Modeling the COVID-19 incorporating oil futures

doi:10.21203/rs.3.rs-4997929/v1

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Modeling the COVID-19 incorporating oil futures

https://doi.org/10.21203/rs.3.rs-4997929/v1

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The COVID-19 pandemic necessitated the production of mathematical models that were able to explain and thoroughly study various aspects and features of the pandemic. In this work, we provide a well-stated mathematical model to explain and simulate the evolution of the COVID-19 pandemic. To test our model’s performance and validity, we utilize actual surveillance data from the pandemic, capturing the results of this empirical investigation. According to the results, our model is valid, since all estimates are statistically significant, and the coefficient explains the evolution of the pandemic.

COVID-19

mathematical modeling

pandemic

data analysis

oil prices

The COVID-19 pandemic was an extreme event that caused panic and fear in the worldwide community [1]. Since the COVID-19 pandemic was followed by unprecedented health impacts for the global society, many governments and policymakers took extreme measures to limit its spread. Based on these facts, it should not be surprising that the research community also focused on the analysis of various aspects and characteristics of the COVID-19 pandemic. Thus, the modeling of the COVID-19 spread is imperative, utilizing many models and techniques, to derive most of the information that could be extracted [2].

In this context, the list of techniques and approaches includes, but is not limited to, the Caputo-Fabrizio derivative [3, 4], fuzzy models [5], Kalman filters [6], chaotic modeling [7, 8], or even combination of stochastic models with machine learning techniques [9]. Although many approaches have tried to model, simulate, or even forecast the COVID-19 spread, the literature has not concluded a certain model or a family of models that is outperforming the others. For example, Busari and Samson [10] tested many models, some of them being regression models, machine learning approaches, and autoregressive integrated moving average models. According to the authors, certain regression models provided interesting results. Similarly, Middya and Roy [11] tested many models and also concluded that no model is superior in all aspects to all the others. According to Nixon et al. [1], after an extensive literature review regarding models for the pandemic simulation, many papers have demonstrated uncertainty and limitations, and some of them did not evaluate properly the model’s performance. In this context, the literature lacks adequate and robust research approaches that can efficiently model the COVID-19 spread and the pandemic’s evolution. To fill this gap in the literature, we propose a mathematical model that has stochastic inherent properties, able to model the COVID-19 spread, as validated by the utilization of surveillance data. Our model is oriented on the mathematical modeling of the pandemic, using simple models i.e. regression models. Moreover, it is tested on real data, proving in this way its superiority since the results indicate statistical significance.

Therefore, the novelty of our proposed model in modeling the COVID-19 spread stems from its distinct attributes highlighted by existing studies. Particularly, while many approaches delve into complex techniques like fractional operators or chaotic modeling, our model stands out by relying on simpler yet robust regression models, as indicated by studies from Busari and Samson [10] and Middya and Roy [11]. Moreover, our approach emphasizes empirical validation against real data, addressing a gap highlighted by Nixon et al. [1] and echoed in studies by El Koufi and El Koufi [12], Matveeva and Leonenko [13], and Kalachev et al. [14], while unlike models focusing on isolated aspects, our model presents a comprehensive framework, aligning with the call for a single superior model by Nixon et al. [1] and Middya and Roy [11]. The statistical significance of coefficients in our model, showcased in Table 1 and Table 3, underscores the efficacy of employing simpler regression techniques compared to more complex models utilizing fractional operators, emphasizing the interpretability and significance of our results within the complex landscape of COVID-19 modeling. Ultimately, incorporating one of the most pivotal commodities, namely oil, enriches this approach by considering multiple dimensions of global conditions, regarding health and the economy, during the COVID-19 pandemic.

The remainder of the paper is structured as follows: section 2 presents the literature review; section 3 describes the methodology and the models employed; section 4 analyzes the results; and finally, section 5 concludes the paper.

