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Global stability of two models with incomplete treatment for tuberculosis

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  • Yang, Yali
  • Li, Jianquan
  • Ma, Zhien
  • Liu, Luju

Abstract

Two tuberculosis (TB) models with incomplete treatment are investigated. It is assumed that the treated individuals may enter either the latent compartment due to the remainder of Mycobacterium tuberculosis or the infectious compartment due to the treatment failure. The first model is a simple one with treatment failure reflecting the current TB treatment fact in most countries with high tuberculosis incidence. The second model refines the simple one by dividing the latent compartment into slow and fast two kinds of progresses. This improvement can be used to describe the case that the latent TB individuals have been infected with some other chronic diseases (such as HIV and diabetes) which may weaken the immunity of infected individuals and shorten the latent period of TB. Both of the two models assume mass action incidence and exponential distributions of transfers between different compartments. The basic reproduction numbers of the two models are derived and their intuitive epidemiological interpretations are given. The global dynamics of two models are all proved by using Liapunov functions. At last, some strategies to control the spread of tuberculosis are discussed.

Suggested Citation

  • Yang, Yali & Li, Jianquan & Ma, Zhien & Liu, Luju, 2010. "Global stability of two models with incomplete treatment for tuberculosis," Chaos, Solitons & Fractals, Elsevier, vol. 43(1), pages 79-85.
    • Handle: RePEc:eee:chsofr:v:43:y:2010:i:1:p:79-85
    DOI: 10.1016/j.chaos.2010.09.002
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    References listed on IDEAS

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    1. Li, Guihua & Zhen, Jin, 2005. "Global stability of an SEI epidemic model with general contact rate," Chaos, Solitons & Fractals, Elsevier, vol. 23(3), pages 997-1004.
    2. Christie Y Jeon & Megan B Murray, 2008. "Diabetes Mellitus Increases the Risk of Active Tuberculosis: A Systematic Review of 13 Observational Studies," PLOS Medicine, Public Library of Science, vol. 5(7), pages 1-11, July.
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    Cited by:

    1. Li, Yong & Liu, Xianning & Yuan, Yiyi & Li, Jiang & Wang, Lianwen, 2022. "Global analysis of tuberculosis dynamical model and optimal control strategies based on case data in the United States," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    2. Kumar, Sunil & Chauhan, R.P. & Momani, Shaher & Hadid, Samir, 2021. "A study of fractional TB model due to mycobacterium tuberculosis bacteria," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    3. Kuddus, Md Abdul & Rahman, Azizur, 2022. "Modelling and analysis of human–mosquito malaria transmission dynamics in Bangladesh," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 123-138.
    4. Zafar, Zain Ul Abadin & Zaib, Sumera & Hussain, Muhammad Tanveer & Tunç, Cemil & Javeed, Shumaila, 2022. "Analysis and numerical simulation of tuberculosis model using different fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    5. Kuddus, Md Abdul & McBryde, Emma S. & Adekunle, Adeshina I. & White, Lisa J. & Meehan, Michael T., 2021. "Mathematical analysis of a two-strain disease model with amplification," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    6. Ullah, Ihsan & Ahmad, Saeed & Rahman, Mati ur & Arfan, Muhammad, 2021. "Investigation of fractional order tuberculosis (TB) model via Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    7. Liu, Yue, 2022. "Extinction, persistence and density function analysis of a stochastic two-strain disease model with drug resistance mutation," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    8. Ullah, Saif & Khan, Muhammad Altaf & Farooq, Muhammad & Gul, Taza, 2019. "Modeling and analysis of Tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 181-199.
    9. Md Abdul Kuddus & Michael T Meehan & Lisa J White & Emma S McBryde & Adeshina I Adekunle, 2020. "Modeling drug-resistant tuberculosis amplification rates and intervention strategies in Bangladesh," PLOS ONE, Public Library of Science, vol. 15(7), pages 1-26, July.
    10. Ullah, Saif & Altaf Khan, Muhammad & Farooq, Muhammad, 2018. "A fractional model for the dynamics of TB virus," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 63-71.

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