Abstract
We propose a new mathematical model to study the effect of stem cells transplantation on HIV-1 patient as a system of first order nonlinear ordinary differential equations. We include in this model five populations: stem cells, free virus, productively infected cells, infected cells in the eclipse stage and non-infected T cells. The model admits two equilibrium states: a non-infected point and an endemically infected one. The study of the stability of the system before treatment is already known: If the “basic reproduction number” is smaller than 1, then the free disease point is asymptotically stable. If the “basic reproduction number” is greater than 1, then the endemic point is asymptotically stable. The study of the stability of the system after treatment proves that, for a patient with an initial “basic reproduction number”, greater than 1, if it is possible for the therapy to lower this ratio under 1, then the patient will recover. If it is not possible to lower the ratio under 1, the HIV infection persists, but, it is possible to offer to the patient a better life.