Exogenous re-infection does not always cause backward bifurcation in TB transmission dynamics

Models for the transmission dynamics of mycobacterium tuberculosis (TB) that incorporate exogenous re-infection are known to induce the phenomenon of backward bifurcation, a dynamic phenomenon associated with the existence of two stable attractors when the reproduction number of the model is less than unity. This study shows, by way of a counter example, that exogenous re-infection does not always cause backward bifurcation in TB transmission dynamics. In particular, it is shown that it is the transmission ability of the re-infected individuals, and not just the re-infection process, that causes the backward bifurcation phenomenon. When re-infected individuals do not transmit infection, the disease-free equilibrium of the model is shown to be globally-asymptotically stable (GAS) when the associated reproduction number is less than unity. The model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. It is shown, using a Lyapunov function, that the unique endemic equilibrium is GAS for the special case with no disease-induced mortality and no transmission by re-infected individuals. It is further shown that even if re-infected individuals do transmit infection, backward bifurcation only occurs if their transmissibility exceeds a certain threshold. Sensitivity analyses, with respect to the derived backward bifurcation threshold, show that the phenomenon of backward bifurcation is more likely to occur if the rates of re-infection and transmissibility of re-infected individuals are sufficiently high. Furthermore, it is likely to occur if the fraction of slow progressors (to active TB) is increased or if the rates of treatment (of symptomatic cases) and disease-induced mortality are increased. On the other hand, backward bifurcation is less likely to occur for increasing rates of endogenous re-activation of latent TB cases.