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登革热传播模型的最优综合控制研究

李雅芝 刘利利

李雅芝,刘利利. 登革热传播模型的最优综合控制研究 [J]. 应用数学和力学,2022,43(4):445-452 doi: 10.21656/1000-0887.420258
引用本文: 李雅芝,刘利利. 登革热传播模型的最优综合控制研究 [J]. 应用数学和力学,2022,43(4):445-452 doi: 10.21656/1000-0887.420258
LI Yazhi, LIU Lili. Study of the Optimal Integrated Control of a Dengue Transmission Model[J]. Applied Mathematics and Mechanics, 2022, 43(4): 445-452. doi: 10.21656/1000-0887.420258
Citation: LI Yazhi, LIU Lili. Study of the Optimal Integrated Control of a Dengue Transmission Model[J]. Applied Mathematics and Mechanics, 2022, 43(4): 445-452. doi: 10.21656/1000-0887.420258

登革热传播模型的最优综合控制研究

doi: 10.21656/1000-0887.420258
基金项目: 国家自然科学基金(11901326;11601293);山西省面上青年基金(201901D211160);贵州省科学技术基金(黔科合基础-ZK[2021]一般002);贵州省教育厅2019年度科技拔尖人才项目(黔教合KY 字[2019]063)
详细信息
    作者简介:

    李雅芝(1990—),女,博士(E-mail:lyz900101@126.com)

    刘利利(1985—),女,博士(通讯作者. E-mail:liulili03@sxu.edu.cn)

  • 中图分类号: O193

Study of the Optimal Integrated Control of a Dengue Transmission Model

  • 摘要:

    建立了一个登革热在蚊子和人之间传播的模型,引入了Wolbachia、自我保护和杀虫剂三种控制措施,分别从常数控制和时变控制两个方面进行探讨。首先,分析了常数控制对模型基本再生数的影响,研究发现:Wolbachia有助于减小基本再生数,且基本再生数与自我保护和杀虫剂呈负相关。其次,以使得感染数最少且实施成本最低为目标,使用Pontryagin极值原理讨论最优控制。最后,通过数值模拟展示了最优控制的效果。

  • 图  1  常数控制对登革热感染人数的影响

    Figure  1.  Effects of constant control on the number of dengue infections

    图  2  控制策略对登革热感染人数的影响

    Figure  2.  Effects of the control strategy on the number of dengue infections

    图  3  最优综合控制与单一控制的比较:(a) 两种控制策略下$u_{1}$的变化图;(b) 两种控制策略下$u_{2}$的变化图

    Figure  3.  Comparison between the optimal integrated control and the single control: (a) the variation diagram of $u_{1}$ under 2 control strategies; (b) the variation diagram of $u_{2}$ under 2 control strategies

    4  不同权重对控制变量的影响:(a) 改变$B_{1}$$u_1$的影响,$A=1, B_{2}=1$;(b) 改变$B_{2}$$u_2$的影响,$A=1, B_{1}=1$;(c) A=1时,$u_{1}, u_{2}$的变化图,$B_{1}=1, B_{2}=1$;(d) $A=5$时,$u_{1}, u_{2}$的变化图,$B_{1}=1, B_{2}=1$

    4.  The influences of different weights on control variables: (a) the influence of $B_{1}$ on $u_{1}$, $A=1, B_{2}=1$; (b) the influence of $B_{2}$ on $u_{2}$, $A=1, B_{1}=1$; (c) the variation diagram of $u_{1}\; {\rm{and}} \;u_{2},$ $A=1, B_{1}=1, B_{2}=1$; (d) the variation diagram of $u_{1}\; {\rm{and}} \;u_{2},$ $A=5$, $B_{1}=1, B_{2}=1$

    表  1  模型(1)中各状态变量的含义

    Table  1.   The meanings of variables in model (1)

    state variable biological meaning
    $S_{{\rm{mi}}}$ mosquito population infected with W and susceptible to D
    $S_{{\rm{mu}}}$ mosquito population not infected with W and susceptible to D
    $E_{{\rm{mi}}}$ mosquito population infected with W and in the incubation period of D
    $E_{{\rm{mu}}}$ mosquito population not infected with W and in the incubation period of D
    $I_{{\rm{mi}}}$ mosquito population infected with W and D
    $I_{{\rm{mu}}}$ mosquito population not infected with W but infected with D
    $S_{{\rm{h}}}$ human population susceptible to D
    $I_{{\rm{h}}}$ human population infected with D
    下载: 导出CSV

