Ulam-hyres stability analysis and fractional operator implications on the Covid-19 virus dynamics with long-term vaccination effects

The COVID-19 pandemic has necessitated the development of highly efficient mathematical models to manage its spread, particularly concerning vaccination strategies. Traditional models, however, often fail to account for memory effects observed in real-world scenarios, which can be effectively captured using fractional derivatives. This paper introduces a novel COVID-19 model incorporating fractional-order derivatives to reflect better the dependence of the pandemic's growth on historical events. By approximating the non-locality of fractional derivatives through a generalized Mittag-Leffler kernel, the model effectively captures long-term vaccination effects. To enhance the accuracy of numerical results, the model also integrates the concept of a two-step Lagrange polynomial. The local asymptotic stability of the disease-free equilibrium point is examined through sensitivity and qualitative analyses, particularly using the basic reproduction number, R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_0 $$\end{document}. The proposed scheme is rigorously validated through fixed point theory, ensuring the model is evidence-based. A case study conducted in Saudi Arabia demonstrates the effectiveness of the proposed model, yielding successful verification through real-world numerical analysis. The results indicate that incorporating fractional derivatives leads to a more robust model, particularly in assessing the long-term impact of vaccination on the COVID-19 epidemic. The findings underscore the significant role of vaccination in reducing R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_0 $$\end{document} and achieving a disease-free state. This work highlights the potential of fractional operators in epidemiological modeling, providing crucial insights into strategies for preventing the transmission of COVID-19 and similar diseases.