A mathematical analysis of the tuberculosis epidemic

Abstract

Long latency period has been seen as a major setback in the control of tuberculosis (TB), because an individual infected of TB can be latent for years. In this paper, a four compartment deterministic model is presented to change the narrative. It is established that making sure every individual infected of TB pass through the latent stage would actually reduce the TB incident rate. Also established from the model is that prevention against TB re-infection has no significant contribution to the disease incident rate, which implies the money meant to guard against TB reinfection should not be wasted. The formulated model was shown to be locally asymptotically stable at the disease free equilibrium (DFE) whenever R_0 < 1 and the global asymptotic stability (GAS) at this point was as well established. It is as well shown that the endemic equilibrium point (EEP) is locally asymptotically stable with the aid of phase portrait and the GAS of the EEP was established by defining a suitable Lyapunov function.