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Research Article

Open Access Peer-reviewed

Herick Laiton Kayange^{ }

Received July 19, 2022; Revised August 21, 2022; Accepted September 01, 2022

The exportation of products is very important for developing World Economies. Countries produce and export products not far from their production structure since they will require the same requisite capabilities. Proximity is the possibility of exporting one product, given that the Economy is exporting another. The higher the Proximity means, the higher the likelihood of two products being exported in tandem. The United States dollar is still the dominant currency in international trade transactions. Since most transactions are invoiced in the U.S dollar, the United States dollar index has been used to trace the value of the U.S dollar against other global currencies. Therefore optimization of controls on proximities and variation in USDX is essential in maximizing export rates. The outbreak of COVID-19 in December 2019 resulted in the decline of International trade activities and exportation in particular. This study used the Lotka Volterra model to examine the effect of COVID-19 on International trade; it was found that the significant impact of COVID-19 was in 2020. Lastly, Pontryagin's Maximum principle was used to optimize Proximity and USDX control to see the maximum possible achievement of exports each year. It was found that optimizing USDX control is more efficient in increasing export rate than optimizing Proximity control. However, optimization of both control is most effective for product exports in the International Trade Network.

Major economic crises have continuously affected International trade. Recently the World has experienced significant economic problems such as The Federal Reserve and the financial crisis of 2007-2008 ^{ 3}, The effect of the COVID-19 pandemic ^{ 2}, and the effects of the Russia-Ukraine conflict ^{ 4}. These crises resulted in unstable states for exporting products to World Market. The exportation of products is essential for the economic growth of many countries.

Over the years, countries have been exporting different products. The network showing the extent of relatedness of the Product Space Network is termed "Product Space Network". Countries move through the product space by developing goods close to those they currently produce ^{ 5}. Product space is a new perspective to studying countries' patterns and dynamic evolution in the global trade pattern under the evolutionary economic geography ^{ 6}. The product space shows the Proximity among products, which in turn captures the similarity of the requisite capabilities to produce them ^{ 7}. Therefore Proximity captures the possibility of producing and exporting two or more products in tandem. The products with a higher chance of being exported together are expected to have higher proximity index values.

For the past eight decades, the United States Dollar has maintained its largely unchallenged status as the World's currency ^{ 8}. Therefore most transactions in International trade are still invoiced in the U.S dollar. The United States dollar index (USDX) in this study was a weighted geometric mean of the dollar's value compared only with a "basket" of 6 other major currencies, which are the euro, Japanese yen, Canadian dollar, British pound, Swedish krona and Swiss franc ^{ 9}. USDX is saving as the value of the U.S dollar against other currencies. If the index is above 100, the export sector is disadvantaged through overvaluation. Conversely, if the index is below 100, it is at an advantage through undervaluation ^{ 10}. Therefore the index is said to have a negative relationship with the change in the number of products exported in the international trade market.

**Figure 1****.**Product space Network for 2019, whereby Nodes represent exported Products and Edges represents proximities between products (Source: Author's simulation)

Product space Networks are Complex network systems. The evolution and dynamics of complex network systems can be represented in the forms of ordinary differential equations. Lotka-Volterra models were initially introduced to study the dynamics of Natural systems ^{ 11, 12, 13}. The uses of Lotka Volterra were recently further extended to study other systems such as Banking systems ^{ 14}, Technological dimensions ^{ 15}, and Currency Exchange rates ^{ 16}. Most studies used the Lotka-Volterra model to study the dynamics of current systems. Less is known about optimizing product space Networks using the Lotka-Volterra model. This study will employ the optimization methods for ordinary differential equations using Pontryagin's maximum principle, as Pontryagin et al. ^{ 17} have recently studied its usefulness. The developed model will introduce two control variables: Proximity control and USDX. Furtherly, each control variable will be examined to see which one had a greater effect on increasing exports of products.

