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

  • * Corresponding authorr: Shanjing Ren

    * Corresponding authorr: Shanjing Ren
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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  • 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.

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


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  • Figure 1.  Here is the Model of TB

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

    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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