Abstract
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.
Highlights
1 Introduction Human immunodeficiency virus (HIV), which leads to acquired immunodeficiency syndrome (AIDS), destroys the human body’s ability to fight infections
According to UNAIDS data reported in June 2016 [3], around 18.2 million people living with HIV were receiving antiretroviral therapy (ART)
7 Conclusions In this article, a Caputo–Fabrizio fractional differential equation model for HIV/AIDS with an antiretroviral treatment compartment has been investigated. This fractional model is based on the use of the non-singular exponentially decreasing kernels appearing in the Caputo–Fabrizio fractional derivative
Summary
Human immunodeficiency virus (HIV), which leads to acquired immunodeficiency syndrome (AIDS), destroys the human body’s ability to fight infections. At the end of 2016, approximately 36.7 million people were living with HIV, and approximately 1.8 million new infections were occurring globally each year [2]. According to UNAIDS data reported in June 2016 [3], around 18.2 million people living with HIV were receiving antiretroviral therapy (ART). The US Center for Disease control and Prevention (CDC) [4] reported in 2017 that without treatment of HIV/AIDS with antiretroviral medicine, HIV infection advances through several stages and individuals with AIDS typically survive about 3 years. Antiretroviral HIV/AIDS therapy involves the simultaneous management of two or more antiviral drugs which can assist patients to live longer and restore their immune system [5]. In 2016, the regions worst affected by HIV/AIDS were located in eastern and southern Africa with nearly 10.3 million people being treated for
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