The product space network is developed with the intuitive idea that the Economy's production structure is based on comparative advantages in developing new products. In a Network, nodes are the products, and edges are proximity values showing connectivity in the exportation of relatively two or more products**.**

In a product space Network structure, some products are more connected (blue nodes) than others (red nodes). Examples of More connected products are Tobacco and Tobacco Manufactures (12), feeding stuff for animals (8), Manufactures of Metal (69), Beverages (11), and cork and wood (24). Many countries export these products and give higher possibilities to export other related products.

Products that are not produced and exported in connection with many other products are such as crude rubbers (23), Pulp and waste papers (25), Coal coke and briquettes (32), Gas Natural, and Manufactures (34). Most of these products are produced by fewer countries since they require high Technology.

Moutsinas and Guo ^{ 18} used Lotka-Volterra mode; to explain the Node-level loss in Complex networks. This study will further develop their model to portray the dynamics of growth rates of exports of products, whereby stands as the growth rate of product exports . This study will additionally include the effect of USDX in the model.

The dynamics of the growth rate of exports rate is in the form of

(1) |

Whereby is the change of export rate of product with time.

is the self-regulation dynamics of the export rate of the product .

is the influence of the change of export rate of product on the Export rate of .

is the proximity index between product and .

The self-dynamics of the export rate of product can be written as

(2) |

Where the intrinsic growth of growth rate of a commodity is represented by , The maximum capacity for the Economy to carry the growth rate of export of Commodity is represented by , the point above which the Economy starts to have negative growth is represented by (Alee effect) and is the USDX as the function of mean USDX and the coefficient of variation of USDX .

The proximity index represents the probability of exporting a commodity given that an Economy is already exporting another commodity . The proximity index will serve as the interaction index between Commodity and . Hidalgo et al. ^{ 5} developed the formula for Proximity index as;

(3) |

Where RCA stands for the Revealed comparative advantage

(4) |

shows whether a country exports more product as the share of its total output than the average Country; for a country to have an advantage in Transportation of output, then RCA>1 (not RCA<1). The overall interaction dynamics between commodities will be

(5) |

Where , and are saturation rates of the response function

Generally, when USDX is considered, the overall dynamics of the product space network (1) are written as follows;

(6) |

For analysis purposes, equation (v) can be written in the form of the reduced model as

(7) |

Such that is the effective growth rate of product exportation and is the practical Proximity, and

Optimal control theory is a branch of mathematics developed to find optimal ways to control a dynamic system ^{ 11}. The control problem is of the form where =time, =state variables, and =control variables with Both control and state variables can affect the Objective function.

In this study, we will introduce the control variables and to the original dynamical model equation (vi), namely the control measure of USDX variation and control measure of efficiency methods applied to increase Proximity among products

The control problem becomes

(8) |

Equation (viii) can be linearized into

(9) |

Using the Pontryagin Maximum principle ^{ 17}, we are required to minimize the cost function

(10) |

Subject to the equation . Variables and are initial variables and free variables, respectively. *A* is a positive weight and are the cost of intervention. Therefore, we are required to find optimal and such that Pontryagin maximum principle will further be used to suggest the optimality condition through the Hamiltonian function;

(11) |

Where is the Co-state variable for equation (9).

**Theorem***: There exists a pair of optimal **controls** ** and optimal solution** ** which minimizes** ** **also there is a presence of adjoint function** ** for Hamiltonian function** ** such that*

(12) |

*The Transversality condition is*

Kayange et al. ^{ 19} suggested that optimality conditions and can be used to solve the pair of optimal controls for and

(13) |

(14) |

Equation (13) and (14) gives the solutions for and to be

(15) |

By standard argument control, as suggested by ^{ 19},

(16) |

International data is often classified as Standard International trade classification (SITC). The SITC provides a hierarchical system for disaggregated trade statistics ^{ 20}. This study uses international trade data for four years from 2018-2021 as classified into SITC-REVISION 4, published in https://comtrade.un.org/data/. The United States dollar index data are derived from the daily updated data from the Marketwatch website https://www.marketwatch.com/investing/index/dxy/download.

