# American Institute of Mathematical Sciences

October  2017, 14(5&6): 1337-1360. doi: 10.3934/mbe.2017069

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

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China 2 School of Mathematics and Computer Science, Guizhou Education University, Guiyang 550018, China

* Corresponding authorr: Shanjing Ren

Received  June 04, 2016 Revised  December 30, 2016 Published  May 2017

Fund Project: This research was supported by the National Natural Science Foundation of China(N0.11371161), the Special Fund of Provincial Governor for Excellent Scientific Technology and Educational Talents(Grand No.QKJB[2012]19)

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.

Citation: Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1337-1360. doi: 10.3934/mbe.2017069
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##### References:
Here is the Model of TB
The time series of $S(t)$ and $I(t)$, and the age distributions of $e(t, a)$ and $r(t, c)$ when $\tau=12$
he time series of $S(t)$ and $I(t)$, and the age distributions of $e(t, a)$ and $r(t, c)$ when $\tau=1$
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