7 Mathematical basis

The mathematical basis of the Digital Twin is a set of models, algorithms, and calibration methods built on actual data. If the previous chapter explains how a management decision is prepared, this chapter shows the calculation mechanisms that make such a decision testable.

The models connect socio-economic development goals with baseline indicators, external factors, investment decisions, and infrastructure constraints. The demographic block sets the population dynamics, system dynamics describes how the territory changes over time, balance models capture flows of resources and economic activity, and applied models translate projects, portfolios, public infrastructure, migration, and the technical condition of networks into measurable effects.

The practical purpose of the chapter is to make scenario comparison explicit. Each decision can be decomposed into input data, calculation dependencies, assumptions, and output indicators. This is why the digital twin can be used not as an illustration, but as a decision-support instrument: it helps estimate what changes after a project is implemented, which constraints are triggered, where resource deficits appear, and how stable the expected result is.

The mathematical models are implemented as calculation modules in R and Java. These modules can be integrated into higher-level information and analytical systems through Docker containers or accessed through REST API methods.

How to read this chapter:

• for the general logic of forecasting, start with demography, system dynamics, and the intersectoral balance;
• for investment-project analysis, move to financial performance, project impact, and project portfolios;
• for urban environment and infrastructure tasks, use the sections on parks, transport projects, public infrastructure provision, commuting migration, and technical network condition;
• to understand whether a model is applicable to a concrete task, look at the input data, assumptions, calibration, and application-result sections.

Clickable chapter contents:

• Population forecasting model — demographic forecasts and restoration of incomplete time series.
• City system dynamics model — links between the state of the city, management actions, and indicator dynamics.
• Intersectoral balance and economic forecasting model — intersectoral balance and sectoral forecasts.
• Model for assessing the financial performance of an investment project — cash flows, budget performance, and commercial performance.
• Model for assessing the impact of an investment project on socio-economic indicators — project effects on target city indicators.
• Model for assessing the impact of a portfolio of investment projects on socio-economic indicators — combined effects of multiple projects.
• Model for assessing the impact of city parks on socio-economic development and ESG indicators — social, economic, and environmental effects of green infrastructure.
• Model for scenario calculation of the socio-economic performance of transport projects using the GIH methodology — transport scenarios and their impact on the city.
• Model for assessing the provision of public infrastructure — availability and deficits of social infrastructure.
• Model for assessing commuting migration — daily movement of population and territorial load.
• Model for assessing the technical condition, wear and tear, and terms of safe operation of infrastructure — diagnostics, wear, and remaining useful life of engineering systems.