Fractional-calculus analysis of the dynamics of $$\text {CD4}^{+}$$ T cells and human immunodeficiency viruses

Combination antiretroviral Therapy (cART) is the standard treatment approach for human immunodeficiency virus (HIV), involving the use of various antiretroviral drugs to effectively suppress the virus’s replication in the body. The objective of cART is to decrease the viral load in the blood to undetectable levels, enhance the immune system’s function, and ultimately prolong the patient’s life by preventing the progression to AIDS and associated opportunistic infections. In this work, we formulated the dynamics of HIV infection, including the effects of cART, within a fractional framework. This paper presents a numerical study that investigates the complex dynamics of HIV infection in \(\text {CD4}^{+}\) T cells. The proposed HIV model incorporates the impact of antiretroviral medication via the Caputo–Fabrizio derivative. To comprehend the dynamics of the proposed HIV infection model, a numerical approach is employed. The dynamic behavior of the system is illustrated by examining the influence of various input parameters, aiming to capture the system’s sensitivity to these factors. Furthermore, this modeling approach highlights the interaction between the immune system and the virus. Through numerical simulations utilizing specific input values, we explore the chaotic and periodic behavior of HIV infection and provide insights into its intricate dynamics.