Persistent Topological Structures and Dynamics in Tuberculosis Delay Model

This study investigates the nonlinear and delayed dynamics of tuberculosis transmission using a mathematical model based on delay differential equations. The focus is placed on examining the influence of the reinfection rate and the delay associated with the waning of post-recovery immunity. A novel analytical framework is introduced, combining classical tools from the theory of dynamical systems with modern techniques from topological data analysis in order to rigorously identify and classify the system's long-term behaviors into three regimes: steady, periodic, and chaotic. A composite diagnostic measure is constructed by combining the number of one-dimensional topological loops with the Shannon entropy of the infectious population. This joint indicator is used to explore the global parameter space and to reveal zones where qualitative transitions and bifurcations occur. The study further employs multiple complementary analyses, including time-domain trajectories, frequency-domain patterns through Fourier transform, persistent topological features such as barcodes and Betti curves, as well as geometric reconstructions of the system's trajectories in three-dimensional phase space and in reduced dimensions using principal component analysis. Together, these tools uncover robust signatures that distinguish the different behavioral regimes and demonstrate how reinfection and delayed immunity loss shape the complexity of disease dynamics. The consistency of regime classification is further confirmed through unsupervised clustering techniques. The results emphasize the crucial role of topological methods in identifying hidden structures, attractor geometries, and nonlinear transitions that cannot be captured by classical linear stability analysis. This approach provides new insights for designing adaptive and predictive strategies for public health interventions, particularly in managing diseases characterized by delayed feedback and reinfection.