Change in stability direction induced by temporal interventions: a case study of a tuberculosis transmission model with relapse and reinfection

This article presents a mathematical model of tuberculosis (TB) that incorporates non-linear incidence rate, relapse, and reinfection to capture the complexity of TB transmission dynamics. The non-linear incidence rate is introduced to capture the significant impact of population ignorance on the dangers of TB, which can lead to its rapid spread. In this study, the existence and stability of equilibrium points are analyzed both analytically and numerically. Our findings indicate that a basic reproduction number less than one is not sufficient to ensure TB elimination within a population. The model exhibits complex dynamics, including forward and backward bifurcation with hysteresis, as well as the potential for multiple stable equilibria (bistability) due to the effects of nonlinear incidence rates and reinfection. Bistability is a common phenomenon in Tuberculosis transmission models, characterized by unique features such as relapse and reinfection processes. Bistability enables both TB-free and TB-endemic equilibria to coexist, even when a stable TB-free equilibrium exists. The occurrence of three endemic equilibria adds complexity to the model, illustrating the challenges in TB control. When bistability occurs, we analyzed the potential shifts in stability trajectories from the endemic equilibrium to the disease-free equilibrium through specific interventions. Our global sensitivity analysis of the infected population emphasizes that primary infection and recovery rates are crucial parameters for reducing TB transmission. These insights highlight the importance of controlling primary infection through the use of preventive measures and optimizing recovery strategies to support the efforts taken toward TB eradication. This analysis offers a nuanced perspective on the challenges of achieving TB eradication, particularly in settings with high relapse and reinfection risks, and underscores the need for the implementation of comprehensive intervention strategies in public health programs. A numerical simulation using an adjustable infection rate step function was conducted to explore the optimal combination of intervention intensity, timing, and duration required for effective TB elimination. We illustrate how optimal timing and intervention intensity can shift the solution trajectory from a TB-endemic to a TB-free equilibrium when bistability occurs.