Stability, sensitivity, and bifurcation analysis of a fractional-order HIV model of CD\(4^+\) T cells with memory and external virus transmission from macrophages

Abstract

This paper presents a fractional-order mathematical model in the Caputo sense to study the dynamics of HIV infection, focusing on the interactions among uninfected T cells, infected T cells, and HIV. The inclusion of fractional derivatives allows for the incorporation of memory effects, providing a more accurate representation of disease progression. We begin by establishing fundamental mathematical properties such as existence and uniqueness, positivity and boundedness of the solutions. The Caputo fractional derivative: Equilibrium points of the system are identified, and the basic reproduction number, \(\mathcal {R}_0\), is calculated to assess the potential for infection spread. Sensitivity analysis of \(\mathcal {R}_0\) is performed to determine the most influential parameters, both positively and negatively. The local and global stability analyses have been carried out for all equilibrium points. We also derive the transcritical and Hopf bifurcation conditions, revealing the onset of oscillatory behavior. Numerical simulations are carried out using the fractional forward Euler method, demonstrating how the fractional-order \(\alpha \) significantly influences the disease dynamics. Our results indicate that higher values of \(\alpha \) are associated with oscillatory dynamics. In contrast, a reduction in \(\alpha \) contributes to the stabilization of the system, as evidenced by the bifurcation diagrams for various fractional orders. Additionally, we provide biological interpretations of all numerical results and compare every result with existing related models to highlight its novelty and relevance. The paper concludes with a summary of findings and potential future research directions.