Global analysis of a fractional-order viral model with lytic and non-lytic adaptive immunity

In this paper, we suggest a new fractional virus model with two routes of infection. The first is the usual virus-to-cell infection and the other is the direct transmission of cell-to-cell. The proposed model integrates the effect of the fractional derivative and the influence of adaptive immunity in the studied viral dynamics. Adaptive immunity is represented by cellular and humoral immune responses. The lytic and non-lytic immunological mechanisms that prevent viral reproduction and reduce cells infection are also included in our model. Caputo fractional derivatives are incorporated into each compartment of the model. We begin by showing that our suggested model is well-posed in terms of solution existence, uniqueness, non-negativity and boundedness. We prove that there are five equilibria in our enhanced viral model: virus-clear steady point E circle\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{\circ }$$\end{document}, immunity-free steady point E1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{1}$$\end{document}, infection steady point with only cellular response E2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{2}$$\end{document}, infection steady point with only humoral response E3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{3}$$\end{document} and infection steady point with adaptive immunity E4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{4}$$\end{document}. By defining five kinds of reproduction numbers, we demonstrate the equilibria's global stability by utilizing the Lyapunov method and LaSalle's invariance principle. In addition, many numerical simulations are given to validate our theoretical results regarding the stability of steady points. The numerical outcomes also show that the long-term memory effect, represented by the fractional order derivative, does not affect the stability of the steady points. However, when the fractional order derivative is decreasing, solutions tend to reach equilibrium faster.