The COVID-19 pandemic was an unprecedented event with many repercussions worldwide. Thus, it should not be surprising that the modeling of the COVID-19 pandemic was imperative, making many scientists examine thoroughly this difficult situation, from many aspects. To do so, researchers have employed many different models and techniques from various fields and disciplines. To begin with, many approaches try to model the COVID-19 virus transmission. To give some examples, Najarzadeh et al. [5] used a fuzzy model to simulate the spread of the virus, while another approach implemented the Kalman filter, based on a stochastic epidemiological model [6] to simulate the spread of the pandemic. Furthermore, Matveeva and Leonenko [13] utilized a Gaussian process in a regression model to simulate the COVID-19 spread, highlighting in this way the usefulness of these models. Similarly, El Koufi and El Koufi [12] provided a stochastic model for the modeling of the COVID-19 spread, for the case study of Pakistan. Kalachev et al. [14] utilized three formulations of the Susceptible-Infected-Removed model, a well-established framework in the investigation of epidemics, with application in the case study of the influenza outbreak and the COVID-19 pandemic, providing insights into the model’s performance.

Moreover, Bilgram et al. [9] used a combination of stochastic and machine-learning models to examine the safe and near-optimal strategies for virus exposure. Similarly, Middya and Roy [11] implemented and tested the performance of various deep learning models, regarding the health outcomes of the COVID-19 spread for the case study of India. According to the authors, no model outperforms all other models, implying that each model has its strengths and weak points.

In the same context, Farman et al. [3] modeled the dynamical transmission of COVID-19 utilizing Caputo-Fabrizio fractional operators, proposing a model capable of performing for integer and non-integer scenarios, while Pandey et al. [4] through a similar approach, utilizing the same technique, examined the mathematical modeling of COVID-19 spread for the case-study of India. Abbes et al. [15] utilized the same technique, the Caputo fractional difference operator, to investigate the COVID-19 spread, incorporating non-linear dynamic behaviors. The authors also examine commensurate and incommensurate fractional orders, through several numerical techniques, including phase attractors, maximum Lyapunov exponents, bifurcation diagrams, and C₀ algorithm. While Khairulbahri [16] used the same technique, also incorporating behavioral measures, asymptomatic cases, and lockdown measures, to render the model multi-faceted.

Another strand of literature tries to model the chaotic components of the virus spread. For example, Al-Basyouni and Khan [7] utilized linear stability theory to model the chaotic behavior of the pandemic, while Farman et al. [17] used fractal fractional techniques to examine the spread of the Omicron variant COVID-19 virus. Similarly, Khan et al. [8] utilize fractal techniques to model the COVID-19 pandemic. Moreover, Arshad et al. [18] employ a fractional order epidemic model to mathematically assess the spread and its characteristics. Additionally, Li and Guo [19] used the optimal control and cost-effectiveness analysis to create a sufficient model regarding the Omicron strain.

Furthermore, Chen et al. [20] modeled the effect of surveillance and physical distancing measures on the COVID-19 dynamics at a local aspect. We must point out that van der Vegt et al. [21] introduced a package in the R statistical software to model the spread of the COVID-19 pandemic, unveiling the importance of the spread’s modeling. More importantly, there is also a comparison of techniques regarding the modeling and the forecasting of the spread of the pandemic, by investigating and predicting the new cases, for the case study of Nigeria, employing various models, i.e. regression techniques, autoregressive integrated moving average models and machine learning approaches [10]. The overall results provide evidence of the models’ superiority, some of them being also certain regression models.

Moreover, some studies examine the various features and events related to the pandemic, and not the spread itself. More precisely, Song et al. [22] used a stability and optimal control model, also examining the vaccination effect and the isolation delays, regarding the COVID-19 pandemic. Similarly, Tchoumi et al. [23] focused on vaccination and its effect on the limitation of the pandemic [23]. As regards the quarantine effect investigation, Singh et al. [24] propose a dynamical transmission model of a coupled non-linear fractional differential equation in the Atangana-Baleanu Caputo, to examine the COVID-19 transmission in a dynamic framework. Ojo et al. [25] utilized several nonlinear optimal control strategies to develop a robust mathematical model for the investigation of the co-infection of both COVID-19 and influenza.

