Assessing Effectiveness of COVID-19 Vaccine in India Through SE${}^{\textbf{2}}$I${}^{\textbf{6}}$R${}^{\textbf{4}}$D${}^{\textbf{4}}$V CompartmentalModel Using a Cohort Study

India's approval of COVID-19 vaccines such as Covishield and Covaxin, along with ongoing updates on health guidelines and boosters, underscores the importance of understanding vaccine efficacy and real-world vaccine effectiveness (VE). While efficacy indicates trial performance, effectiveness reflects actual impact across varied conditions. Precise VE estimation through mathematical modeling is crucial for optimizing vaccination strategies and resource distribution. In this study, we enhance the SEI<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}_{3}$</tex-math></inline-formula>R<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}_{2}$</tex-math></inline-formula>D<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}_{2}$</tex-math></inline-formula>V model, segmented by reporting status, into the SE<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{2}$</tex-math></inline-formula>I<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{6}$</tex-math></inline-formula>R<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{4}$</tex-math></inline-formula>D<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{4}$</tex-math></inline-formula>V model. This model further subdivides infected, recovered, deceased, and exposed groups by vaccination status, improving VE analysis. This granularity is essential for comparing attack rates among vaccinated and unvaccinated populations, aligning closely with real-time data, and accounting for transmission variations. Our findings show that VE fluctuates, initially dropping to 41% due to transmission and variants but rising to 95% with higher vaccination, reducing transmission and severity. Using cohort analysis, our model estimates time-varying VE while factoring in distinct recovery and mortality rates by vaccination and reporting status, and also compares various VE definitions with findings from similar studies. We projected VE under scenarios of waning immunity, new variants, and age group impacts. Sensitivity analysis fine-tuned parameters to assess VE's dependence on vaccination coverage, transmission rates, and incubation periods. From March 2020 to April 2024, our SE<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{2}$</tex-math></inline-formula>I<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{6}$</tex-math></inline-formula>R<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{4}$</tex-math></inline-formula>D<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}^{4}$</tex-math></inline-formula>V model showed close alignment with real data, achieving a relative error of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.87\boldsymbol{\times}10^{-2}$</tex-math></inline-formula> in infections, outperforming SEIRDVB's <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3.53\boldsymbol{\times}10^{-2}$</tex-math></inline-formula> and SEI<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}_{3}$</tex-math></inline-formula>R<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}_{2}$</tex-math></inline-formula>D<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${}_{2}$</tex-math></inline-formula>V's <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$0.97\boldsymbol{\times}10^{-2}$</tex-math></inline-formula>. The model has also been validated across various countries, demonstrating its applicability in real-world contexts.