    表  2  模型(1)中各参数的含义

    Table  2.   The meanings of parameters in model (1)

    parameter biological meanings
    $w$ recruitment rate of human beings
    $b_{{\rm{m}}}$ birth rate of mosquitoes
    $d_{{\rm{h}}}$ natural death rate of human beings
    $d_{{\rm{m}}}$ natural death rate of mosquitoes
    $\beta_{{\rm{h}}}$ D infection rate of human beings bitten by W-free mosquitoes
    $e_{{\rm{m}}}$ rate at which exposed mosquitoes become infectious
    $\overline{\beta}_{{\rm{h}}}=q\beta_{{\rm{h}}}$ D infection rate of human beings bitten by W-infected mosquitoes (q<1, which reflects the inhibition effect of W)
    $\beta_{{\rm{m}}}$ D infection rate of mosquitoes biting D-infected human beings
    $\gamma_{{\rm{h}}}$ recovery rate of infectious human beings
    下载: 导出CSV
  • [1] World Health Organization. Dengue and severe dengue: treatment[EB/OL]. [2021-09-15]. https://www.who.int/health-topics/dengue-and-severe-dengue#tab=tab_3.
    [2] World Health Organization. Dengue and severe dengue: symptoms[EB/OL]. [2021-09-15]. https://www.who.int/health-topics/dengue-and-severe-dengue#tab=tab_2.
    [3] AGUSTO F B, KHAN M A. Optimal control strategies for dengue transmission in Pakistan[J]. Mathematical Biosciences, 2018, 305: 102-121. doi: 10.1016/j.mbs.2018.09.007
    [4] PRASETYO T A, SARAGIH R, HANDAYANI D. Optimal control on the mathematical models of dengue epidemic by giving vaccination and repellent strategies[J]. Journal of Physics: Conference Series, 2020, 1490(1): 012034.
    [5] XUE L, REN X, MAGPANTAY F, et al. Optimal control of mitigation strategies for dengue virus transmission[J]. Bulletin of Mathematical Biology, 2021, 83(2): 8. doi: 10.1007/s11538-020-00839-3
    [6] FISTER K R, MCCARTHY M L, OPPENHEIMER S F, et al. Optimal control of insects through sterile insect release and habitat modification[J]. Mathematical Biosciences: an International Journal, 2013, 244(2): 201-212. doi: 10.1016/j.mbs.2013.05.008
    [7] MASUD M A, KIM B N, KIM Y. Optimal control problems of mosquito-borne disease subject to changes in feeding behavior of Aedes mosquitoes[J]. Biosystems, 2017, 156/157: 23-39. doi: 10.1016/j.biosystems.2017.03.005
    [8] DORIGATTI I, MCCORMACK C, NEDJATI-GILANI G, et al. Using Wolbachia for dengue control: insights from modelling[J]. Trends in Parasitology, 2018, 34(2): 102-113. doi: 10.1016/j.pt.2017.11.002
    [9] CHITNIS N, HYMAN J M, CUSHING J M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model[J]. Bulletin of Mathematical Biology, 2008, 70: 1272-1296. doi: 10.1007/s11538-008-9299-0
    [10] 王小娥, 蔺小林, 李建全. 具有Holling IV型功能反应捕食系统的状态反馈控制[J]. 应用数学和力学, 2021, 41(12): 1369-1380. (WANG Xiaoe, LIN Xiaolin, LI Jianquan. State feedback control of predator-prey systems with holling Ⅳ functional responses[J]. Applied Mathematics and Mechanics, 2021, 41(12): 1369-1380.(in Chinese)
    [11] 王昕炜, 彭海军, 钟万勰. 具有潜伏期时滞的时变SEIR模型的最优疫苗接种策略[J]. 应用数学和力学, 2019, 40(7): 701-712. (WANG Xinwei, PENG Haijun, ZHONG Wanxie. Optimal vaccination strategies for a time-varying SEIR epidemic model with latent delay[J]. Applied Mathematics and Mechanics, 2019, 40(7): 701-712.(in Chinese)
    [12] PONTRYAGIN L S, BOLTYANSKII V G, GAMKRELIDZE R V, et al. The Mathematical Theory of Optimal Processes[M]. New Jersey: Willey, 1962.
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出版历程
  • 收稿日期:  2021-08-30
  • 录用日期:  2021-09-15
  • 修回日期:  2021-09-15
  • 网络出版日期:  2022-03-16
  • 刊出日期:  2022-04-01

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