A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment

• Published in: Advances in difference equations 2019-05, Vol.2019 (1), p.1-20, Article 200
• Main Authors: Moore, Elvin J., Sirisubtawee, Sekson, Koonprasert, Sanoe
• Format: Article
• Language: English
• Subjects: Acquired immune deficiency syndrome; AIDS; Analysis; Caputo–Fabrizio fractional derivative; Computer memory; Computer simulation; Difference and Functional Equations; Differential equations; Epidemics; Equilibrium; Fixed points (mathematics); Functional Analysis; HIV; HIV/AIDS epidemic model; Human immunodeficiency virus; Iterative methods; Mathematical models; Mathematics; Mathematics and Statistics; Non-singularity; Ordinary Differential Equations; Partial Differential Equations; Three-step fractional Adams–Bashforth scheme
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-019-2138-9

In recent years, many new definitions of fractional derivatives have been proposed and used to develop mathematical models for a wide variety of real-world systems containing memory, history, or nonlocal effects. The main purpose of the present paper is to develop and analyze a Caputo–Fabrizio fractional derivative model for the HIV/AIDS epidemic which includes an antiretroviral treatment compartment. The existence and uniqueness of the system of solutions of the model are established using a fixed-point theorem and an iterative method. The model is shown to have a disease-free and an endemic equilibrium point. Conditions are derived for the existence of the endemic equilibrium point and for the local asymptotic stability of the disease-free equilibrium point. The results confirm that the disease-free equilibrium point becomes increasingly stable as the fractional order is reduced. Numerical simulations are carried out using a three-step Adams–Bashforth predictor method for a range of fractional orders to illustrate the effects of varying the fractional order and to support the theoretical results.