The simulation of the control problem shows that USDX and proximities strongly affect the growth dynamics of export rates. Using optimization methods provided by Pontryagin's maximum principle, this study will show the dynamics of export rates and the possible achievement of export rates each year for four years from 2018 to 2021. The study portrays the dynamics when no control is optimized, only Proximity is optimized, only USDX is optimized, and when both proximities and Usdx are optimized each year.

Figure 2 shows the dynamics of growth rates of exports in the original dynamical model (blue line) and control problem (green line) when neither the USDX nor the Proximity index is optimized. The control problem shows the rate which could be achieved by the system if optimized.

Results show that the Economy's export growth is highly affected by USDX and Proximity among products, regardless of other parametric values. Optimization of Proximity indices and USDX gives the maximum increase of Exports that an Economy could reach. Optimization of the USDX could be done by minimization of USDX values, while for optimizing proximity indices, we are required to maximize its values. The whole optimization is done by optimizing the control measures due to the efficiency of USDX minimization policies and control measures due to the ability of an Economy to Export commodities that are highly related in their course of production.

Considering Figure 2, Figure 2(a) shows that all control variables are at a minimum (zero), which means that no deliberate action is taken to control export growth each year.* *Figure 2(b) shows that in 2018 there was a decline in the export growth rate at the same rate for both (dynamic and control problems). Zhang et al. ^{ 21} postulated that in 2018 there was a rise in oil prices caused by Trade disputes among exporting countries and Consumers. The increase in prices negatively affected the growth of global exports. In 2019 there was a rise in export; the dynamic and control problems were decreasing simultaneously. In December 2019, pneumonia of unknown cause jolted Wuhan city of Hubei province in China and spread across Asia and the World like wildfire. By January 2020, the WHO declared a public health emergency of international concern ^{ 2}. Therefore Figure 2(c) shows that during the year 2019, the global economy was experiencing a recovery from the effect of the rise in oil prices in 2018. Figure 2(d) shows that the major impact of COVID-19 was during the year 2020. Heiland and Ulltveit-Moe ^{ 22} stated that Country Lockdowns and Restrictions led to declining global export rates of products as the major pandemic effect was vivid during this year. Figure 2(e) shows the recovery of Global Export rates as the Economies softened restrictions for exports, and the introduction of vaccination policies for populations made it easier for International trade activities to be carried on.

**Figure 2****.**Growth of exports in each year when there was no control optimized: (a) Control profile (b) 2018 (c) 2019 (d) 2020 (e) 2021

In general, from Figure 2, the nature of the graphs for both dynamical systems and the control model will be almost the same as there were no deliberate actions taken to reduce or maximize exports each year.

Figure 3 compares the standard dynamical systems for Product space Network (blue line) and possible growth rates when Proximity between products is optimized. Proximity can be optimized by improving Technology, inputs, infrastructure, and institutions essential for producing available exported products.

Figure 3 (a) shows the control profile graph where only Proximity between products is optimized (maximum at one), whereas no action has been taken to optimize the USDX (minimum at zero).

All graphs of dynamical systems before the control variables were introduced (blue lines) from Figure 3(b-e) show the growth trends of exports for each year, as explained in Figure 2. In contrast, the graphs for a controlled model with optimized Proximity are in green lines.

Figure 3(b) shows that after optimizing annual Proximity for 2018, there was an improvement in growth rates. Although the exportation of products was still decreasing after optimizing Proximity, the export growth rate was expected to be slightly higher. Figure 3(c) shows that besides the recovery happening in 2019, the export rate would have increased faster with optimizing Proximity. Figure 3(d) shows that the effect of the COVID-19 pandemic was the decrease in export rate, but with optimal control of Proximity, the rate of decline would have been slower. Therefore more exportation of products would have been higher. Figure 3(e) shows the recovery phase of export growth in the year 2021; specifically, it shows that with optimal Proximity, the recovery would have faster.

