Exploring the stability of equilibria and Hopf-bifurcation in a tuberculosis model with delayed treatment

Abstract Tuberculosis poses a worldwide threat, particularly to low and middle-income nations. The timely screening, identification, and management of TB become problematic in these regions due to resource constraints, inadequate education and awareness, stigma and discrimination, weak healthcare infrastructure, and concurrent health conditions. This delay may lead to postponed diagnosis and treatment. This work presents and examines a mathematical model for tuberculosis that incorporates exogenous reinfection, slow-fast progression, and delay in detection and treatment. More precisely, we examine a scenario where a certain proportion of people with active tuberculosis receive prompt diagnosis and treatment, while the rest experience a delay in detection and appropriate treatment. First, we establish the non-negativity and boundedness of the solutions, and discuss the criteria for the existence of different equilibria. Then, the local stability of the equilibria is analyzed. The threshold value of the time delay which decides the local stability of the disease-free equilibrium (DFE) is obtained. We observe that the model exhibits a Hopf-bifurcation when the time delay crosses a certain threshold value, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> . This reveals that the EEP is stable for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>τ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> . However, as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>τ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , the EEP loses its stability via Hopf-bifurcation, leading to sustained periodic oscillations. Further, by applying center manifold theory, we analytically determine the direction and stability of the Hopf-bifurcation and validate the results numerically. It is observed that the Hopf-bifurcation is supercritical and the bifurcating periodic oscillations are stable. Global stability of the endemic equilibrium point (EEP) under certain conditions is established by choosing a suitable Lyapunov function. Additionally, the system also exhibits a Hopf-bifurcation in the absence of delay and numerical simulations are performed to determine the critical parameter values at which the bifurcation occurs. The impact of treatment delay is significant, as it not only contributes to the emergence of complex dynamical behavior but also significantly influences the stability properties of the system. This delay can lead to critical transitions, such as changes in equilibrium stability and the onset of oscillatory dynamics, depending on the interaction of system parameters.