ECONOMIC AND MATHEMATICAL MODELS OF DEMOGRAPHIC FORECASTING UNDER CONDITIONS OF ACTIVE URBANIZATION

INTERNATIONAL SYMPOSIUM “ADVANCED RESEARCH IN ECONOMICS AND BUSINESS MANAGEMENT”, SEPTEMBER 19, 2025 400 ECONOMIC AND MATHEMATICAL MODELS OF DEMOGRAPHIC FORECASTING UNDER CONDITIONS OF ACTIVE URBANIZATION Daliev Akhtam Sharafutdinovich PhD in Economics, Associate Professor. National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan. E-mail: [email protected] https://doi.org/10.5281/zenodo.17266576 Abstract. The article discusses economic and mathematical models for forecasting population size and the number of families on the housing waiting list, as well as planning and designing residential micro-districts with a fixed population size, taking into account demographic processes. Forecasting housing construction based on demographic models, which make it possible to consider the complex structure of the population, the dynamics of urbanization, the distribution of families by type, and other parameters, is a highly relevant task. It is important not only to estimate the total population size but also to determine the necessary parameters of a micro-district: apartment structure, building area, sizes of infrastructure facilities, communal service objects, etc., ensuring optimal housing distribution. Two algorithms are considered. The first is based on constructing a probability transition matrix for families moving from one state to another, taking into account urbanization processes. The second algorithm is based on the Markov chain method, which allows analyzing the dynamics of demographic changes over several years and forming the corresponding stochastic matrix. Keywords: urbanization, forecasting, optimization, dynamics. In conditions of accelerating urbanization, unstable demographic dynamics, and the growth of households of various types, the development of housing construction acquires strategic importance. Demographic transformation, including shifts in the age and family structure of the population, has a direct impact on housing demand and requires a revision of traditional approaches to housing policy [1,2]. The main goals of regional economies are the development of human potential and the improvement of quality and life expectancy. Demographic characteristics directly influence: – the growth of gross regional product and the development of social infrastructure (healthcare, education, etc.); – the scale of housing construction, transport infrastructure, and other socio-economic parameters of regional development. Uzbekistan, as one of the most populous countries in Central Asia, demonstrates stable population growth. According to forecasts, by 2030 the population will exceed 41 million people, by 2040 – 46 million, and by 2050 – 50 million. These changes will be accompanied by an increase in the working-age population, growth in the number of young people under 18, as well as an increasing share of the population over 65 years of age. INTERNATIONAL SYMPOSIUM “ADVANCED RESEARCH IN ECONOMICS AND BUSINESS MANAGEMENT”, SEPTEMBER 19, 2025 401 According to statistical data, as of January 1, 2025, the population of Tashkent amounted to 3,112.8 thousand people, increasing by 75,232 in 2024 alone. Urbanization processes and natural population growth drive the rising housing demand, especially in large cities such as Tashkent, Samarkand, Bukhara, and Fergana. If in 2010–2016, 400,000 new houses were built in the country, then in 2017–2024 this number reached 1.65 million, reflecting a fourfold increase in housing construction rates. In 2024 alone, more than 100,000 apartments were built, while the annual increase in the housing stock reached 7.6 million square meters. With the population rising from 28 million in 2010 to 37 million in 2024, the housing availability indicator increased from 14.3 to 45.8 units per 1,000 people. As of January 1, 2025, Tashkent’s housing stock amounted to 703.8 million square meters, 1.3 times larger than in 2021. Thus, the city’s housing stock represents a complex socioeconomic system, the development of which depends on multiple factors. Demographic changes and urbanization necessitate the development of scientifically grounded mechanisms for forecasting and assessing the impact of demographic processes on housing construction using economic and mathematical methods. As is well known, the demographic process consists of three main parameters that determine housing demand: – natural population dynamics (births, deaths); – migration; – changes in family structure (average family size, complete and incomplete families). These parameters influence: – the required volume of housing stock; – the structure of housing (apartment types, building height); – the infrastructure of a micro-district. Regulation in the field of housing construction can ensure the balance of supply and demand, making it essential to consider demographic factors when planning projects. Modern trends point to the need for long-term housing development programs as a strategic sector of regional economies, designed to meet the growing needs of the population. Of particular importance is the structure of families on the housing waiting list, which differs significantly from the general urban population structure and from that of reconstructed areas. For planning purposes, it is necessary to account for changes occurring in waiting-list families by the time they actually receive housing. The study proposes two algorithms. The first treats the waiting list as a set of families G, whose number changes over time. The set G is divided into six subsets, each representing families of one to six members. Each state subset 2–4 is divided into subsets of complete families Пi, (i = ), having a marital couple, and incomplete families НПi (i = ), without a marital couple. Over time, a family can increase, decrease, or remain in the same size. Events leading to family expansion include: – birth of a child; – marriage with the arrival of a new family member; – joining of other relatives (most often parents, less frequently siblings, grandparents, or grandchildren). INTERNATIONAL SYMPOSIUM “ADVANCED RESEARCH IN ECONOMICS AND BUSINESS MANAGEMENT”, SEPTEMBER 19, 2025 402 Events leading to family reduction include: – death of a family member; – marriage with departure from