A nonlinear compartmental model with several types of detections is proposed. Analysis reveals that local asymptotic stability for disease-free equilibrium (DFE) can be achieved if: (i) random screening is sufficiently effective; (ii) infection by detected HIV-positive individuals is minimal. If the average number of infections by a known infective exceeds unity, the endemic equilibrium is always unstable and the total number of infectives could increase without bound, provided that the initial infective population sizes are sufficiently large. On the other hand, if the number of infections by a known infective is less than one, then either the DFE or the endemic equilibrium is globally asymptotically stable, leading to a more manageable epidemic, even if the disease is not eradicated. We make use of the Cuban HIV/AIDS data to fit the model during two separate time periods in 1986-2008 to reflect the implementation of different types of detections. The reproduction numbers for each time period are then computed from the two sets of estimated parameter values.