Since the late 1990s, thanks to innovations in several decisions and globalization, running a business and the public’s way of life have changed significantly. ¹ New opportunities for fulfilling one’s potential become available to active people. The development of telecommunications systems allows you to organize virtual development teams, while distance learning also reduces the need to be in specific places. The mobility of the labor force became possible due to the elimination of border and customs barriers, and the transport infrastructure was additionally developed, especially in Europe. Therefore, people, especially the younger generation, have become less attached to a settled way of life and choose dynamic movement between locations. At the same time, alpha cities and megacities, which provide maximum opportunities for self-realization, become centers of attraction. The movement of mobility of the labor force within Europe and between other states has led to an increase in the dynamics of the structure of employment of the population. International transport services of all kinds have become available over the world (the proposed mathematical model is equally acceptable for any regions of the world.).

The authors consider the coronavirus crisis as a test of whether smart city technology may mitigate the negative impacts. The COVID-19 pandemic was a black swan event that disrupted the established chains of interaction and completely reshaped the economic landscape of most regions and cities of the world, which are centers of attraction for human flows. ² The researchers could define the cities with a preparedness resiliency plan as ‘smarter’ using the Internet-of-Things (IoT) sensors, cameras etc. ¹ Digital transformation allows many businesses to develop seamless working-from-home experiences. The degree of equipment with technical means for analyzing current information about the state of urban infrastructure, its processing for different cities and regions is different. This often depends on the budget of the administration and on legislative restrictions. The most advanced control systems today are in Southeast Asia, such as China, but during the COVID-19 pandemic, telecommunications and data processing systems have been rapidly developing throughout Europe. The importance of IoT systems and machine-to-machine interaction has also increased manifold. First of all, this is the automatic collection of data on the migration of people, traffic on the roads, the expenditure of all types of resources, and the general situation in cities. In addition, BigData technologies based on predictive algorithms make it possible to make decisions with an effect extended over time. The anti-pandemic measures have accelerated broadband consumption growth by over 60%. ³

The problem of post-pandemic governance could be described from the point of view of organizing business interaction within the consumer value chain. The idea of forming a consumer value chain was initially proposed by M. Porter in 1985. ⁴ This considers the term ‘echelon’ which relates to the theoretical approaches of modeling multi-echelon supply chain design ⁵ and social sustainability assessment of upstream (a set of connected firms that involve raw material extraction and transformation), midstream (production and assembly), and downstream (sales and services) echelons of supply chains. ^{6 – 9} Some researchers explore solid waste management systems from the logistics and supply chains theory ¹⁰ and multi-echelon supply chains under stochastic and fuzzy environments. ¹¹

The upcoming removal of restrictive quarantine measures will have a shocking impact on all services and will require the administrations of regions and cities to make informed and calculated decisions. ¹² The lack of theoretical propositions regarding sustainable development goals supporting the quality of life in the post-pandemic period refers to the research gap. The authors are attempting to investigate environmental and social goals from the point of view of administrative measures to conquer the negative consequences of coronavirus disease, considering an ecological policy. The authors propose exploring digital transformation in smart cities as a tool for sustainable development, making the logical bridge between the theoretical framework of sustainability concept and common property resources as an environmental asset in a digital era.

This article aims to provide an economic-mathematical model of smart cities externalities’ impact from the point of view of achieving social and environmental goals. The result is a set of formalisms ^{13 , 14} that allow the application of scientific methods to find optimal solutions. ^{15 , 16} The proposed approach allows the development of the fundamental basis for making administrative decisions within individual megapolises and in environmental policy fora territory of any scale.

The construction of a mathematical model is necessary to select optimal solutions. The new industry 4.0 technologies (e.g., cyber physics systems, big data, IoT, etc.) allow for extending the field of financial logistics implementation based on the digital transformation of logistics ^{18 , 27} and digital twins. ²⁶ We used the technology of developing a digital twin, which allowed us to apply the methods developed in mathematics to find the optimal result. The authors used the concept of digital twin being developed in previous works regarding digital logistics. ^{26 , 28} This research aims to develop digital twins of the considered process. The authors introduce several notations for the arguments used. ²⁹

The dependence of the load on time t is Λ t . Since Λ t changes unevenly throughout the year, reflecting the load acting on the resources available in the region or the megapolis, we will represent its decomposition as a set λ i t (where i = 1 , 2 , … ). The authors introduce a periodicity T characteristic of the load on resources. T is equal to the annual cycle. The researchers define a set ν i t for the values of the probabilities of the use of the common resources of the territory. Next, we introduce E k t functions that define a set of system states together with a set of their probabilities: p i t = φ i t + α i t .

