Alexander V. Sokolov

The aim of this study is to develop models for analyzing the dynamics of copper and nickel pollution in small lakes within the influence zone of the Pechenga Nickel Plant (up to 100 km) in the past, present, and future (without emissions).The research focuses on modeling the dynamics of nickel and copper concentrations in water, soil, and lake sediments caused by atmospheric emissions from the Pechenga Nickel Plant (Kola Peninsula) from 1946 to 2050. The model is built upon heterogeneous data collected during over 30 years of research on pollution effects in the Kola Peninsula. Until recently (2020), the data reflected the state of lakes, rivers, soils, and sediments under significant atmospheric emissions of pollutants. New data, collected in 2023 under drastically reduced emissions, allowed refinement of several parameters and modifications to the model to describe a new phenomenon—the recovery of the region's natural environment.The use of balanced identification techniques enabled the selection of a model of appropriate complexity for the available heterogeneous dataset (over 10 sources), the identification of unknown parameters (both numerical and functional), and the generation of results. The specialized software employed in this study (available at https://github.com/distcomp/SvF) includes examples of various problem-solving scenarios (https://github.com/distcomp/SvF/tree/main/Examples). The programs and corresponding databases used in this work can also be found in the repository.The developed model matches the complexity of the experimental data and reflects the new reality—a slow recovery of ecosystems under drastically reduced emissions. The obtained forecast is reliable: under the scenario of zero emissions, water concentrations are determined by the release (transition to soluble forms) of Ni and Cu from reserves in the soil and sediments. This process is very slow, resulting in a noticeable reduction in water concentrations on the one hand, but precluding hopes for rapid further improvement on the other. The estimated "half-leaching" period (analogous to "half-life") of these reserves is on the order of several hundred years.Keywords: atmospheric transport, pollution transformation, nickel, copper, subarctic aquatic and terrestrial ecosystems, mathematical modeling, balanced identification, forecasting

The article deals with modeling the dynamics of nickel concentration in soils, water and bottom sediments of lakes, caused by emissions into the atmosphere of the Pechenganickel plant (Kola Peninsula) during the entire period of its operation. The technology of balanced identification is used, which allows, based on a mathematical description of heterogeneous geochemical processes occurring in ecosystems, to combine heterogeneous experimental data and build a computer model with an optimal balance of complexity and closeness to data. The results are presented and discussed, including estimates of the retrospective state of the simulated objects (before the start of intensive studies) and a forecast of their dynamics until 2030. According to model calculations, the intensity of Ni accumulation in soil was 2.4 and 2 times higher in bottom sediments during maximum precipitation periods (1980–2005), while according to the model forecast, after the shutdown of the plant, Ni accumulation in bottom sediments will begin to decrease and Ni will slowly leach from the soil with an intensity of 0.2 mg/m2 year.

Modeling of the dynamics of the nickel concentration in soils, water, and bottom sediments of lakes, caused by emissions into the atmosphere from the Pechenga Nickel Plant (Kola Peninsula) over the entire period of its operation, is considered. The technology of balanced identification, which makes it possible based on mathematical description of inhomogeneous geochemical processes occurring in ecosystems to combine heterogeneous experimental data and to develop a computer model with an optimal balance of complexity and closeness to the data, is used. The results including estimates of the retrospective state of the simulated objects and a forecast of their dynamics until 2030 are presented and discussed. According to model calculations, the intensity of Ni accumulation in the soil was 2.4 and it was twice as high as in bottom sediments during the periods of maximum deposition (1980–2005). According to the forecast model, after the shutdown of the plant, Ni accumulation in bottom sediments will begin to decrease and Ni will slowly leach from the soil with an intensity of 0.2 mg/(m² yr).

The paper discusses modeling the dynamics of nickel concentration in soils, water, and the bottom sediments of lakes caused by atmospheric emissions from the Pechenganickel plant, Kola Peninsula, throughout its whole operation period. The applied technology of balanced identification makes it possible to use a mathematical description of heterogeneous geochemical processes in ecosystems to combine heterogeneous experimental data and build up a computer model with an optimal balance of its complexity and fitting quality of the data. The model is used to analyze the spatial and temporal variability of natural objects in the zone of distribution of atmospheric pollution (nickel) from the Pechenganickel plant. The paper presents and discusses results of this study, including estimates of the retrospective state of the simulated objects (before the start of the intense studies) and a forecast of their dynamics until 2030. According to the model calculations, the intensity of Ni accumulation in the soil and bottom sediments was 2.35 and 4.48 mg/(m² year) during the maximum deposition periods (1980–2005), whereas the model predicts a decrease in the intensity of Ni accumulation in the bottom sediments (0.23 mg/(m² year)) and slow Ni leaching from the soil (0.19 mg/(m² year)) after the shutdown of the plant.

