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A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 295)

Abstract

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.

Keywords

  • Tuberculosis
  • Reproductive number
  • Delay differential equation
  • Global stability
  • Lyapunov functional

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Correspondence to Jay Michael R. Macalalag .

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Macalalag, J.M.R., De Lara-Tuprio, E.P., Teng, T.R.Y. (2019). A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays. In: Mohd, M., Abdul Rahman, N., Abd Hamid, N., Mohd Yatim, Y. (eds) Dynamical Systems, Bifurcation Analysis and Applications. DySBA 2018. Springer Proceedings in Mathematics & Statistics, vol 295. Springer, Singapore. https://doi.org/10.1007/978-981-32-9832-3_6

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