An optimal scheme to boost immunity and suppress viruses for HIV by combining a phased immunotherapy with the sustaining antiviral therapy.

Despite many approaches to treat HIV virus, the endeavor, due to the inability of therapy to eradicate HIV infection, has been aroused to formulate rational therapeutic strategies to establish sustained immunity to suppress viruses after stopping therapy. In this paper, incorporating the time lag of the expansion of immune cells, we propose an explicit model with continuous antiretroviral therapy (CATT) and an intermittent immunotherapy to describe an interaction of uninfected cells, HIV virus and immune response. Two kinds of bistability and the sensitivities of the amplitude and period of the periodic solution with respect to all of parameters indicate that both and relating to the therapy are scheduled to propose an optimal treatment tactics. Furthermore, taking a patient performed a CATT but with an unsuccessful outcome as a example, we inset a phased immunotherapy into the above CATT and then adjust the therapeutic session as well as the inlaid time to quest the preferable therapeutic regimen. Mathematically, we alter the solution of system from the basin of the attraction of the immune-free equilibrium to the immune control balance when the treatment is ceased, meanwhile minimize the cost function through a period of combined therapy. Due to the particularity of our optimal problem, we contribute a novel optimization approach by meshing a special domain on the antiretroviral and immunotherapy parameters and $\bar{b}$, to catch an optimal combined treatment scheme. Simulations exhibit that early mediating immunotherapy suppresses the load of virus lower while shortening the combined treatment session does not reduce but magnify the cost function. Our results can provide some insights into the design of optimal therapeutic strategies to boost sustained immunity to quell viruses.