The optimization of Proximity between products is essential for the improvement of export rates of products.

**Figure 3****.**Growth of exports in each year when only proximity values between products are optimized: (a) Control profile (b) 2018 (c) 2019 (d) 2020 (e) 2021

Figure 4 shows the comparison of the original dynamics of export rates against the expected growth rates of exports if USDX only was to be optimized. USDX could be optimized by the global Economy's deliberate actions to control the exchange rate. At a country level, short-term solutions such as operational hedging are standard. Keefe and Shadmani ^{ 23} stated that keeping the currency undervalued significantly promotes economic activities.

In Figure 4, the Control profile (figure 4(a)) shows that only USDX is optimal each year while Proximity for each year is kept at a minimum . Figure 4(b) shows that besides the decrease in export rates in 2018, optimizing USDX control would increase export rates. Figure 4(c) shows that in 2019 there was a recovery phase in Export rates, but optimizing the USDX would lead to faster growth of export rates compared to when it wasn't optimized. Figure 4(d) shows that in 2020 the major effect of COVID-19 in the reduction of export was apparent, but optimizing USDX would lead to an increase in export rate. Figure 4(e) shows that the recovery of global export would be faster with optimal USDX compared to when there was no controlled USDX.

**Figure 4****.**Growth of exports in each year when only USDX values are optimized: (a)Control profile (b) 2018 (c) 2019 (d) 2020 (e) 2021

Figure 5 shows the trends of exportation before action is taken into consideration and possible achievements if USDX control and proximity controls have to be optimized.

**Figure 5****.**Growth of exports in each year when both (Proximity values and USDX values) are optimized: (a) Control profile (b) 2018 (c) 2019 (d) 2020 (e) 2021

Figure 5(a) shows the control profile whereby both USDX control and Proximity Control are optimized ( and ). Optimizing both controls would be compared to when no deliberate action is taken to optimize global Export rates.

Figure 5(b-e) shows that optimization of both controls leads to the highest growth rates of exports than optimization of individual controls separately. By optimizing USDX and Proximity for any year, Simulations show that the exportation of products in the International Trade Network will always Increase.

Exportation of products can be affected by many factors such as Exchange rates, Country and international borders policies, Technology, institutions, Skilled personnel, etc. Each Country can export certain products depending on the availability of resources that are essential for exporting such products. Introducing new products that do not relate to the available produced and exported products is very rare. Thus, many countries introduce new products for exportation that are not far from the general production structure to use the available resources Proximity between products is the probability of exporting one product given that the Country is exporting another product.

In international trade networks, most international trade transactions are invoiced in the U.S dollar. So we need a standard measure of the exchange rate of countries' currency against the U.S dollar, known as USDX. The volatility of USDX influence transaction negatively in the International Trade Network. Therefore to increase the export rate of commodities, it is crucial to optimize the control of proximities and USDX. In this study, the effect of each and both control is studied.

The optimization of USDX control was only found to be more effective than the optimization of the Proximity index. However, both would lead to improvement in export rates of products each year. Both USDX control and Proximity control are supposed to be optimized to get the maximum growth of export rates of the products.

As much as this study is beneficial, there were some challenges. The major challenge was the variation in the number of reporting countries for export data for each year; in 2018, 2019, 2020, and 2021 the number of countries which reported exports were 155, 132, 66, and 88, respectively. In the latter two years (2020 and 2021), fewer countries reported due to the COVID-19 epidemic. To be relevant, each year, the analysis was done depending on the reported countries, whereby the implication of the results still was valid. There were not many challenges for the United States dollar indices as the reported data from the website were timely and continuously updated.

In the future, this study can be improved by introducing innovation parameter, which is essential for introducing and exporting new products. This study aimed to study the exportation of products at the Global level, and more can be studied at the country level and compare exports with their related exchange rates.

This study has no conflict of interest.

