Mathematical Biosciences and Engineering (MBE)

Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse

Pages: 1337 - 1360, Volume 14, Issue 5/6, October/December 2017      doi:10.3934/mbe.2017069

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Shanjing Ren - School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China (email)

Abstract: In this paper, an $SEIR$ epidemic model for an imperfect treatment disease with age-dependent latency and relapse is proposed. The model is well-suited to model tuberculosis. The basic reproduction number $R_{0}$ is calculated. We obtain the global behavior of the model in terms of $R_{0}$. If $R_{0}<1$, the disease-free equilibrium is globally asymptotically stable, whereas if $R_{0}>1$, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present.

Keywords:  Age-structure, Liapunov function, tuberculosis, infection equilibrium, global stability.
Mathematics Subject Classification:  Primary: 35L60, 92C37; Secondary: 34K20.

Received: June 2016;      Accepted: December 2016;      Available Online: May 2017.