具有复发和年龄结构的肺结核病传播模型的参数辨识性研究

数学物理学报 ›› 2025, Vol. 45 ›› Issue (1): 269-278.

具有复发和年龄结构的肺结核病传播模型的参数辨识性研究

武子艺^1,2, 杨俊元^1,2,*

¹山西大学复杂系统研究所太原 030006;
²山西省疾病防控的数学技术与大数据分析重点实验室太原 030006

收稿日期:2024-01-16 修回日期:2024-04-09 出版日期:2025-02-26 发布日期:2025-01-08
通讯作者: *杨俊元,E-mail:yjyang66@sxu.edu.cn
基金资助:
教育部人文社科一般项目 (22YJAZH129)、国家自然科学基金 (12271143, 61573016, 12001339)、山西省回国留学人员科研教研资助项目 (2023-024) 和山西省自然科学基金 (20210302123454)

Study on Parameter Identifiability of an Age-Structured Tuberculosis Model with Relapse

Wu Ziyi^1,2, Yang Junyuan^1,2

¹Complex Systems Research Center, Shanxi University, Taiyuan 030006;
²Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Taiyuan 030006

Received:2024-01-16 Revised:2024-04-09 Online:2025-02-26 Published:2025-01-08
Supported by:
Humanities and Social Foundation of Ministry of Education (22YJAZH129), the National Natural Science Foundation of China (12271143, 61573016, 12001339), the Shanxi Scholarship Council of China (2023-024) and the Shanxi Province Science Foundation (20210302123454)

摘要/Abstract

摘要： 模型的参数辨识性是判断模型预测准确与否的关键. 依赖可辨识性结果的模型预测更为科学和准确. 相较于常微分方程模型, 具有初边值条件的年龄结构传染病模型参数辨识问题存在较大挑战. 该文利用公共卫生科学数据中心报告数据探讨具有年龄结构和复发的肺结核病模型的参数辨识问题. 首先利用特征值法得到模型参数结构辨识可能性的先后顺序, 其次通过蒙特卡洛实验计算各参数的平均相对误差发现模型参数是实用可辨识的. 进一步, 通过计算 Fisher 信息矩阵及偏秩相关性分析讨论模型中参数的不确定性对肺结核病传播的影响.

关键词: 肺结核, 辨识性分析, 敏感性分析

Abstract: The identifiability of model parameters plays a crucial role in determining the precision of model predictions. Additionally, predictions based on identifiable outcomes exhibit a higher degree of scientific rigor and accuracy. Unlike ordinary differential systems, achieving parameter identifiability in age-structured models with initial-boundary conditions poses considerable challenges. This paper aims to investigate the structural and practical identifiability of an age-structured tuberculosis model with relapse. First, we employ the eigenvalue method to ascertain the order of unidentifiable parameters. In conjunction with data provided by the Public Health Science Data Center, we employ Monte Carlo simulation to explore the practical identifiability of the proposed model. By calculating the Average Relative Error (ARE) for each parameter and utilizing the Fisher information matrix, we determine that all parameters are identifiable. Furthermore, we assess how uncertainty in these parameters affects tuberculosis transmission by analyzing the Fisher information matrix and partial rank correlation coefficient.

Key words: tuberculosis, identifiable analysis, sensitivity analysis

中图分类号:

武子艺, 杨俊元. 具有复发和年龄结构的肺结核病传播模型的参数辨识性研究[J]. 数学物理学报, 2025, 45(1): 269-278.

Wu Ziyi, Yang Junyuan. Study on Parameter Identifiability of an Age-Structured Tuberculosis Model with Relapse[J]. Acta mathematica scientia,Series A, 2025, 45(1): 269-278.