Open source data, https://comtrade.un.org/data/ and https://www.investing.com/indices/usdollar-historical-data.

[1] | G. Gopinath, E. Boz, C. Casas, F. J. Díez, P.-O. Gourinchas, and M. Plagborg-Møller, “Dominant currency paradigm,” American Economic Review, vol. 110, pp. 677-719, 2020. | ||

In article | View Article | ||

[2] | S. Umair, U. Waqas, and M. Faheem, “COVID-19 pandemic: stringent measures of Malaysia and implications for other countries,” Postgraduate medical journal, vol. 97, pp. 130-132, 2021. | ||

In article | View Article PubMed | ||

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In article | |||

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In article | View Article PubMed | ||

[6] | S. Li, X. Li, W. Lang, H. Chen, and X. Huang, “The Spatial and Mechanism Difference in the Export Evolution of Product Space in Global Countries,” Sustainability, vol. 13, p. 2255, 2021. | ||

In article | View Article | ||

[7] | L. Fraccascia, I. Giannoccaro, and V. Albino, “Green product development: What does the country product space imply?,” Journal of cleaner production, vol. 170, pp. 1076-1088, 2018. | ||

In article | View Article | ||

[8] | K. Vierelä, “Unites States dollar as an international currency,” 2018. | ||

In article | |||

[9] | M. M. Charme, “Estimating the United States Dollar Index Returns' Value at Risk: Empirical Evidence from RiskMetrics and Simultaneous Bootstrap Quantile Regression Methods,” Global Journal of Management And Business Research, 2020. | ||

In article | View Article | ||

[10] | A. Bigsten, P. Collier, S. Dercon, M. Fafcharnps, B. Gauthier, J. Willern Gunning, et al., “Exports of African manufactures: macro policy and firm behaviour,” Journal of International Trade & Economic Development, vol. 8, pp. 53-71, 2006. | ||

In article | View Article | ||

[11] | Q. Din, “Dynamics of a discrete Lotka-Volterra model,” Advances in Difference Equations, vol. 2013, pp. 1-13, 2013. | ||

In article | View Article | ||

[12] | H. Matsuda, N. Ogita, A. Sasaki, and K. Satō, “Statistical mechanics of population: the lattice Lotka-Volterra model,” Progress of theoretical Physics, vol. 88, pp. 1035-1049, 1992. | ||

In article | View Article | ||

[13] | C. Zhu and G. Yin, “On competitive Lotka–Volterra model in random environments,” Journal of Mathematical Analysis and Applications, vol. 357, pp. 154-170, 2009. | ||

In article | View Article | ||

[14] | S. Mao, M. Zhu, X. Wang, and X. Xiao, “Grey–Lotka–Volterra model for the competition and cooperation between third-party online payment systems and online banking in China,” Applied Soft Computing, vol. 95, p. 106501, 2020. | ||

In article | View Article | ||

[15] | I. Ivanova, Ø. Strand, and L. Leydesdorff, “An eco-systems approach to constructing economic complexity measures: endogenization of the technological dimension using Lotka–Volterra equations,” Advances in Complex Systems, vol. 22, p. 1850023, 2019. | ||

In article | View Article | ||

[16] | Y. D. Marinakis, R. White, and S. T. Walsh, “Lotka–Volterra signals in ASEAN currency exchange rates,” Physica A: Statistical Mechanics and its Applications, vol. 545, p. 123743, 2020. | ||

In article | View Article | ||

[17] | L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze, E. Mishchenko, K. Trirogoff, and L. Neustadt, LS Pontryagin Selected Works: The Mathematical Theory of Optimal Processes: Routledge, 2018. | ||

In article | |||

[18] | G. Moutsinas and W. Guo, “Node-level resilience loss in dynamic complex networks,” Scientific reports, vol. 10, pp. 1-12, 2020. | ||

