Modeling the mechanics of viral kinetics under immune control during primary infection of HIV-1 with treatment in fractional order

In this paper, we propose and analyze a fractional order model of viral kinetics for primary infection of HIV-1 in presence of immune control with treatment, as the classical model of target-cell-limited model is unable to predict long term viral kinetics unless a delayed immune effect is assumed. Further, the reverse transcriptase treatment can be incorporated by reducing the target cell infection rate. The existence of equilibria and their asymptotical stability results using fractional Routh–Hurwitz stability criterion will be discussed. A threshold value ^′R₀ ^′ known as basic reproduction number has been found to ensure the extinction or persistence of the infection. The numerical solution using generalized Adams–Bashforth–Moulton method to the proposed HIV-1 fractional model is obtained. Also, the sufficient conditions that guarantee the asymptotic stability of endemic equilibrium point is presented. Meanwhile, global asymptotic stability of the endemic equilibrium point is investigated by constructing a suitable Lyapunov functions. The fractional derivative is described in Caputo sense. The obtained numerical results of the proposed model show the effectiveness and strength of the Adams–Bashforth–Moulton method. Some numerical simulations are given to illustrate the analytical results.