Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: Geometric branching reproduction Markov processes, Authors: Assen Tchorbadjieff, Penka Mayster , We present a model of a continuous-time Markov branching process with the infinitesimal generating function defined by the geometric probability distribution. It is proved that the solution of the backward Kolmogorov equation is expressed by the composition of special functions – Wright function in the subcritical case and Lambert-W function in the critical case. We found the explicit form of conditional limit distribution in the subcritical branching reproduction. In the critical case, the extinction probability and probability mass function are expressed as a series containing Bell polynomial, Stirling numbers, and Lah numbers.

In this work, we study a linear birth–death process starting from random initial conditions. First, we consider these initial conditions as a random number of particles following different standard probabilistic distributions – Negative-Binomial and its closest Geometric, Poisson or Pólya–Aeppli distributions. It is proved analytically and numerically that in these cases the random number of particles alive at any positive time follows the same probability law like the initial condition, but with different parameters depending on time. The random initial conditions cannot change the critical parameter of branching mechanism, but they impact the extinction probability. Finally, the numerical model is extended to an application for studying branching processes with more complex initial conditions. This is demonstrated with a linear birth–death process initialised with Pólya urn sampling scheme. The obtained preliminary results for particle distribution show close relation to Pólya–Aeppli distribution.

The COVID-19 pandemic has had a very serious impact on societies and caused large-scale economic changes and death toll worldwide. The first cases were detected in China, but soon the virus spread quickly worldwide and the intensity of newly reported infections grew high during this initial period almost everywhere. Later, despite all imposed measures, the intensity shifted abruptly multiple times during the two-year period between 2020 and 2022 causing waves of too high infection rates in almost every part of the world. To target this problem, we assume the data heterogeneity as multiple consecutive regime changes. The research study includes the development of a model based on automatic regime change detection and their combination with the linear birth-death process for long-run data fits. The results are empirically verified on data for 38 countries and US states for the period from February 2020 to April 2022. Finally, the initial phase (conditions) properties of infection development are studied.

This study aims to reveal the arsenic dynamics in groundwater of а river floodplain contaminated with mine tailings under temperate climate conditions and natural river hydrodynamics. Arsenic concentrations were monitored in the primary morphological units of the floodplain in the upper stretch of the Ogosta River in NW Bulgaria. Iron, lead-silver, and gold mining heavily affected the river valley in the second half of the 20th century. We used groundwater monitoring data from 21 piezometers for the period 2016-2020. Based on the geochemical and geomorphological conditions in the valley, the piezometers were grouped into three clusters. Regression models were developed for each cluster and representative piezometers to predict arsenic concentrations. In the active floodplain, seasonal fluctuations in arsenic concentrations followed the river and groundwater regime. In this part of the valley floor, we determined two periods of elevated arsenic concentrations during the spring and autumn/winter seasons that coincide with high river water stages. Arsenic content in the groundwater of the higher floodplain was less dependent on the water level fluctuations but followed changes in redox potential, electrical conductivity, and water temperature. The obtained results showed the elaborated models as valuable tools for studying arsenic dynamics in alluvial aquifers of contaminated river floodplains. The suggested models could be coupled with groundwater monitoring systems to monitor arsenic concentrations and identify periods of the year with levels below and above threshold values.