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

Shanjing Ren

doi:10.3934/mbe.2017069

Article Contents

2017, Volume 14, Issue 5&6: 1337-1360. Doi: 10.3934/mbe.2017069

This issue Previous Article An SEI infection model incorporating media impact Next Article A bacteriophage model based on CRISPR/Cas immune system in a chemostat

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

Shanjing Ren^1,2, ,

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
^* Corresponding authorr: Shanjing Ren

Received: June 03, 2016

Revised: December 29, 2016

Published: November 2017

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).

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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.

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Mathematics Subject Classification: Primary: 35L60, 92C37; Secondary: 34K20.

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Figure 1. Here is the Model of TB

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Figure 2. The time series of $S(t)$ and $I(t)$, and the age distributions of $e(t, a)$ and $r(t, c)$ when $\tau=12$

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Figure 3. 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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