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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 43 ›› Issue (1): 77-89.doi: 10.3969/j.issn.1005-3085.2026.01.005cstr: 32411.14.cjem.CN61-1269/O1.2026.01.005

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虫媒传染病爆发初期的Caputo-Hadamard分数阶传播动力学模型

杨胤超1,  任翠萍1,  李建全2,  马润年1   

  1. 1. 西安欧亚学院，西安 710065

    2. 陕西科技大学数学与数据科学学院，西安 710016

  • 收稿日期:2023-03-09 接受日期:2023-10-16 出版日期:2026-02-15 发布日期:2026-04-15
  • 基金资助:
    国家自然科学基金 (11971281)；陕西省“十三五”课题 (SGH20Z021).

Caputo-Hadamard Dynamic Behavior of Insect-borne Infectious Diseases in the Initial Stage of Outbreak

YANG Yinchao1,  REN Cuiping1,  LI Jianquan2,  MA Runnian1   

  1. 1. Xi'an Eurasia University, Xi'an 710065

    2. School of Mathematics & Data Science, Shaanxi University of Science  Technology, Xi'an 710016

  • Received:2023-03-09 Accepted:2023-10-16 Online:2026-02-15 Published:2026-04-15
  • Supported by:
    The National Natural Science Foundation of China (11971281); the 13th Five-Year Plan Project of Shaanxi Province (SGH20Z021).

摘要：

研究虫媒传染病爆发初期的传播规律有助于从根本上抑制疾病的大范围传播。疾病在爆发初期传播速度相对较慢，为掌握该时期传染病的流行规律，在经典SEIR模型的基础上，提出Caputo-Hadamard分数阶微分方程，深入研究了模型解的性质，给出解的存在性、唯一性定理。求出模型的基本再生数和无病平衡点、地方病平衡点。理论分析和仿真结果表明，当基本再生数$R_0<1$时，分数阶系统在无病平衡点处局部渐近稳定；当$R_0>1$时，分数阶系统在地方病平衡点处渐近稳定。其次，由2018年法国留尼汪岛登革热疫情提供的数据做出参数估计，得到相关灵敏度分析的结果以及抑制疫情发展的策略，通过调整治愈率、因病死亡率和人群接触率等参数，实现对虫媒传染病扩散的有效控制。最后，求出模型的数值解，据此证明Caputo-Hadamard分数阶模型能够较好地反映真实情况，能够为虫媒传染病的防治提供理论参考。

关键词: Caputo-Hadamard分数阶导数, 虫媒传染病, 登革热, 灵敏度分析, 稳定性分析

Abstract:

The insect-borne infectious diseases spread relatively slowly at the initial stage of the outbreak. Caputo-Hadamard fractional differential model was proposed on the basis of the classical SEIR model. The existence and uniqueness theorems of the solution were given. The basic reproduction number, disease free equilibrium point and endemic equilibrium point of the model are obtained. Based on the data provided by the dengue fever epidemic in Reunion Island, France in 2018, the theoretical analysis and simulation results show that the fractional order system is locally asymptotically stable at the disease-free equilibrium when $R_0<1$; The fractional order system was asymptotically stable at the endemic equilibrium point when $R_0>1$. In the mean time the results of sensitivity analysis and strategies to control the development of the epidemic are obtained. Finally, the fractional order algorithm is used to obtain the solution of the model, which proves that the model presented in this paper can reflect the real situation.

Key words: Caputo-Hadamard fractional derivative, insect-borne disease, dengue fever, sensitivity analysis, stability analysis

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