MATHEMATICAL ANALYSIS OF A NEW TUBERCULOSIS EPIDEMIC MODEL AND THE INFLUENCE OF BACKWARD BIFURCATION

The paper presents a deterministic model for the transmis-sion dynamics of Mycobacterium tuberculosis (TB) in a population in thepresence of Directly Observed Therapy Short-course (DOTS). The model,which allows for the detection and treatment of individuals with symptomsand uses standard incidence function for the infection rate, is rigorouslyanalysed to gain insight into its dynamical features. The analysis revealsthat the model undergoes a backward bifurcation, where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium whenthe associated reproduction threshold RC is less than unity. This phenom-enon resulted due to the exogenous re-infection property of TB disease. Itis shown that, in the absence of such re-infection, the model has a globally-asymptotically stable DFE when RC is less than unity. Further, the modelhas a unique endemic equilibrium, for a special case, whenever the associ-ated threshold quantity exceeds unity. This endemic equilibrium is shownto be globally-asymptotically stable, for a special case, using a non- linearLyapunov function of Goh-Volterra type. The model provides a reason-able t to a data set for the TB transmission data in Nigeria. Additionalnumerical simulations show that exogenous re-infection increases the cu-mulative number of new TB cases regardless of whether or not the DOTSprogram is implemented (but the corresponding cumulative number of newcases decreases for the case with DOTS program, in comparison to the casewithout DOTS).

MATHEMATICAL ANALYSIS OF A NEW TUBERCULOSIS EPIDEMIC MODEL AND THE INFLUENCE OF BACKWARD BIFURCATION Keywords:

Tuberculosis , reproduction number , stability