Modeling visceral leishmaniasis and tuberculosis co-infection dynamics

The co-infection of visceral leishmaniasis (VL) and tuberculosis (TB) patients pose a major public health challenge. In this study, we develop a mathematical model to study the transmission dynamics of VL and TB co-infection by first analyzing the VL and TB sub-models separately. The dynamics of these sub-models and the full co-infection model are determined based on the reproduction number. When the associated reproduction number ( R 1 ) for the TB-only model and ( R 2 ) for the VL-only are less than unity, the model exhibits backward bifurcation. If m a x { R 1 , R 2 } = R 1 , then the TB-VL co-infection model exhibits backward bifurcation for values of R 1 . Furthermore, if m a x { R 1 , R 2 } = R 2 , and by choosing the transmission probability, β L as the bifurcation parameter, then the phenomenon of backward bifurcation occurs for values of R 2 . Consequently, the full model, whose associated reproduction number is R 0 , also exhibits backward bifurcation when R 0 = 1 . The equilibrium points and their stability for the models are determined and analyzed based on the magnitude of the respective reproduction numbers. Finally, some numerical simulations are presented to show the reliability of our theoretical results.