Global Dynamics of a TB Model with Classes Age Structure and Environmental Transmission

  • Yan-Xia Dang
  • Juan Wang
  • Xue-Zhi LiEmail author
  • Mini Ghosh
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 302)


In this article, an age structured SVEIR epidemic model for TB is formulated and analyzed by considering three types of ages e.g., latent age, infection age and vaccination age. The presented model also incorporates the environmental transmission of TB. The dynamics of the disease is governed by a system of differential-integral equations. We assume that the vaccines for TB are partially effective. Some vaccinated individuals get permanent immunity to this disease, but some vaccinated individuals lose its protective power over a time and become susceptible again. The dynamical property of the model is established by using LaSalle’s invariance principle and constructing suitable Lyapunov functions. It has been shown that the dynamics of the model is governed by basic reproductive number \(\mathcal {R}(\xi )\). The disease-free equilibrium is globally stable if the basic reproductive number \(\mathcal {R}(\xi )<1\). The endemic equilibrium is locally and globally stable if \(\mathcal {R}(\xi )>1\). As the basic reproduction number plays an important role in determining the stability of the system, reducing this number below one through vaccination can lead to decrease in the transmission of this disease. Additionally, contaminated environment also contributes to the increase in \(\mathcal {R}(\xi )\), so we also need to keep the environment clean to decrease the basic reproduction number \(\mathcal {R}(\xi )\) below one. These types of control measures are easy to implement in our society and certainly this will improve the well-being of the society.


TB model Age-since-latency Age-since-infection Age-since-vaccination Environmental transmission Reproduction number Global stability Lyapunov function 

AMS subject classifications




The authors thank the handling editor and anonymous referees for their valuable comments and suggestions which led to an improvement of our original manuscript.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Yan-Xia Dang
    • 1
  • Juan Wang
    • 2
  • Xue-Zhi Li
    • 3
    Email author
  • Mini Ghosh
    • 4
  1. 1.Department of Public EducationZhumadian Vocational and Technical CollegeZhumadianChina
  2. 2.Department of MathematicsXinyang Normal UniversityXinyangChina
  3. 3.College of Mathematics and Information Sciences, Henan Normal UniversityXinxiangChina
  4. 4.School of Advanced Sciences, Vellore Institute of TechnologyChennaiIndia

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