Additionally, some other approaches utilize advanced modeling [26], while some others develop reactive–diffusion epidemic models to examine the human mobility during COVID-19 [27]. Finally, some studies are limited to even more specific aspects of the COVID-19 pandemic, for example, Rayegan et al. [28] examined the airborne transmission of the virus, only indoors. On the other hand, some studies use mathematical modeling to predict the COVID-19 mortality rate [29].

In this point of reference, the modeling of the COVID-19 confirmed cases, and thus, the evolution of the pandemic is not adequately examined, with many approaches failing to provide a well-stated mathematical model with an empirical evaluation of its validity and superiority. This is the gap in the literature we aim to fill with our approach.

Let X (t) be the level of COVID19 spread, which is modeled as a geometric Brownian motion to capture the the exponential growth behaviour. That is, $\:X\left(t\right)=X\left(0\right){e}^{\left(\mu\:-\frac{1}{2}{\sigma\:}^{2}\right)t+\sigma\:W\left(t\right)}$, where µ is the expected growth rate, σ is the volatility of growth rate, W(t) is a standard Brownian motion. Thus, $\:E\:X\left(t\right)=X\left(0\right){e}^{\mu\:t}$.

We also can portray the dynamics of X (t) in terms of a phase diagram analysis as follows:

$$\:g\left(t\right)=\alpha\:-\beta\:X\left(t\right)+\epsilon\:$$

where g is the growth rate of X at time t, α and β are positive constants, ε is an error with Eε = 0. This specification is consistent with the reality of COVID19. That is, it insures that (for each realization) g is strictly concave in X. That is, initially and at early stages .the growth is positive, then it later reaches zero, then it becomes negative. It follows that

$$\:Eg\left(t\right)=\alpha\:-\beta\:EX\left(t\right)$$

Therefore, at equilibrium (when $\:Eg\left(t\right)=0$), we have

$$\:EX{\left(t\right)}^{*}=\frac{\alpha\:}{\beta\:}=X\left(0\right){e}^{\mu\:{t}^{*}}$$

where t^* is the time at which the growth rate g is zero; for t < t^*, g > 0, for t > t^*, g < 0. This is intuitive.

Equation (1) can be readily estimated using the following regression

$$\:g\left(t\right)={B}_{0}-{\beta\:}_{1}X\left(t\right)+\epsilon\:$$

to obtain estimates for α and β; $\:{\widehat{B}}_{0}$ is an estimate of α and $\:{\beta\:}_{1}$is an estimate of β.

From the previous section:

$$\:dX\left(t\right)=X\left(t\right)(\mu\:dt+\sigma\:dW\left(t\right))$$

Thus µ can be estimated using the following regression:

$$\:\frac{\varDelta\:X\left(t\right)}{\varDelta\:t}={\beta\:}_{2}X\left(t\right)+\epsilon\:$$

Where $\:\epsilon\:$ is the error; thus $\:{\beta\:}_{2}$ is an estimate of µ. To calculate t^*, we use (2) as follows

$$\:X\left(0\right){e}^{{\widehat{\beta\:}}_{2}{t}^{*}}=\frac{{\widehat{B}}_{0}}{{\widehat{\beta\:}}_{1}}$$

That is $\:{t}^{*}=\frac{\text{ln}\left(\frac{{\widehat{B}}_{0}}{{\widehat{\beta\:}}_{1}}\right)-\text{ln}\left(X\left(0\right)\right)}{{\widehat{\beta\:}}_{2}}$

We also can forecast the value of X (t) at any future time using

$$\:X\left(0\right){e}^{{\widehat{\beta\:}}_{2}T}=\frac{{\widehat{B}}_{0}}{{\widehat{\beta\:}}_{1}}$$

where T is a specific future time.

The data used in this work were derived from the Oxford database, and refer to worldwide COVID-19 confirmed cases, in daily frequency. The data spans the period from 22 January 2020 to 12 May of the year 2022. In what follows, we present and describe the empirical results.

