Optimal control analysis of a tuberculosis model with exogenous re-infection and incomplete treatment

A new mathematical model of tuberculosis (TB) featuring exogenous re-infection and incomplete treatment is presented and analyzed. The model divides total population into susceptible, latently infected, actively infected (uninformed and enlightened), and treatment classes. The model with or without incomplete treatment exhibits backward bifurcation phenomenon, which is caused by the presence of exogenous re-infection. However, further investigation reveals that the absence of incomplete treatment has the potential to reduce the backward bifurcation range significantly. The global dynamics of the TB model without exogenous re-infection is completely determined by the basic reproduction number under certain modifications on parameters. Furthermore, the model is extended to include three time-dependent control functions, such as public awareness campaign, treatment effort, and preventive control against incomplete treatment. The existence of the optimal control for the nonautonomous model is proven and the three controls are characterized using Pontryagin's maximum principle. Numerical simulations are performed to show the significance of singular implementation of each of the controls and combination of the three controls in minimizing the TB burden in the population.