In this expression, φ i t is the T / 2 πi periodic, and the term α i t satisfies the condition lim t → ∞ α i t = 0 , and ∀ i . In this interpretation, φ i t reflects the stationary probability p i t . Next, we can consider the sequence: p 0 i ′ t = − λ i t p 0 i t + ν i t p 1 i t , p 1 i ′ t = − ν i t p 1 i t + λ i t p 0 i t together with the complete group condition: ∑ i p i t = 1 .

After carrying out an algebraic transformation of the form: p 0 i ′ t = − λ i t + ν i t p 0 i t + ν i t at zero conditions p 0 i 0 = α i ≤ 1 , p 1 i 0 = 1 − α i ≤ 1 , it becomes possible to obtain the desired system already in quadratures in the form: p 0 i t = e − ∫ 0 t λ i x + ν i x dx α i + ∫ 0 t ν i x e − ∫ 0 x λ i z + ν i z dz dx .

The authors denote φ i t = ∫ 0 t λ i x + ν i x dx , as well as k , which satisfies the inequality: kT ≤ t < k + 1 T . The researchers suggest an expression for the calculation of φ i t : φ i t = ka i + ∫ 0 τ λ i z + ν i z dz ,

Here, it is assumed that: a i = ∫ 0 T λ i x + ν i x dx , τ = t − kT .

Similarly, the authors express: ∫ 0 t ν i x e φ i x dx = ∑ l = 1 k ∫ l − 1 T lT ν i x e φ i x dx + ∫ kT t ν i x e φ i x dx , and this gives us p 0 i t = e − φ i τ b i e a − 1 + ∫ 0 τ ν i x e φ i x dx + e − ka − φ i τ α i − b i e a − 1 ,

In this case, the designation for the parameter is introduced: b i = ∫ 0 T ν i x e φ i x dx .

The authors search for the optimal solution on the obtained basis ³⁰ using the common approach of digital twin concept. ^{13 , 31 , 32} To do this, we formulate the problem in a general form. The authors then define the number of users of the resources as n . In this case, the load l i , where i = 1 … n , reflects the of each user’s externalities for the vector: L ¯ = l 1 l 2 … l n .

The total load L ∗ = ∑ i = 1 n l i .

The researchers introduce the weighted average ^{14 , 30 , 33 – 35} variable costs U i and the dependence g i l i of the revenue of each user. Due to the limited capacity of the shared resource, starting from L ¯ > L ¯ R , the profitability drops. This corresponds to the fulfillment of the condition g i ′ l i < 0 . Moreover, due to the negative impact of the externalities of the specified n users, the second derivative is subject to the condition g i ″ l i < 0 .

Then, the profit of the i − th user is R i = l i g l 1 + l 2 + … + l n − U i l i = l i g i L − U i l i . Based on the Nash equilibrium principle, ³⁶ there is a value l i ∗ for i = 1 … n , at which R i reaches a maximum for any components L ¯ : L ¯ i ∗ l 1 ∗ l 2 ∗ … l i − 1 ∗ l i + 1 ∗ … l n ∗ . We find the extremum points from the conditions (for the calculation): ∂ R i ∂ l i = 0 , at i = 1 … n . Denoting the sum l − i ∗ = ∑ k ≠ i l k ∗ , the authors obtain g l i + l − i ∗ + l i g ′ l i + l − 1 ∗ − U = 0 for i = 1 … n , which gives the desired value of L ∗ = n U − g L ∗ g ′ L ∗ . Consequently, the extremum is reached at L 0 = U − g L 0 g ′ L 0 . Since the condition n > 1 is satisfied, it follows that L ∗ > L 0 . This proves the obtained result.

