In a first part, we consider a Tuberculosis (TB) model with time delays in both state and control variables, representing the time delay on the diagnosis and commence- ment of treatment of individuals with active TB infection, respectively. The stability of the disease free and endemic equilibriums is investigated for any time delay. Correspon- ding optimal control problems, with time delays in both state and control variables, are formulated and studied. In the second part, we propose a mathematical model for the transmission dynamics of human immunodeficiency virus (HIV).

We prove the global stability of the equilibrium points. Based on data provided by the Progress Report on the AIDS response in Cape Verde 2014, we calibrate our model to the cumulative cases of infection by HIV and AIDS from 1987 to 2014 and we show that our model predicts well this reality. A sensitivity analysis is done for the case study in Cape Verde.