In article | View Article PubMed | ||

[19] | H. L. Kayange, E. S. Massawe, D. O. Makinde, and L. S. Immanuel, “Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives,” International Journal of Systems Science and Applied Mathematics, vol. 5, p. 43, 2020. | ||

In article | View Article | ||

[20] | W.-F. Hungerland and C. Altmeppen, “What is a product anyway? Applying the Standard International Trade Classification (SITC) to historical data,” Historical Methods: A Journal of Quantitative and Interdisciplinary History, vol. 54, pp. 65-79, 2021. | ||

In article | View Article | ||

[21] | Z. Zhang, M. He, Y. Zhang, and Y. Wang, “Geopolitical risk trends and crude oil price predictability,” Energy, p. 124824, 2022. | ||

In article | View Article | ||

[22] | I. Heiland and K. H. Ulltveit-Moe, “11 An unintended crisis is sea transportation due to COVID-19 restrictions,” COVID-19 and trade policy: Why turning inward won't work, p. 151, 2020. | ||

In article | |||

[23] | H. G. Keefe and H. Shadmani, “Foreign exchange market intervention and asymmetric preferences,” Emerging Markets Review, vol. 37, pp. 148-163, 2018. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2022 Herick Laiton Kayange

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Herick Laiton Kayange. Modeling and Optimal Control of Product Space Network during Covid-19 Pandemic with the Effect of Variation in USDX. *American Journal of Modeling and Optimization*. Vol. 9, No. 1, 2022, pp 6-14. http://pubs.sciepub.com/ajmo/9/1/2

Kayange, Herick Laiton. "Modeling and Optimal Control of Product Space Network during Covid-19 Pandemic with the Effect of Variation in USDX." *American Journal of Modeling and Optimization* 9.1 (2022): 6-14.

Kayange, H. L. (2022). Modeling and Optimal Control of Product Space Network during Covid-19 Pandemic with the Effect of Variation in USDX. *American Journal of Modeling and Optimization*, *9*(1), 6-14.

Kayange, Herick Laiton. "Modeling and Optimal Control of Product Space Network during Covid-19 Pandemic with the Effect of Variation in USDX." *American Journal of Modeling and Optimization* 9, no. 1 (2022): 6-14.

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[1] | G. Gopinath, E. Boz, C. Casas, F. J. Díez, P.-O. Gourinchas, and M. Plagborg-Møller, “Dominant currency paradigm,” American Economic Review, vol. 110, pp. 677-719, 2020. | ||

In article | View Article | ||

[2] | S. Umair, U. Waqas, and M. Faheem, “COVID-19 pandemic: stringent measures of Malaysia and implications for other countries,” Postgraduate medical journal, vol. 97, pp. 130-132, 2021. | ||

In article | View Article PubMed | ||

[3] | S. G. Cecchetti, “Crisis and responses: the Federal Reserve and the financial crisis of 2007-2008,” National Bureau of Economic Research2008. | ||

In article | View Article | ||

[4] | I. Liadze, C. Macchiarelli, P. Mortimer-Lee, and P. S. Juanino, “The economic costs of the Russia-Ukraine conflict,” NIESR Policy Paper, vol. 32, 2022. | ||

In article | |||

[5] | C. A. Hidalgo, B. Klinger, A.-L. Barabási, and R. Hausmann, “The product space conditions the development of nations,” Science, vol. 317, pp. 482-487, 2007. | ||

In article | View Article PubMed | ||

[6] | S. Li, X. Li, W. Lang, H. Chen, and X. Huang, “The Spatial and Mechanism Difference in the Export Evolution of Product Space in Global Countries,” Sustainability, vol. 13, p. 2255, 2021. | ||

In article | View Article | ||

[7] | L. Fraccascia, I. Giannoccaro, and V. Albino, “Green product development: What does the country product space imply?,” Journal of cleaner production, vol. 170, pp. 1076-1088, 2018. | ||

In article | View Article | ||

[8] | K. Vierelä, “Unites States dollar as an international currency,” 2018. | ||

