  We propose and study a new mathematical model of the human immunodeficiency
virus (HIV). The main novelty is to consider that the antibody growth depends
not only on the virus and on the antibodies concentration but also on the
uninfected cells concentration. The model consists of five nonlinear
differential equations describing the evolution of the uninfected cells, the
infected ones, the free viruses, and the adaptive immunity. The adaptive immune
response is represented by the cytotoxic T-lymphocytes (CTL) cells and the
antibodies with the growth function supposed to be trilinear. The model
includes two kinds of treatments. The objective of the first one is to reduce
the number of infected cells, while the aim of the second is to block free
viruses. Firstly, the positivity and the boundedness of solutions are
established. After that, the local stability of the disease free steady state
and the infection steady states are characterized. Next, an optimal control
problem is posed and investigated. Finally, numerical simulations are performed
in order to show the behavior of solutions and the effectiveness of the two
incorporated treatments via an efficient optimal control strategy.
