Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

In this paper, we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals, in which only susceptible individuals can travel freely between the patches. The model has multiple equilibria. We determine conditions that lead to the appearance of a backward bifurcation. The results show that the TB model can have exogenous reinfection among the treated individuals and, at the same time, does not exhibit backward bifurcation. Also, conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained. In case without reinfection, the model has four equilibria. In this case, the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations (FDEs). Numerical simulations confirm the validity of the theoretical results.