsupervised learning approches to cluster variables across countries and the methods we proposed to validate our results were K-means clustering, Hierarchy clustering, and Principal Component Analysis (PCA). We also performed correlation calculations among parameters used in the modeling step and we chose an optimization method called Ordinary Least Square (OLS) for the socio-economic determinants of COVID-19. Eventually, the deep learning methods we used were Neural Network (NN) and Multi-Layer Perceptron (MLP) regressor, which is a class of feedforward Artificial Neural Network (ANN).

We have been able to develop new approaches to the socio-economic determinants for the modeling of the COVID-19 pandemic during the exponential phase. Some of these determinants have shown high correlation with epidemiologic parameters as it can be seen in the heatmap diagrams in Figures 2G, 8A, 11A, 14A, explaining the role of each variable thanks to these correlations.

For developed and developing countries, the lasso regression reduced the correlation between the social fracture index and the 10% highest income, while for OECD countries, the correlation between the Gini index and social fracture index was reduced to zero. Some of our variables were not used in the optimization method-OLS due to multicollinearity observed on results summary. For the two sets of countries, consumer confidence index, opposite of the initial autocorrelation slope averaged on 6 days for the first and second wave, 10% lowest income, and 10% highest income were not used in the modeling. The R ² for OLS results for developed, developing, and OECD countries are 0.76 and 0.90, respectively, which shows a high significance rate (Figures 8E,F, 12).

The principal component analysis shows high correlation for the numbers of new cases we used in this research. The social fracture index has high correlation in PC1 for both cases, while in PC2, percentage of GDP devoted to health expenditure was dominant for OECD countries, and maximum R o for the first wave was dominant for both developed and developing countries (see Figures 8C, 14C). We can deduce from all these observations that the socio-economic determinants are a key to the modeling of infectious diseases like COVID-19 as these parameters give high signals on the trend during the spread of the pandemic for various countries ([22–30]).

The systematic study of the correlations between socio-economic variables (Gini and Theil indices, percentage of GDP devoted to heath expenditure, etc.) and epidemiological variables (reproduction rate, opposite of the slope of autocorrelation to origin, etc.) shows a disparity between developed and developing countries, as well as between epidemic waves. Developed countries with high indices of social divide, but high health expenditure, did not, for the first wave, react better to the COVID-19 epidemic than developing countries. On the other hand, the rapid implementation of isolation and vaccination measures enabled them to anticipate and reduce the effects of the second wave. In a subsequent work, we will study the evolution of this disparity between developed and developing countries during subsequent waves of SARS CoV-2.