https://doi.org/10.1140/epjp/s13360-026-07309-0
Regular Article
Modeling tuberculosis transmission with kernel functions: effects of vaccination and time delay
1
Department of Mathematics, S. A. Engineering College, 600077, Chennai, India
2
School of Mathematical Sciences, Chengdu University of Technology, 610059, Chengdu, Sichuan, China
3
Department of Mathematics, Vasavi College of Engineering (A), 500031, Hyderabad, Telangana, India
4
Department of Management, Polytechnic University of Marche, Piazza Martelli 8, I60121, Ancona, Italy
5
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, 632014, Vellore, Tamil Nadu, India
6
Department of Mathematics, Aditya University, Surampalem, Andhra Pradesh, India
a
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Received:
9
December
2025
Accepted:
11
January
2026
Published online:
25
January
2026
Abstract
This study investigates the dynamics of tuberculosis (TB) transmission in high-density populations with suboptimal hygiene conditions, focusing on India as a representative setting. We develop a transmission model incorporating both weak and strong kernel functions to represent distributed time delays in TB latency, along with vaccination effects. By analyzing stability around the TB-free equilibrium with respect to the time delay arising from the latent phase, we identify critical bifurcation thresholds for local stability. The model exhibits a Hopf bifurcation at the tuberculosis-existence equilibrium when the latency delay exceeds this critical value, indicating oscillatory disease dynamics. Numerical simulations validate our analytical findings and provide insights into the interplay between vaccination parameters, latency periods, and TB control. The results offer valuable guidance for optimizing vaccination strategies and timing interventions in TB-endemic regions.
Key words: Tuberculosis transmission model / Hopf bifurcation / Time delay / Vaccination / Kernel
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
