Mathematical analysis on the transmission dynamics of delta and omicron variants of COVID-19 in the United States

The COVID-19 pandemic has undergone significant changes due to the emergence of new variants. The Delta and Omicron variants, in particular, have posed unique challenges because of their increased transmissibility and potential for vaccine escape. Understanding the transmission dynamics of these variants is crucial for public health planning and response. Consequently, this paper introduces a new deterministic mathematical model to understand the transmission dynamics of the Delta and Omicron variants of COVID-19, considering re-infection and imperfect vaccination. The analysis begins with the computation of the basic reproduction number for the Delta and Omicron variants and the examination of the local stability of the disease-free equilibrium using the Routh-Hurwitz criterion. An in-depth analysis of the COVID-19 model highlights that both variants demonstrate a phenomenon called backward bifurcation, which is characterized by the co-existence of a stable disease-free equilibrium and a stable endemic equilibrium when their basic reproduction number falls below one. This property poses challenges in effectively controlling COVID-19 within the population. However, assuming flawless vaccine efficacy and zero re-infection, these variants have a globally asymptotically stable disease-free equilibrium. We conducted a sensitivity analysis on the basic parameters of the Delta and Omicron reproduction numbers to identify influential factors contributing to the transmission of these variants. Additionally, we calculated the Omicron invasion reproduction number and developed analytical expressions to determine the necessary percentage of vaccinated individuals required for COVID-19 eradication, even with an imperfect vaccine. The model was validated by fitting it with the daily confirmed cases of COVID-19 in the United States during the period coinciding with the emergence of the Omicron variant. It was determined that with a COVID-19 vaccine offering 60% protection against the Omicron variant, a vaccination rate of at least 97.67% among the susceptible population is required to attain the herd immunity threshold. Numerical simulations indicate that increasing both the vaccination rate and the efficacy of the vaccine against the Delta and Omicron variants significantly reduces the number of hospitalized individuals.