Impact of delayed diagnosis and treatment on uninformed actively infected populations in tuberculosis: Stability and Hopf-bifurcation

Abstract Many people are unaware of the symptoms of tuberculosis (TB) or how the disease is spread. In some areas, availability of healthcare is limited and people may not have easy access to diagnostic tests or treatment for TB within a timely manner. These factors can lead to delays in seeking medical attention and receiving a diagnosis. In this paper, a TB model incorporating a delay in the detection and treatment of the uninformed actively infected population is presented and analyzed. The delay under consideration here is the total delay, which takes into account both health system delay and patient delay. Local stability analysis of the model is performed. Additionally, the critical value for the delay that determines the local stability of the disease-free equilibrium (DFE) is derived. The presence of a Hopf-bifurcation, along with its direction as supercritical or subcritical, is discussed both analytically and numerically. According to the analysis, when the delay exceeds a critical threshold, a Hopf-bifurcation occurs, resulting in instability of the unique endemic equilibrium point (EEP). A suitable Lyapunov function in the presence of delay is constructed in order to examine the global stability of the unique EEP. Further, the occurrence of stable closed periodic orbits, also known as limit cycles, is observed. In addition to this, the presence of Hopf-bifurcation for the non-delayed scenario is also confirmed. These findings are validated numerically under various parameter settings. The analysis highlights that both the presence and absence of delay contribute to the diverse and complex nature of the model.