Mathematical study of a fractional order HIV model of CD\(4^+\) T-cells with recovery

Human immunodeficiency virus (HIV) infection involves complex interactions between the virus and the immune system, including latency, viral reservoirs, and variable replication rates. Fractional-order models can more accurately describe these dynamics compared to conventional integer-order models. In this paper, we consider E. Beretta et al.’s marine bacteriophage infection model and transform this framework to formulate an HIV infection model. Our main objective is to observe the complex dynamics of the proposed HIV model through the application of fractional calculus. We examine the dynamics of the fractional-order model, addressing aspects such as non-negativity, boundedness, and the existence and uniqueness of the solution. We conduct a comprehensive mathematical analysis of the model through various steady-states and Hopf-bifurcation analyses. We explore rich dynamics through memory effects by using the fractional order \(\alpha \) $\alpha$ for \(\hbox {CD}_4^+\) ${\text{CD}}_4^{+}$ T-cell populations. Extensive numerical simulations are conducted using the Fractional Forward Euler Method, and the numerical outcomes indicate that the memory effects characterized by \(\alpha \) $\alpha$ significantly influence the system’s stability. When the infection rate is exceedingly high, the system exhibits instability; however, reducing the value of \(\alpha \) $\alpha$ stabilizes the system, whereas increasing \(\alpha \) $\alpha$ induces destabilization. We generate a bifurcation diagram, which confirms that the system possesses periodic solutions and demonstrates stability transitions when certain key parameters are altered. Finally, we discuss our numerical findings and elucidate the potential biological implications, thereby concluding our manuscript.