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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 ›› 2019, Vol. 36 ›› Issue (1): 85-98.doi: 10.3969/j.issn.1005-3085.2019.01.007

• • 上一篇    下一篇

一类非标准离散霍乱动力学模型

廖   书,   杨炜明   

  1. 重庆工商大学数学与统计学院，重庆  400067
  • 收稿日期:2017-04-01 接受日期:2017-09-27 出版日期:2019-02-15 发布日期:2019-04-15
  • 基金资助:
    重庆市自然科学基金(cstc2017jcyjAX0067; cstc2018jcyjAX0823)；重庆市教委科学技术研究项目(KJ1600610; KJ1706163)；经济社会应用统计重庆市重点实验室.

A Nonstandard Numerical Methods for a Mathematical Model for Cholera

LIAO Shu,   YANG Wei-ming   

  1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067
  • Received:2017-04-01 Accepted:2017-09-27 Online:2019-02-15 Published:2019-04-15
  • Supported by:
    The Natural Science Foundation of CQ (cstc2017jcyjAX0067; cstc2018jcyjAX0823); the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1600610; KJ1706163); Chongqing Key Laboratory of Social Economy and Applied Statistics.

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摘要： 本文旨在利用非标准有限差分方法离散并求解一个包含预防接种的霍乱传染病模型．该离散模型具有和对应的原连续模型一致的平衡点，正性和有界性等性质．其次本文证明当基本再生数小于 1 时，无病平衡点是局部渐近稳定和全局渐近稳定的；当基本再生数大于 1 时，通过构造适当的 Lyapunov 函数，地方病平衡点也是全局渐近稳定的．最后利用离散模型可以成功模拟 2008 年津巴布韦霍乱，并可数值证明离散模型的稳定性，且与步长和初始条件等无关，再与其他离散方法比较验证 NSFD 方法的优势所在．

关键词: 霍乱, 非标准有限差分, 稳定性, Lyapunov函数

Abstract: By applying a nonstandard finite difference scheme, we construct and solve a discretized cholera epidemic model. The scheme can ensure that equilibrium points, the positivity and boundedness of solutions to the discrete model is the same as the original mathematical model. We have proved that when the basic reproduction number is less than 1, the disease-free equilibrium is locally and globally asymptotically stable. When the basic reproduction number is greater than 1, we prove that the endemic equilibrium is globally asymptotically stable by constructing a suitable Lyapunov function. Finally, the NSFD scheme can be well suited to numerically solve the cholera outbreak in Zimbabwe.

Key words: Cholera, nonstandard finite difference, stability, Lyapunov function

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