The effect of time delay on the dynamics of a fractional-order epidemic model

This study establishes a novel time-delay fractional SEIHR infectious disease model to investigate the effects of saturated incidence rates and time delays on different populations, including susceptibles, infected individuals, recovered individuals, and latent infected individuals. First, the existence and boundedness of the model's solutions are verified, confirming its well-posedness. Subsequently, the existence of equilibria is analyzed, and the impact of parameter variations on the system is explored by examining the equilibria & varepsilon;0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon _{0} $\end{document} and & varepsilon;& lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon _{*} $\end{document}, as well as the basic reproduction number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0} $\end{document}. Additionally, the global dynamics of the equilibria are further analyzed using the Lyapunov method, while Hopf bifurcation theory is applied to explore the conditions under which the system shifts from stability to oscillatory behavior. Numerical simulations further validate the theoretical analysis, showing that time-delay effects significantly influence the system's responsiveness and the rate of disease transmission. Moreover, when the time delay tau crosses the critical threshold tau 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau _{0} $\end{document}, the system exhibits periodic oscillations. By predicting periodic fluctuations and incorporating memory effects and persistent influences, we can better control epidemics, emphasizing the importance of time-delay adjustments and enhancing the system's biological realism.