The dynamics of an age‐structured TB transmission model with relapse

Cao, Hui; Gao, Xiaoyan; Yan, Dongxue; Zhang, Suxia

doi:10.1002/mma.6156

Cited by 9 publications

(9 citation statements)

References 29 publications

(65 reference statements)

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“…Remark Note that the bounded feasible region Ω defined in Cao et al 1 missed the boundedness of T . For completeness, here, we add it to the invariant set.…”

Section: Previous Resultsmentioning

confidence: 99%

“…Various interesting and important mathematical models were developed to describe the underlying transmission mechanisms of TB. In this paper, we revisit the following age‐structured TB transmission model proposed 1

({, \begin{matrix} left \\ array \frac{d S (t)}{d t} = Λ - d S (t) - β S (t) I (t), \\ array \frac{\partial E (t, a)}{\partial t} + \frac{\partial E (t, a)}{\partial a} = - (d + α (a) + γ_{1}) E (t, a), \\ array \frac{d I (t)}{d t} = {true\int}_{0}^{\infty} α (a) E (t, a) d a - (d + μ + γ_{2}) I (t) + k T (t), \\ array \frac{d T (t)}{d t} = γ_{1} {true\int}_{0}^{\infty} E (t, a) d a + γ_{2} I (t) - (d + k) T (t), \\ array E (t, 0) = β S (t) I (t), \end{matrix})

where S ( t ), E ( t , a ), I ( t ), and T ( t ) are the numbers of susceptible, latent individuals with infection age a , infective individuals, and recovered individuals at time t , respectively. Here, all individuals have the same natural death rate d .…”

Section: Introductionmentioning

confidence: 99%

“…Remark It should be noted that Cao et al 1 omitted the expression of I ∗ . For completeness, here, we recompute the existence of the unique ESS in detail.…”

Section: Introductionmentioning

confidence: 99%

“…The authors provided the through analysis of system (); however, the global stability of ESS was unresolved in Cao et al 1 if ℜ 0 > 1. The aim of this paper is to show that the ESS is always global asymptotic stability (GAS) whenever it exists.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the global stability of an age‐structured tuberculosis transmission model with relapse

Liu

Zhang

et al. 2022

Math Methods in App Sciences

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where the global asymptotic stability (GAS) of the endemic steady state (ESS) was unresolved when ℜ 0 > 1. We recompute the existence of ESS in detail, re-establish the GAS of the disease-free steady state (DFSS) in a simple and direct manner, and, furthermore, resolve the GAS of ESS, which was left as an open problem in the above paper. We adopt the method of Lyapunov functional with a key skill of selecting some novel appropriate kernel functions.

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“…Remark Note that the bounded feasible region Ω defined in Cao et al 1 missed the boundedness of T . For completeness, here, we add it to the invariant set.…”

Section: Previous Resultsmentioning

confidence: 99%

({, \begin{matrix} left \\ array \frac{d S (t)}{d t} = Λ - d S (t) - β S (t) I (t), \\ array \frac{\partial E (t, a)}{\partial t} + \frac{\partial E (t, a)}{\partial a} = - (d + α (a) + γ_{1}) E (t, a), \\ array \frac{d I (t)}{d t} = {true\int}_{0}^{\infty} α (a) E (t, a) d a - (d + μ + γ_{2}) I (t) + k T (t), \\ array \frac{d T (t)}{d t} = γ_{1} {true\int}_{0}^{\infty} E (t, a) d a + γ_{2} I (t) - (d + k) T (t), \\ array E (t, 0) = β S (t) I (t), \end{matrix})

Section: Introductionmentioning

confidence: 99%

“…Remark It should be noted that Cao et al 1 omitted the expression of I ∗ . For completeness, here, we recompute the existence of the unique ESS in detail.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the global stability of an age‐structured tuberculosis transmission model with relapse

Liu

Zhang

et al. 2022

Math Methods in App Sciences

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“…Thus, based on previous discussions of disease transmission (see Refs. 30–39) and above, we propose SIRE epidemic model

\{\begin{matrix} left \frac{d S}{d t} = normalΛ - β_{1} f (S) E - g (S) \int_{0}^{\infty} β_{2} (a) i (t, a) d a - μ S, \\ left \frac{\partial i false(t, a false)}{\partial t} + \frac{\partial i false(t, a false)}{\partial a} = - (μ + σ false(a false) + α false(a false)) i (t, a), \\ left \frac{d E}{d t} = \int_{0}^{\infty} θ (a) i (t, a) d a (1 - E) - γ E, \\ left \frac{\partial r false(t, b false)}{\partial t} + \frac{\partial r false(t, b false)}{\partial b} = - (μ + δ false(b false)) r (t, b), \\ left i (t, 0) = β_{1} f (S) E + g (S) \int_{0}^{\infty} β_{2} (a) i (t, a) d a + \int_{0}^{\infty} δ (b) r (t, b) d b, \\ le... \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of an age‐structured SIRE epidemic model with two routes of infection in environment

Chen

Teng

et al. 2021

Stud Appl Math

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In this study, an SIRE epidemic model including infection age, relapse age, and effect of protection for susceptible in environmental virus infectious diseases is investigated. We use the function ffalse(Sfalse)$f(S)$ to represent the protection of susceptible so as to reduce the probability that susceptible are exposed to the polluted environment and become infected, and the function gfalse(Sfalse)$g(S)$ represents the protection of susceptible so as to reduce the probability that susceptible are directly contact with infected persons and become infected. We obtain threshold R0$R_{0}$ and the basic properties of solutions. Further, we see that when R0<1$R_{0}<1$ the disease‐free equilibrium exists and is unique and globally asymptotically stable, and when R0>1$R_{0}> 1$, besides the disease‐free equilibrium, the model has a unique endemic equilibrium that is locally asymptotically stable. Furthermore, we only obtained the system is uniformly persistence without the global asymptotical stability of endemic equilibrium when R0>1$R_0>1$. The numerical examples show that there exists a stable manifold of endemic equilibrium when R0>1$R_0>1$.

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Hopf bifurcation of the recurrent infectious disease model with disease age and two delays

Jia,

Tan,

Cao

2024

Chaos, Solitons & Fractals

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The dynamics of an age‐structured TB transmission model with relapse

Cited by 9 publications

References 29 publications

On the global stability of an age‐structured tuberculosis transmission model with relapse

On the global stability of an age‐structured tuberculosis transmission model with relapse

Dynamical analysis of an age‐structured SIRE epidemic model with two routes of infection in environment

Hopf bifurcation of the recurrent infectious disease model with disease age and two delays

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