Mathematical modeling of SARS-CoV-2 variant substitutions in European countries: transmission dynamics and...

Countries across Europe have faced similar evolutions of SARS-CoV-2 variants of concern, including the Alpha, Delta, and Omicron variants.

We used data from GISAID and applied a robust, automated mathematical substitution model to study the dynamics of COVID-19 variants in Europe over a period of more than 2 years, from late 2020 to early 2023. This model identifies variant substitution patterns and distinguishes between residual and dominant behavior. We used weekly sequencing data from 19 European countries to estimate the increase in transmissibility  $\left(\Delta \beta \right)$   between consecutive SARS-CoV-2 variants. In addition, we focused on large countries with separate regional outbreaks and complex scenarios of multiple competing variants.

Our model accurately reproduced the observed substitution patterns between the Alpha, Delta, and Omicron major variants. We estimated the daily variant prevalence and calculated $\Delta \beta$ between variants, revealing that: ( $i$ ) $\Delta \beta$ increased progressively from the Alpha to the Omicron variant; ( $ii$ ) $\Delta \beta$ showed a high degree of variability within Omicron variants; ( $iii$ ) a higher $\Delta \beta$ was associated with a later emergence of the variant within a country; ( $iv$ ) a higher degree of immunization of the population against previous variants was associated with a higher $\Delta \beta$ for the Delta variant; ( $v$ ) larger countries exhibited smaller $\Delta \beta ,$  suggesting regionally diverse outbreaks within the same country; and finally ( $vi$ ) the model reliably captures the dynamics of competing variants, even in complex scenarios.

The use of mathematical models allows for precise and reliable estimation of daily cases of each variant. By quantifying $\Delta \beta ,$  we have tracked the spread of the different variants across Europe, highlighting a robust increase in transmissibility trend from Alpha to Omicron. Additionally, we have shown that the geographical characteristics of a country, as well as the timing of new variant entrances, can explain some of the observed differences in variant substitution dynamics across countries.