In this article, we present a mathematical model for the transmission of HIV with the compartment of
individuals in remission and vertical transmission describing the dynamics of the spread of the HIV/AIDS epidemic in
a community. In the mathematical analysis of the model, we compute the basic reproduction number R0 and study the
existence and stability of the disease-free equilibrium point. We also formulate an appropriate optimal control problem
and study the conditions necessary for disease control to determine the role of preventive measures and treatment
in reducing the spread of HIV/AIDS. Indeed, we study the impact of these control variables taken separately and
combined. So we find that treatment is more cost-e ective in reducing the spread of HIV than preventive measures.
Finally, the numerical results conform to the theoretical analysis.

HIV/AIDS epidemiology; mathematical model; local and global stability; numerical results; Optimal control