A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays

  • Jay Michael R. MacalalagEmail author
  • Elvira P. De Lara-Tuprio
  • Timothy Robin Y. Teng
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 295)


In this paper, a Susceptible-Exposed-Infectious-Treated (SEIT) epidemic model with two discrete time delays for the disease transmission of tuberculosis (TB) is proposed and analyzed. The first time delay \(\tau _1\) represents the time of progression of an individual from the latent TB infection to the active TB disease, and the other delay \(\tau _2\) corresponds to the treatment period. We begin our mathematical analysis of the model by establishing the existence, uniqueness, nonnegativity and boundedness of the solutions. We derive the basic reproductive number \(R_0\) for the model. Using LaSalle’s Invariance Principle, we determine the stability of the equilibrium points when the treatment success rate is equal to zero. We prove that if \(R_0<1\), then the disease-free equilibrium is globally asymptotically stable. If \(R_0>1\), then the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results.


Tuberculosis Reproductive number Delay differential equation Global stability Lyapunov functional 


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Jay Michael R. Macalalag
    • 1
    • 2
    Email author
  • Elvira P. De Lara-Tuprio
    • 2
  • Timothy Robin Y. Teng
    • 2
  1. 1.Department of MathematicsCaraga State UniversityButuan CityPhilippines
  2. 2.Department of MathematicsAteneo de Manila UniversityQuezon CityPhilippines

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