Optimal Control Design for a HIV-1 Model With Time Delay and Functional Responses

The main objective of this study is to model the HIV-1 infection using ordinary differential equations (ODEs) by considering factors that include infectivity rate, antiretroviral therapy (ART), logistic growth, intracellular time delay for the incidence of the uninfected cells and the infected cells, latency period, and immune responses. The theoretical frameworks such as stability, bifurcation, and the suitable control parameter are investigated to treat the disease progression for the HIV-1 infection model. Based on the stability theory, the steady-state analysis for the HIV-1 model is performed for three types of equilibrium: (1) disease-free equilibrium, (2) immune-free equilibrium, and (3) equilibrium with infection. Utilizing a next-generation matrix approach, the reproduction number is theoretically derived to ensure the virulence of the spread. The Routh–Hurwitz criterion is employed to perform the stability analysis for the model with and without delay for the infection equilibrium point. The qualitative changes in the behaviors of the system called bifurcations, investigating intracellular delay as a bifurcation parameter, Hopf bifurcation conditions, are derived. The threshold value for the delay is analytically obtained; below the threshold value, the model remains stable; if the threshold value exceeds, there are oscillations in the cell population. Furthermore, the model is structured as an optimal control problem choosing therapy as a control parameter based on the Hamilton–Lagrangian approach and the Pontryagin maximum principle. Numerical simulations are conducted to validate the effectiveness of the proposed stability conditions, bifurcation conditions, and optimal control design for the therapy.