  We introduce delays in a tuberculosis (TB) model, representing the time delay
on the diagnosis and commencement of treatment of individuals with active TB
infection. The stability of the disease free and endemic equilibriums is
investigated for any time delay. Corresponding optimal control problems, with
time delays in both state and control variables, are formulated and studied.
Although it is well-known that there is a delay between two to eight weeks
between TB infection and reaction of body's immune system to tuberculin, delays
for the active infected to be detected and treated, and delays on the treatment
of persistent latent individuals due to clinical and patient reasons, which
clearly justifies the introduction of time delays on state and control
measures, our work seems to be the first to consider such time-delays for TB
and apply time-delay optimal control to carry out the optimality analysis.
