Mathematical simulations and sensitivity visualization of fractional order disease model describing human immunodeficiency
Muhammad Asad Ullah, Nauman Raza, Talat Nazir
Abstract
In this study, we explore a mathematical model of human immunodeficiency virus type 1 (HIV-1) infection in CD4+ T-cells, considering fractional-order dynamics in the Caputo sense. Fractional models are of great significance due to their capacity to predict disease outbreaks while incorporating memory and non-local characteristics. To establish the existence and uniqueness of solutions for the fractional model, a fixed-point theorem and an iterative technique are employed. Furthermore, it is confirmed that the solutions of the model are both positive and bounded. The basic reproduction number, denoted as R0, is calculated using the next-generation matrix approach, and the equilibrium points of the model, including disease-free and endemic equilibrium points, are presented. A sensitivity analysis of R0 is performed by varying different parameters. To analyze the local stability of equilibrium points, the Routh-Hurwitz technique is employed. The global stability of equilibrium points is demonstrated using the Lyapunov function and LaSalle's principle, as well as through UlamHyers (UH) stability and the generalized UH stability conditions. The proposed numerical scheme Adams-Bashforth predictor-corrector is validated by comparing it with the fourth-order Runge-Kutta (RK4) technique. We examine the influence of the fractional order (ξ) by conducting numerical simulations for different values of ξ using the Adams-Bashforth predictor-corrector approach. Furthermore, graphical presentations illustrate how different crucial model parameters impact disease dynamics. The results obtained from this study demonstrate that the adopted strategies significantly improve prediction accuracy.