The role of social media in a tuberculosis compartmental model: Exploring Hopf-bifurcation and nonlinear oscillations

The information shared by users on social media, including symptoms and the current status of their illness potentially aids in identifying and forecasting disease outbreaks. In this work, we introduce and analyze a deterministic compartmental model for tuberculosis (TB) which considers exogenous reinfection, relapse or treatment failure, and also incorporates the impact of media. The threshold quantity R0T and the equilibria for the model are obtained. Stability analysis of the disease-free equilibrium (DFE) is performed. The direction of supercritical (forward) and subcritical (backward) bifurcations is also obtained at R0T=1. The existence of multiple endemic equilibrium point (EEP) in the absence of exogenous reinfection (p=0) is examined. The existence of Hopf-bifurcation is discussed for the model analytically. Numerical simulations are conducted to support and broaden our analysis. Our numerical simulation reveals the presence of a subcritical bifurcation even in the absence of exogenous reinfection. The occurrence and disappearance of periodic oscillations through Hopf-bifurcation are observed under various parameter settings. More specifically, we observe Hopf-bifurcation by selecting the following parameters as bifurcation parameters: the disease transmission coefficient (β), the coefficient that decides how effectively TB information can influence the transmission rate (a), the rate at which the individuals leave the treated class (ρ), and the modification parameter for relative infectiousness of treated people (δ). Furthermore, we observe a stability switch phenomenon of the unique EEP with changes in a, giving rise to an interesting structure known as an endemic bubble. The presence of a limit cycle (closed periodic curve) around an unstable EEP is noticed. Thus, due to the significant nonlinearity of the model, it revels complex and diverse dynamics.