Table 1

Regression results for the first equation.
Variable	Coefficient	Std. Error	t-Statistic	P-Value
$\:{\widehat{{\rm\:B}}}_{0}$	13.27558	0.155063	85.61414	0
$\:{\widehat{\beta\:}}_{1}$	8.38E-08	3.95E-09	21.24558	0

According to the results in Table 1, the coefficients of both estimates are statistically significant. The R² is of medium intense (Table 2), approximately 35%

Table 2

Regression results for the first equation.
R-squared	0.34953
Adjusted R-squared	0.348756
S.E. of regression	2.935657
Sum squared resid	7239.188
Log likelihood	-2100.521
F-statistic	451.3748
Prob(F-statistic)	0
Mean dependent var	15.77221
S.D. dependent var	3.637755
Akaike info criterion	4.994112
Schwarz criterion	5.005361
Hannan-Quinn criter.	4.998423
Durbin-Watson stat	0.000685

Table 3

Regression results for the second equation.
Variable	Coefficient	Std. Error	t-Statistic	Prob.
$\:{\widehat{\beta\:}}_{2}$	0.002615	0.000125	20.94887	0

Similarly, according to the results in Table 3, the coefficients of both estimates are statistically significant. The overall results show that our proposed model is sufficient in examining the pandemic’s evolution over time, providing a well-stated mathematical model that is also proved by the empirical investigation. Likewise, in Table 4 we provide the statistical properties of the second model.

Table 4

Regression results for the second equation.
R-squared	0.02663
Adjusted R-squared	0.02663
S.E. of regression	142367.5
Sum squared resid	1.62E + 13
Log likelihood	-10654.15
Durbin-Watson stat	0.331381
Mean dependent var	102650.4
S.D. dependent var	144301.8
Akaike info criterion	26.57146
Schwarz criterion	26.5773
Hannan-Quinn criter.	26.5737

In the next phase of our analysis, we obtained data for the daily adjusted price of crude oil futures during the same period 22 January 2020 until 11 May 2022. We aim to explore the presence of a potential correlation between this future price and the number of confirmed COVID-19 cases. Consequently, we derived data regarding crude oil future prices, from Yahoo.Finance, in daily frequency. In this work we investigate crude oil prices since this commodity is of utmost importance for the global economy, driving many sectors’ performance. This specific commodity affects the macroeconomic conditions globally [30], and geopolitical stability, since countries producing it are considered to have a global power over the world market [31], the energy sector which in turn impacts all the other sectors of the economy [32], and finally, the financial sector through various dynamics due to its preference for investment portfolios after its financialization [33]. Especially during COVID-19, oil prices were significantly affected by the pandemic [34, 35], mainly negatively, even citing negative bubble prices [36]. In this regard, due to its undisputed importance, and since the literature has already commented on the effect of the pandemic on oil prices, we utilize both oil prices and pandemic data to test the linear and non-linear characteristics of this relationship, apart from proving this relationship. Initially, we apply a logarithmic transformation to ensure stationarity and comparability of all data. Subsequently, we employ the Spearman correlation to assess the relationship. The findings reveal a strong correlation of 0.926, indicating a significant link between the confirmed COVID-19 cases and crude oil prices. Additionally, we explore the presence of non-linear patterns in this relationship using a non-linear correlation. The results indicate the existence of a non-linear relationship, with the correlation strength increasing to 0.937. This is further supported by Fig. 1, which visually illustrates the presence of a non-linear pattern in this relationship.

The results reveal a significant relationship between the prevalence of confirmed COVID-19 cases and the fluctuations in oil future prices. Utilizing both linear and non-linear Spearman correlation methods, our findings robustly indicate a discernible impact of the COVID-19 pandemic on the trajectory of oil prices, which highlights a dynamic interplay wherein the progression of confirmed cases directly influences the movements in oil prices, while this association can be attributed to the pandemic's multifaceted effects on global economic activities, supply chain disruptions, and shifts in consumer behavior. Furthermore, our methodology accounts for both linear and non-linear correlations, capturing nuanced relationships that transcend mere statistical linearities and enabling a more comprehensive understanding of the intricate connections between pandemic-induced factors and the volatility in oil markets.