the family; – divorce; – departure of a family member (adult children, parents, or other relatives). We introduce the following notations: G(T)=||Pij (, t) || – probability transition matrix of a family from state i to j ; Q=||qij|| –probability density matrix ; –probability density matrix for incomplete families; –the same for complete families; –transition probability matrix for incomplete families; –the same for complete families; – vector of the number of people in incomplete families at the start of forecasting; – the same for complete families; – forecast vector of incomplete families; –the same for complete families. Suppose that at some moment ξ a family is in state i. Under the influence of factors (birth, death, etc.), at moment t>ξ the family may randomly move to state j (j=i+1, i−1, i) with some probability Pij. For such cases, a special form of the Chapman–Kolmogorov equation applies:  ,T (1) We assume that Pij for t>0 have derivatives, since these functions are continuous with respect to t. Differentiating the Chapman–Kolmogorov equation (1) with respect to each variable ξ and t, under the condition ξ=0 and t=0, after certain transformations we arrive at the forward and backward systems of differential equations of the Markov chain: (2) where: As the initial conditions for these differential equations, we assume: (3) In (2), qi and qij denote probability densities, which are determined by the formulas: INTERNATIONAL SYMPOSIUM “ADVANCED RESEARCH IN ECONOMICS AND BUSINESS MANAGEMENT”, SEPTEMBER 19, 2025 403 The probability density matrix Q is a tridiagonal matrix. Solving (1)–(3) we obtain: where E is the identity matrix. In this study, the probability density of an event is proposed to be determined using statistical data by the formula: , (4) where A(t) – number of individuals for whom a given event may occur; K(t) – number of events that occurred in the population during the period Δt; Δt – time interval during which the events occurred. On the basis of this formula, the probability density matrix Q is determined. At the same time, some of its elements equal zero. For example, the probability densities of marriage for minors, as well as the probabilities of childbirth for minors and for women belonging to the third age subgroup, take zero values. Using formula (1), we calculate the transition probability matrix. Multiplying the initial family-size vector by the transition probability matrix, we obtain the projected family structure at the end of the period Δt. To illustrate the method, let us give an example. The initial quantitative data on family composition, as well as the recorded events that occurred in them over the time interval Δt, are presented in the following tables. Initial data on the demographic state of families (age groups: 18–35; 36–49; >49) Number of members Family type Number of families by age groups Total families Required number of apartments 1 male 60, 70, 27 337 157 (1A) female 75, 85, 20 180 (1B) 2 singleparent 135, 60, 30 563 225 (2A) full 218, 90, 30 338 (2B) 3 singleparent 212, 95, 30 675 337 (3A) full 215, 91, 32 338 (3B) 4 singleparent 55, 35, 22 225 112 (4A) full 48, 42, 23 113 (4B) 5 full 150, 90, 30 270 270 (5B) 6 full 110, 60, 10 180 180 (6B) INTERNATIONAL SYMPOSIUM “ADVANCED RESEARCH IN ECONOMICS AND BUSINESS MANAGEMENT”, SEPTEMBER 19, 2025 404 In the table, the type of apartment required for this family is given in parentheses. Family type Mortality Birth Marriage with exit Member leaving Marriage with entry Member joining Divorce Single-parent families 1 0 0 0 190 210 190 0 2 100 0 42 80 42 80 0 3 50 0 72 126 80 126 0 4 30 0 25 80 20 80 0 Full families 2 90 40 0 15 0 15 40 3 52 120 84 120 95 120 65 4 30 125 20 62 25 62 35 5 60 110 10 20 30 32 18 6 20 35 27 12 32 30 24 Using formula (4), the values of probability densities are determined separately for full and single-parent families. Based on them, the transition probability matrix P is formed, which reflects the probability of change in family state over the period Δt. Let the initial quantitative vector of families be denoted by S(t). Then the projected distribution of families by type at time t+Δt is determined by the expression: S(t+Δt)=S(t)⋅P, where S(t) – vector of the initial distribution of families by state at time t; P – transition probability matrix; S(t+Δt) – projected distribution of families by state at the end of the forecast period. Thus, we obtain a strict formal description of the process of forecasting changes in family structure, taking into account probabilistic transitions between their states. Then, multiplying the initial family-size vector by the transition probability matrix, we determine the projected number of each type in the future: = (337, 225, 337, 112, 0, 0) * Gнп = (255, 250, 364, 134, 0, 0); = (0, 338, 338, 113, 270, 180) * Gп = (0, 284, 376, 193, 296, 144). Based on the obtained data, we calculate the number of apartments by type at the end of the forecast period required for the resettlement of the considered families: Family type, persons Number of families Required apartments Apartment type Difference (initial) (final) 1 157 146 255 1A -11 180 109 1B -71 2 225 250 534 2A +25 338 284 2B -54 3 337 364 740 3A +27 338 376 3B +38 4 112 134 327 4A +22 113 193 4B +80 5 0 0 296 5A 0 270 296 5B +26 6 0 0 144 6A + 180 144 6B -36 Total: 2250 2296 2296 +46 INTERNATIONAL SYMPOSIUM “ADVANCED RESEARCH IN ECONOMICS AND BUSINESS MANAGEMENT”, SEPTEMBER 19, 2025 405 Based on the described algorithms and models, software modules were developed and implemented in urban planning practice [3,4]. REFERENCES 1. Decree of the President of the Republic of Uzbekistan of January 10, 2019 No. UP-5623 “On Measures for the Fundamental Improvement of Urbanization Processes.” 2. S. Mirziyoyeva. National Development Strategies as an Effective Instrument for Realizing Competitive Advantages under Global Competition. In: Iqtisodiyot: tahlillar va prognozlar, No. 7, 2019, pp. 16–23. 3. Daliev A.Sh., Dzhumaboyev Kh.R. A Program for Determining the Forecasted Number of Families on the Housing Waiting List. In: Algorithms, Issue 56, pp. 76–80. 4. Daliev A.Sh., Dzhumaboyev Kh.R. An Algorithm for Forecasting the Demographic Structure of Families Planned for Settlement in a Designed Microdistrict (Rural Settlement). In: Questions of RASU, Issue 38, pp. 132–138.