In article | |||

[9] | M. M. Charme, “Estimating the United States Dollar Index Returns' Value at Risk: Empirical Evidence from RiskMetrics and Simultaneous Bootstrap Quantile Regression Methods,” Global Journal of Management And Business Research, 2020. | ||

In article | View Article | ||

[10] | A. Bigsten, P. Collier, S. Dercon, M. Fafcharnps, B. Gauthier, J. Willern Gunning, et al., “Exports of African manufactures: macro policy and firm behaviour,” Journal of International Trade & Economic Development, vol. 8, pp. 53-71, 2006. | ||

In article | View Article | ||

[11] | Q. Din, “Dynamics of a discrete Lotka-Volterra model,” Advances in Difference Equations, vol. 2013, pp. 1-13, 2013. | ||

In article | View Article | ||

[12] | H. Matsuda, N. Ogita, A. Sasaki, and K. Satō, “Statistical mechanics of population: the lattice Lotka-Volterra model,” Progress of theoretical Physics, vol. 88, pp. 1035-1049, 1992. | ||

In article | View Article | ||

[13] | C. Zhu and G. Yin, “On competitive Lotka–Volterra model in random environments,” Journal of Mathematical Analysis and Applications, vol. 357, pp. 154-170, 2009. | ||

In article | View Article | ||

[14] | S. Mao, M. Zhu, X. Wang, and X. Xiao, “Grey–Lotka–Volterra model for the competition and cooperation between third-party online payment systems and online banking in China,” Applied Soft Computing, vol. 95, p. 106501, 2020. | ||

In article | View Article | ||

[15] | I. Ivanova, Ø. Strand, and L. Leydesdorff, “An eco-systems approach to constructing economic complexity measures: endogenization of the technological dimension using Lotka–Volterra equations,” Advances in Complex Systems, vol. 22, p. 1850023, 2019. | ||

In article | View Article | ||

[16] | Y. D. Marinakis, R. White, and S. T. Walsh, “Lotka–Volterra signals in ASEAN currency exchange rates,” Physica A: Statistical Mechanics and its Applications, vol. 545, p. 123743, 2020. | ||

In article | View Article | ||

[17] | L. S. Pontryagin, V. Boltyanskii, R. Gamkrelidze, E. Mishchenko, K. Trirogoff, and L. Neustadt, LS Pontryagin Selected Works: The Mathematical Theory of Optimal Processes: Routledge, 2018. | ||

In article | |||

[18] | G. Moutsinas and W. Guo, “Node-level resilience loss in dynamic complex networks,” Scientific reports, vol. 10, pp. 1-12, 2020. | ||

In article | View Article PubMed | ||

[19] | H. L. Kayange, E. S. Massawe, D. O. Makinde, and L. S. Immanuel, “Modelling and Optimal Control of Ebola Virus Disease in the Presence of Treatment and Quarantine of Infectives,” International Journal of Systems Science and Applied Mathematics, vol. 5, p. 43, 2020. | ||

In article | View Article | ||

[20] | W.-F. Hungerland and C. Altmeppen, “What is a product anyway? Applying the Standard International Trade Classification (SITC) to historical data,” Historical Methods: A Journal of Quantitative and Interdisciplinary History, vol. 54, pp. 65-79, 2021. | ||

In article | View Article | ||

[21] | Z. Zhang, M. He, Y. Zhang, and Y. Wang, “Geopolitical risk trends and crude oil price predictability,” Energy, p. 124824, 2022. | ||

In article | View Article | ||

[22] | I. Heiland and K. H. Ulltveit-Moe, “11 An unintended crisis is sea transportation due to COVID-19 restrictions,” COVID-19 and trade policy: Why turning inward won't work, p. 151, 2020. | ||

In article | |||

[23] | H. G. Keefe and H. Shadmani, “Foreign exchange market intervention and asymmetric preferences,” Emerging Markets Review, vol. 37, pp. 148-163, 2018. | ||

In article | View Article | ||