The observed correlation between the surge in confirmed COVID-19 cases and the consequential impact on oil future prices signifies a crucial interdependency between public health crises and global economic sectors. This association underscores the vulnerability of commodity markets, particularly oil, to external shocks induced by pandemics, while the pronounced influence of the COVID-19 pandemic on oil prices highlights the intricate web of factors that contribute to market volatility, encompassing not only supply and demand dynamics but also the intricacies of socio-economic disruptions caused by widespread health crises. Understanding this correlation illuminates the need for policymakers, investors, and market analysts to integrate a broader spectrum of variables, including health-related metrics, into their forecasting models and risk assessment frameworks to navigate and mitigate the repercussions of such complex interplays between public health emergencies and economic systems. Consequently, the implications of this correlation extend to various sectors of the economy, influencing investment strategies, government policies, and business operations, while recognizing the sensitivity of oil prices to health crises emphasizes the importance of adaptive strategies for businesses reliant on oil-related industries. Additionally, it underscores the significance of proactive policy interventions aimed at mitigating the economic fallout of pandemics, emphasizing the need for diversified economic models resilient to external shocks, and also highlighting the necessity for a holistic approach that integrates health, economic, and environmental considerations in global decision-making frameworks to foster greater resilience and stability in the face of unforeseen crises.

Our results are in line with the current literature since many approaches have come up with similar results. To provide some of these research approaches, before COVID-19 crude oil was in general, informationally efficient with few deviations, however, during the pandemic, large deviations from efficiency were evidenced [35]. Moreover, Mensi et al. [34] also argue that the behavior of the oil market during COVID-19 changed (compared to before), regarding upward and downward trends. Likewise, according to Gharib et al. [36], a negative financial bubble was evidenced in the price of oil, during the COVID-19 pandemic.

In this work, we propose a mathematical model that is capable of interpreting the evolution of COVID-19. To do so, we specify the models that are adequate in explaining the evolution of the COVID-19 pandemic, and we then empirically test our framework. By utilizing global confirmed COVID-19 cases, in a daily frequency, we test our proposed model’s performance.

The results indicate that the estimates are statistically significant, implying that our model is valid and well-stated. The empirical results prove our model’s validity, giving value to our approach’s superiority. The fact that already some research studies have proven that the regression models if framed over a robust mathematical model, can provide insights into the pandemic’s evolution [10], gives even more value to our work, justifying our approach.

Moreover, our work is in line with the current literature, since Matveeva and Leonenko [13] through a Gaussian regression approach assessed the dynamics of COVID-19 propagation, while Busari and Samson [10] proved that certain regression models provided valid and trustworthy results regarding the new cases of the contracted victims of the COVID-19 pandemic. Finally, we should have in mind that since much effort is made by many researchers, regarding the mathematical modeling of the pandemic, and relevant events, the synergy of these researchers’ vigorous attempts, could be optimized if these approaches could be organized and even combined [37].

Our approach paves the way for the implementation of thorough mathematical models, utilizing simple models in their implementation, to provide results over the evolution of the pandemic. Of course, more variables could be incorporated into the model, to better study the various features of the pandemic.

Author Contribution

M.A. provided all the mathematical formulations and algorithms and chose the case study. C.F. supervised and guided the article preparation, wrote the mathematical formulations, and chose the case study. T.D. wrote the introduction, literature review, and conclusions. K.G. made the empirical analysis.

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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No competing interests reported.

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Modeling the COVID-19 incorporating oil futures

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Abstract

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1. Introduction

2. Literature Review

3. The model

4. Empirical Estimation

5. Empirical Results

6. Case study

7. Conclusion

Declarations

Author Contribution

Data Availability

References

Footnotes

Additional Declarations

Status:

Version 1

Variable	Coefficient	Std. Error	t-Statistic	P-Value
\(\:{\widehat{{\rm\:B}}}_{0}\)	13.27558	0.155063	85.61414	0
\(\:{\widehat{\beta\:}}_{1}\)	8.38E-08	3.95E-09	21.24558	0