考虑扩散的SVEITR肺结核传播模型的最优控制策略

数学物理学报 ›› 2025, Vol. 45 ›› Issue (3): 934-945.

考虑扩散的SVEITR肺结核传播模型的最优控制策略

李雅芝¹,刘利利²,田妍妮^3,^*()

¹黔南民族师范学院数学与统计学院贵州都匀 558000
²山西大学复杂系统研究所太原 030006
³三峡大学理学院湖北宜昌 443002

收稿日期:2024-08-21 修回日期:2025-02-13 出版日期:2025-06-26 发布日期:2025-06-20
通讯作者: *田妍妮，Email: tynswu@163.com
基金资助:
国家自然科学基金(12361102);国家自然科学基金(11901326);国家自然科学基金(12101513);山西省自然科学基金(202303021211003);贵州省科学技术协会青年科技人才托举工程

Optimal Control Strategy for the SVEITR Pulmonary Tuberculosis Transmission Model Considering Diffusion

LI Yazhi¹,Liu Lili²,Tian Yanni^3,^*()

¹SchoolofMathematicsandStatistics,QiannanNormalUniversityforNationalities, Guizhou Duyun 558000
²ComplexSystemResearchCenter,ShanxiUniversity, Taiyuan 030006
³CollegeofScience,ChinaThreeGorgesUniversity, Hubei Yichang 443002

Received:2024-08-21 Revised:2025-02-13 Online:2025-06-26 Published:2025-06-20
Supported by:
NSFC(12361102);NSFC(11901326);NSFC(12101513);Natural Science Foundation of Shanxi Province(202303021211003);Young Elite Scientist Sponsorship Program by GAST

摘要/Abstract

摘要：

该文建立了一个带有扩散因素的肺结核传播模型, 引入提高个人防护意识,及时的检测和治疗, 提高治疗水平以及降低复发率这四个控制变量, 从理论和数值上讨论了肺结核的最优控制问题.首先使用偏微分方程的最优控制理论得到了最优控制的存在性, 其次通过构造灵敏度系统和伴随系统, 计算得出最优控制的特征,最后通过数值模拟展示了最优控制变量随时间变化的情况, 并展示了最优控制效果, 结果发现相比无控制情况,在控制终止时刻,最优控制可以显著降低潜伏者、染病者和接受治疗者的数量; 相比单一控制情况, 最优控制可以降低染病者峰值, 延迟达峰时间,并减少染病者最终感染数量. 所得结论可为一线疾病防控提供一定理论参考.

关键词: 肺结核, 扩散模型, 最优控制, 数值模拟

Abstract:

A tuberculosis transmission model with diffusion factors was established, introducing four control variables: increasing personal protective awareness, timely detection and treatment, improving treatment level, and reducing recurrence rate. The optimal control problem was discussed theoretically and numerically. Firstly, the existence of optimal control was obtained using the optimal control theory of partial differential equations. Secondly, the characteristics of optimal control were calculated by constructing sensitivity systems and adjoint systems. Finally, the variation of optimal control variables over time was demonstrated through numerical simulations, and the optimal control effect was shown. The results showed that compared with the uncontrolled situation, optimal control can significantly reduce the number of latent, infected, and treated individuals at the termination of control; Compared to a single control scenario, optimal control can reduce the peak of infected individuals, delay peak time, and decrease the final number of infected individuals. The conclusion drawn can provide a theoretical reference for frontline disease prevention and control.

Key words: tuberculosis, diffusion model, optimal control, numerical simulation

中图分类号:

O193

李雅芝, 刘利利, 田妍妮. 考虑扩散的SVEITR肺结核传播模型的最优控制策略[J]. 数学物理学报, 2025, 45(3): 934-945.

LI Yazhi, Liu Lili, Tian Yanni. Optimal Control Strategy for the SVEITR Pulmonary Tuberculosis Transmission Model Considering Diffusion[J]. Acta mathematica scientia,Series A, 2025, 45(3): 934-945.

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参考文献 16

[1]	Global Tuberculosis report 2023. Geneva: World Health Organization; 2023. Licence:CC BY-NC-SA 3.O IGO
[2]	Mao K X, Zhen C X, Hong Yan L U, et al. Protective eff ect of vaccination of Bacille Calmette-Gnerin on children. Chin J Contemp Pediatr, 2003, 5(4): 325-327
[3]	Abubakar I, Pimpin L, et al. Systematic review and meta-analysis of the current evidence on the duration of protection by bacillus Calmette-Guerin vaccination against tuberculosis. Health Technol Assess, 2013, 17(37): 1-372 doi: 10.3310/hta17370 pmid: 24021245
[4]	徐瑞, 周凯娟, 白宁. 一类基于游离病毒感染和细胞-细胞传播的宿主体内 HIV-1 感染动力学模型. 数学物理学报, 2024, 44A(3): 771-782
	Xu R, Zhou K J, Bai N. A with-in host HIV-1 infection dynamics model based on virus-to-cell infection and cell-to-cell transmission. Acta Math Sci, 2024, 44A(3): 771-782
[5]	杨俊元, 张晨琳, 杨丽. 年龄结构流感模型综合控制策略研究. 数学物理学报, 2024, 44A(3): 804-814
	Yang J Y, Zhang C L, Yang L. Study on comprehensive control strategies of an age structured influenza model. Acta Math Sci, 44A (3): 804-814
[6]	Liu J L, Zhang T L. Global stability for a tuberculosis model. Math Comput Model, 2011, 54: 836-845
[7]	Li Y, Liu X N, Yuan Y Y, et al. Global analysis of tuberculosis dynamical model and optimal control strategies based on case data in the United States. Appl Math Comput, 2022, 422: 126983
[8]	Yang Y L, Tang S Y, Ren X H, et al. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete Contin Dyn Syst Ser B, 2017, 21(3): 1009-1022
[9]	Gao D P, Huang N J. A note on global stability for a tuberculosis model. Appl Math Lett, 2017, 73: 163-168
[10]	Liu S Y, Bi Y J, Liu Y W. Modeling and dynamic analysis of tuberculosis in mainland China from 1998 to 2017: the effect of DOTS strategy and further control. Theor Biol Med Model, 2020, 17(6): 1-10
[11]	Gao D P, Huang N J. Optimal control analysis of a tuberculosis model. Appl Math Model, 2018, 58: 47-64
[12]	Egonmwan A O, Okuonghae D. Mathematical analysis of a tuberculosis model with imperfect vaccine. Int J Biomath, 2019, 12(7): 1950073
[13]	Bhih A E, Laaroussi A E A, Ghazzali R, et al. An optimal chemoprophylaxis and treatment control for a spatiotemporal tuberculosis model. Commun Math Biol Neurosci, 2021, 2021: Article 40
[14]	Wang Z P, Xu R. Stability and traveling waves of an epidemic model with relapse and spatial diffusion. J Appl Anal Comput, 2014, 4(3): 307-322
[15]	Zhang R, Liu L L, Feng X M, et al. Existence of traveling wave solutions for a diffusive tuberculosis model with fast and slow progression. Appl Math Lett, 2021, 112: 106848
[16]	Lenhart S, Workman J T. Optimal Control Applied to Biological Models. New York: Chapman & Hall/CRC, 2007