Optimal control strategy to reduce HIV-1 infection in the brain

The study of Human Immunodeficiency Virus (HIV) and of its progression to Acquired Immune Deficiency Syndrome (AIDS) has been a challenge by scientists since the early 1980s; the improvements in the knowledge of its transmission and of the evolution of the disease have strongly reduced the mortality, even if up to know an effective vaccine is not available. Nevertheless, the Highly Active Antiretroviral Therapy (HAART) allows a high life expectancy. Some of the mechanisms of HIV reproduction are under investigation and recent observations are suggesting the existence of virus reservoir in the brain, the liver, the lungs and the gut. In particular, due to untreated amounts of virus, neurocognitive disorders as well as motor nerves diseases and weakness in muscles are improved in patients already weakened. In this paper it is considered a mathematical modelling to study the within-host viral dynamics, analysing the dynamics of the CD4-T cells, of the macrophages both inside and outside the brain, uninfected and infected, along with the virions carrying on the infection. An optimal control strategy is proposed and implemented,aiming at maximizing the number of healthy cells, while reducing the number of infected ones, scheduling suitably the therapy.