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Analysis of an age-structured multi-group heroin epidemic model

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  • Wang, Jinliang
  • Wang, Jing
  • Kuniya, Toshikazu

Abstract

This paper is concerned with the mathematical analysis of an age-structured multi-group heroin epidemic model, which can be used to describe the spread of heroin habituation and addiction in heterogeneous environment. Under general assumptions on the different level of susceptibility and the relapse to frequent heroin use, we establish sharp criteria for heroin spreading and vanishing. We rigorously investigate the well-posedness of the model, the existence of equilibria, the asymptotic smoothness of solution orbits, and the global stability of equilibria. Specifically, we rigorously show that the drug-free equilibrium is globally asymptotically stable if a threshold value ℜ0 is less than one, and the unique drug-endemic equilibrium is globally attractive if ℜ0 is greater than one. In the proofs of global stability of equilibria, we construct suitable Lyapunov functions by using a graph-theoretic method.

Suggested Citation

  • Wang, Jinliang & Wang, Jing & Kuniya, Toshikazu, 2019. "Analysis of an age-structured multi-group heroin epidemic model," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 78-100.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:78-100
    DOI: 10.1016/j.amc.2018.11.012
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    References listed on IDEAS

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    1. Fang, Bin & Li, Xue-Zhi & Martcheva, Maia & Cai, Li-Ming, 2015. "Global asymptotic properties of a heroin epidemic model with treat-age," Applied Mathematics and Computation, Elsevier, vol. 263(C), pages 315-331.
    2. Vargas-De-León, Cruz, 2011. "On the global stability of SIS, SIR and SIRS epidemic models with standard incidence," Chaos, Solitons & Fractals, Elsevier, vol. 44(12), pages 1106-1110.
    3. Isaac Mwangi Wangari & Lewi Stone, 2017. "Analysis of a Heroin Epidemic Model with Saturated Treatment Function," Journal of Applied Mathematics, Hindawi, vol. 2017, pages 1-21, August.
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    Cited by:

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    2. Haoxiang Tang & Mingtao Li & Xiangyu Yan & Zuhong Lu & Zhongwei Jia, 2021. "Modeling the Dynamics of Drug Spreading in China," IJERPH, MDPI, vol. 18(1), pages 1-25, January.
    3. Hu, Dandan & Huang, Gang, 2022. "Dynamical analysis on a size-structured population model of Daphnia with delayed birth process," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    4. Wei Wang & Sifen Lu & Haoxiang Tang & Biao Wang & Caiping Sun & Pai Zheng & Yi Bai & Zuhong Lu & Yulin Kang, 2022. "A Scoping Review of Drug Epidemic Models," IJERPH, MDPI, vol. 19(4), pages 1-18, February.
    5. Chen, Yi & Wang, Lianwen & Zhang, Jinhui, 2024. "Global asymptotic stability of an age-structured tuberculosis model: An analytical method to determine kernel coefficients in Lyapunov functional," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    6. Luo, Yantao & Zhang, Long & Teng, Zhidong & Zheng, Tingting, 2021. "Analysis of a general multi-group reaction–diffusion epidemic model with nonlinear incidence and temporary acquired immunity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 428-455.
    7. Yin, Qian & Wang, Zhishuang & Xia, Chengyi & Dehmer, Matthias & Emmert-Streib, Frank & Jin, Zhen, 2020. "A novel epidemic model considering demographics and intercity commuting on complex dynamical networks," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    8. Hu Zhang & Anwar Zeb & Aying Wan & Zizhen Zhang, 2022. "Bifurcation Analysis of a Synthetic Drug Transmission Model with Two Time Delays," Mathematics, MDPI, vol. 10(9), pages 1-21, May.

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