The mathematical model formulated in the article provides an opportunity for developing several software applications (however the authors are attempting to develop the common theoretical approach). Since the calculations are scalable, it is possible to cover most of the critical areas of economic activity that affect the quality of life, the environment, and the economic performance of a given territory. First of all, the authors were guided by publicly available specifications V182 ( https://www.bsigroup.com/en-GB/smart-cities/Smart-Cities-Standards-and-Publication/PAS-182-smart-cities-data-concept-model/ accessed on 27 April, 2022). In this case, we consider the result as an algorithm acceptable for embedding in the smart city digital platform. The proposed approach allows the development of the fundamental basis for making administrative decisions within individual megapolises and in environmental policy for the territory of any scale. The developed mathematical model is abstract by definition and is applied taking into account specific tasks and criteria. The theoretical recommendations are expected to be implemented for designing a customer value chain and digital logistics network in smart cities and smart territories. The governing measures consider planning optimal organization of suppliers in a digital logistics network, taking into account the interests of stakeholders and the priorities of the sustainability concept. The developed method will add value to the field of planning smart cities and smart territories based on the model including the economic interests of customers, profit indicators, the delivery time of goods, and reliability of supplies. The usage of this method allows customer value creation by developing a customer-oriented approach and achieving public consensus on an integrated network structure. The expected value of peer review could provide the best available path for designing smart cities’ logistics networks to gain social, environmental, and governing goals of the sustainable development concept.

The authors suggest discussing the wide area of issues regarding the smart city concept based on digital interaction within a constantly evolving sustainable environment. Due to the publicly available specifications, the decision model has grown significantly. Big data technologies, high-speed mobile means of information transmission, and urban management algorithms have become the norm. However, predictive methods are required for black swan events such as the COVID-19 pandemic. ^{37 – 39} The evolution of smart cities could be considered in the context of the discussion of the developed algorithm compared to existing research. The authors are attempting to investigate the complex task of the sustainable development in post-pandemic situation. The existing approaches consider the COVID-19 pandemic as a continuous crisis having a negative impact on the sustainable development, but do not take into account the difficulties caused by post-pandemic crisis. The researchers aimed to develop an approach considering the digital transformation as a tool for achieving Environmental, Social, Governmental (ESG) goals with supporting measures as a common resource access policy.

The authors believe that decision-makers need predictive methods for the post-COVID-19 transition period. Undoubtedly, removing restrictions will entail serious negative consequences for the economy, ecology, and municipal infrastructure of megapolises. It is necessary to calculate the optimal mode of interaction between the administration, government, and experts in commerce and the service sector that determines the lives of the population and arriving people. The developed model solves the most challenging task of organizing an increase in the volume of services concentrated in a limited location while minimizing negative externalities. The framework of the developed approach is limited by mathematical assumptions and methods of game theory. The authors propose a topic for future research related to digital transformation as a tool for achieving sustainable development goals.

Exploring methods and models of digital logistics relying on different conditions of smart cities’ sustainable development could be a topic for future research in a coupe with several points of view in works. ^{13 , 40 – 42}

The authors considered a systematic approach comprising the sustainable development concept and smart cities’ evolution based on digital technologies. This article obtained an analytical solution in quadratures, using the superposition of the obtained solutions with weights corresponding to the initial decomposition. The proposed approach allowed formalizing the resources of the megapolis in the form of a dynamic task. The researchers achieved results in a group of solutions based on the dependence of the environmental costs. The authors recommend the administrative regulation for decision-making, improving the quality of life in megapolis considering the digital transformation as a tool for achieving ESG goals.

Currently, the administrative regulation measures for supporting sustainable development in megapolises use heuristic methods in their current activities. The severe disruption caused by the pandemic has shown the unsustainability of the current management practices. No dynamic analysis and search for optimal solutions are carried out in the decision-making process. Such difficulties are primarily due to the complexity of considering many factors and the lack of scientifically based theoretical models that consider economic indicators. However, the upcoming release from the restrictive measures of the quarantine regime can lead to much greater shocks. The developed methodology makes it possible to calculate the balanced benefit for all participants in the activity, taking into account externalities and choosing alternative options.

No data are associated with this article.