参考文献

[1] 世界卫生组织.[2022-10-27].https://www.who.int/zh/news-room/fact-sheets/detail/tuberculosis.
WHO World Health Organization. [2022-10-27].https://www.who.int/zh/news-room/fact-sheets/detail/tuberculosis.
[2] 中国疾病预防控制中心. [2022-05-26].https://www.chinacdc.cn/jkzt/crb/zl/jhb/jszl-2221.
Chinese Center for Disease Control and Prevention. [2022-05-26].https://www.chinacdc.cn/jkzt/crb/zl/jhb/jszl-2221.
[3] Van den Driessche P, Zou X. Modeling relapse in infectious diseases. Mathematical Biosciences, 2007, 207(1): 89-103
[4] Wallis R S. Mathematical models of tuberculosis reactivation and relapse. Frontiers in Microbiology, 2016, 7: Article 669
[5] Liu L, Zhang J, Li Y, Liu X.On the global stability of an age-structured tuberculosis transmission model with relapse. Mathematical Methods in the Applied Sciences, 2022, 45(9): 5622-5630
[6] Miao H, Xia X, Perelson A S, Wu H.On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Review, 2011, 53(1): 3-39
[7] Cao H, Gao X, Yan D, Zhang S.The dynamics of an age-structured TB transmission model with relapse. Mathematical Methods in the Applied Sciences, 2020, 43(6): 3807-3826
[8] Tuncer N, Gulbudak H, Cannataro V L, Martcheva M.Structural and practical identifiability issues of immuno-epidemiological vector-host models with application to rift valley fever. Bulletin of Mathematical Biology, 2016, 78: 1796-1827
[9] Guillén-González F, Sevillano-Castellano E, Suárez A. Fitting parameters and therapies of ODE tumor models with senescence and immune system. Journal of Mathematical Biology, 2023, 87(5): Article 67
[10] 国家统计局. [2017. [2017].http://www.stats.gov.cn/tjst/ndsj/.
National Bureau of Statistics. [2017. [2017].http://www.stats.gov.cn/tjst/ndsj/.
[11] 全国结核病学抽样调查技术指导组. 2000 年全国结核病流行病学抽样调查报告. 中国防痨杂志, 2022, 4: 65-66
National Technical Steering Group of the Epidemiological Sampling Survey for Tuberculosis. Report on nationwide random survey for the epidemiology of tuberculosis in 2000. Chinese Journal of Tuberculosis Prevention, 2002, 4: 65-66
[12] 全球结核病报告. [2018. [2018]. http://www.who.int/tb/publications/global{-}report/en/.
Global Tuberculosis Report.[2018[2018]. http://www.who.int/tb/publications/global{-}report/en/.
[13] Tuncer N, Timsina A, Nuno M, et al.Parameter identifiability and optimal control of an SARS-CoV-2 model early in the pandemic. Journal of Biological Dynamics, 2022, 16(1): 412-438
[14] Renardy M, Kirschner D, Eisenberg M. Structural identifiability analysis of age-structured PDE epidemic models. Journal of Mathematical Biology, 2022, 84(1): Article 9
[15] Gulbudak H, Qu Z, Milner F, Tuncer N. Sensitivity analysis in an immuno-epidemiological vector-host model. Bulletin of Mathematical Biology, 2022, 84(2): Article 27
[16] 邓萌, 徐瑞. 一类具有CTL 免疫反应和免疫损害的 HIV 感染动力学模型的稳定性分析. 数学物理学报, 2022, 42A(5): 1592-1600
Deng M, Xu R.Stability analysis of an HIV infection dynamic model with CTL immune response and immune impairment. Acta Math Sci, 2022, 42A(5): 1592-1600
[17] 张洪铎. 个体化治疗耐多药肺结核失败的影响因素分析. 临床合理用药杂志, 2017, 15: 108-109
Zhang H Z.Analysis of the influencing factors of individualized treatment failure for multidrug-resistant pulmonary tuberculosis. Journal of Clinical Rational Drug Use, 2017, 15: 108-109
[18] 张雪元. 利福喷丁与利福平在肺结核治疗中的药效比较及安全性分析. 中国医药指南, 2021, 1: 36-37
Zhang X Y.Comparison of efficacy and safety analysis of rifampicin and rifampicin in the treatment of pulmonary tuberculosis. Guide of China Medicine, 2021, 1: 36-37