On regime changes of COVID-19 outbreak.

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.

Models induced from critical birth-death process with random initial conditions.

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.