The study aims at investigating a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time—qualitative change (bifurcation) in the behavior of the “virus vs. human” system with an increase in contagiousness. Numerical experiments with a model of the COVID-19 epidemic in Moscow have demonstrated that when the reproduction number R₀ is about 4, a qualitative change (bifurcation) occurs in the behavior of the virus–human system. Below this value, the long-term forecast tends toward undamped oscillations; above it, the forecast shows damped oscillations: the amplitudes of epidemic waves decrease gradually, with a constant, very high background level of morbidity that keeps natural immunity near 100%. To confirm this result analytically, we use an original modification of the Euler–Lotka renewal equation, which describes the dynamics of infected patients distributed by disease duration (time since infection) and accounts for immunity. To construct a bifurcation diagram, which illustrates the dependence of the equilibrium stability on the parameter R₀, we linearize the equation in the vicinity of the equilibrium point and examine its numerical approximation (discrete form). This approximation can be interpreted as a Leslie model, with the matrix elements dependent on the parameter R₀. By examining the roots of the corresponding Lotka polynomial, we can assess the stability of the equilibrium point and verify the basic assumption about the change in the properties of the system with increasing R₀—about the transition from undamped oscillations to damped ones. For the bifurcation diagram, we use the functions obtained from the simulation of the COVID-19 epidemic in Moscow. However, observations of the epidemic in other cities and countries support the primary finding of our study regarding the attenuation of epidemic waves.

The paper discusses modeling the dynamics of nickel concentration in soils, water, and the bottom sediments of lakes caused by atmospheric emissions from the Pechenganickel plant, Kola Peninsula, throughout its whole operation period. The applied technology of balanced identification makes it possible to use a mathematical description of heterogeneous geochemical processes in ecosystems to combine heterogeneous experimental data and build up a computer model with an optimal balance of its complexity and fitting quality of the data. The model is used to analyze the spatial and temporal variability of natural objects in the zone of distribution of atmospheric pollution (nickel) from the Pechenganickel plant. The paper presents and discusses results of this study, including estimates of the retrospective state of the simulated objects (before the start of the intense studies) and a forecast of their dynamics until 2030. According to the model calculations, the intensity of Ni accumulation in the soil and bottom sediments was 2.35 and 4.48 mg/(m2 year) during the maximum deposition periods (1980–2005), whereas the model predicts a decrease in the intensity of Ni accumulation in the bottom sediments (0.23 mg/(m2 year)) and slow Ni leaching from the soil (0.19 mg/(m2 year)) after the shutdown of the plant.

We investigate a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time. Numerical experiments with a model of the Covid-19 epidemic in Moscow have demonstrated that when the reproduction number R0 is about 4, a qualitative change (bifurcation) occurs in the behavior of the virus-human system. Below this value, the long-term forecast tends to undamped oscillations; above it, the forecast shows damped oscillations: the amplitudes of epidemic waves decrease gradually, with a constant, very high background level of morbidity that keeps the natural immunity near 100%. To confirm this result analytically, we use an original modification of the Euler-Lotka renewal equation, which describes the dynamics of infected patients distributed by disease duration (time since infection) and accounts for immunity. To construct a bifurcation diagram, which illustrates the dependence of the equilibrium stability on the parameter R0, we linearize the equation in the vicinity of the equilibrium point and examine its numerical approximation (discrete form). This approximation can be interpreted as a Leslie model, with the matrix elements dependent on the parameter R0. By examining the roots of the corresponding Lotka polynomial, we can assess the stability of the equilibrium point. For the bifurcation diagram, we use the functions obtained from the simulation of the Covid-19 epidemic in Moscow. However, observations of the epidemic in other cities and countries support the primary finding of our study regarding the attenuation of epidemic waves.