Exploring Hopf-bifurcations and endemic bubbles in a tuberculosis model with behavioral changes and treatment saturation.

To manage risks and minimize the transmission of contagious diseases, individuals may reduce their contact with each other and take other precautions as much as possible in their daily lives and workplaces. As a result, the transmission of the infection reduces due to the behavioral changes. These behavioral changes are incorporated into models by introducing saturation in disease incidence. In this article, we propose and analyze a tuberculosis model that incorporates saturated exogenous reinfection and treatment. The stability analysis of the model's steady states is rigorously examined. We observe that the disease-free equilibrium point and the endemic equilibrium point (EEP) are globally asymptotically stable if the basic reproduction number (R0) is less than 1 and greater than 1, respectively, only when exogenous reinfection is not present (p=0) and when treatment is available for all (ω=0). However, even when R0 is less than 1, tuberculosis may persist at a specific level in the presence of exogenous reinfection and treatment saturation, leading to a backward bifurcation in the system. The existence and direction of Hopf-bifurcations are also discussed. Furthermore, we numerically validate our analytical results using different parameter sets. In the numerical examples, we study Hopf-bifurcations for parameters such as β, p, α, and ω. In one example, we observe that increasing β leads to the loss of stability of the unique EEP through a forward Hopf-bifurcation. If β is further increased, the unique EEP restores its stability, and the bifurcation diagram exhibits an interesting structure known as an endemic bubble. The existence of an endemic bubble for the saturation constant ω is also observed.