Persistent Topological Structures and Dynamics in Tuberculosis Delay Model

Persistent Topological Structures and Dynamics in Tuberculosis Delay Model | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Persistent Topological Structures and Dynamics in Tuberculosis Delay Model M. A. Elfouly, Reda Abouelenien This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7077315/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract 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. Mathematical and Theoretical Biology Infectious Diseases Applied Mathematics Delay Differential Equations Topological Data Analysis Persistent Homology Infectious Disease Modeling Epidemic Topology Vietoris-Rips Complex Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7077315","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":482544162,"identity":"dabd2782-21c9-43a8-a36d-1820313d08ad","order_by":0,"name":"M. A. 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