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Introduction
By about mid of 2022, the importance of the COVID-19 Pandemic’s issue was not needing any special argumentation 1. Since the first signals for the seriousness of that Pandemic worldwide in the beginning of 2020 2, much has been learned about the SARS-CoV-2 virus itself, though this process is not finished because new virus variants/mutations appear with time, with different properties, including much increased contagiousness sometimes. This is the case with the Omicron variant identified in late November 2021 3. Later, other sub-variants of it were found responsible for provoking new (or secondary) waves spreading out even faster and whose peaks lie at varying time intervals further away from the primary peak, but sometimes also very close 4,5. As matter of fact the first Omicron variant which became of importance (labeled shortly BA.1 in the literature) and the sub-variant labeled BA.2 which became dominant since February 2022 worldwide, were detected nearly simultaneously in South Africa and Botswana (joined also by BA.3). The present work is concerned with some mathematical aspects of addressing and modeling the Omicron-stage of the Pandemic 6 in retros- pective. However, one should never forget that dealing with the development of Epidemics is a multidisciplinary subject. Epidemiologists, specialists in infections, virologists, medics with different specializations, pharmaceutics, immunologists, molecular biologists etc. naturally participate by addressing their relevant specialized issues. Thus, the modeling remains just a more or less reliable quest for mathematical apparatus able for short and/or long term forecasting or a posteriori description. In this way, it is possible to address more efficiently the issue of the health, social and economic price that societies have to pay to stop/control the Pandemic. Very recently, during the finalization of the present work in May 2023, the WHO declared 7 that the COVID-19 pandemic is now an established and ongoing health issue which no longer constitutes a public health emergency of international concern.

A bit more than an year earlier, in the beginning of March 2022, the Omicron wave has reached in many countries a maximum and decreasing trends were observed. Therefore the question of lifting different restrictions then raised again. At that time, there were objective and real preconditions which have been met and which have made reasonable such a program of return to the normal. However, this general trend cannot be a basis for automatism in dealing with the problem, within about the same nature and tempo of lifting restrictions everywhere. On the contrary, this should be done with taking into account the specific features of the situation at every place (country or even region, may be). Later, new developments occurred, to some extent related to the initialized processes of lifting restriction, but mainly to the spread of new sub-variants of Omicron 4,5 described as being even more contagious than the original one. The present work, on the basis of improvements of the tracking model published recently in Ref.8 aims to reproduce in retrospective the data on the new daily infection cases in three European countries: the UK, Germany and France till June 2022. The improvements compared to Ref8, consist in more continuous interconnection between consecutive waves and superposition of the waves related to pathogens with different properties. The paper is built as follows. After the introduction, the method used for the calculations is presented including the newly incorporated and improved features. Then, the results for the three countries are presented, and later discussed and compared, also to similar data for the USA. Finally, the conclusions of the present study are given.