具周期性潜伏期的SEIR传染病模型的动力学

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 527-539.

具周期性潜伏期的SEIR传染病模型的动力学

王双明^1,^2,*(),樊馨蔓²,张明军²,梁俊荣²

¹ 兰州财经大学甘肃省电子商务技术与应用重点实验室兰州 730020
² 兰州财经大学信息工程学院兰州 730020

收稿日期:2018-11-14 出版日期:2020-04-26 发布日期:2020-05-21
通讯作者: 王双明 E-mail:wsm@lzufe.edu.cn
基金资助:
甘肃省科技计划(18JR3RA217);兰州财经大学科研项目(Lzufe2019B-006)

The Dynamics of an SEIR Epidemic Model with Time-Periodic Latent Period

Shuangming Wang^1,^2,*(),Xingman Fan²,Mingjun Zhang²,Junrong Liang²

¹ Key Laboratory of E-Business Technology and Application of Gansu Province, Lanzhou University of Finance and Economics, Lanzhou 730020
² School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou 730020

Received:2018-11-14 Online:2020-04-26 Published:2020-05-21
Contact: Shuangming Wang E-mail:wsm@lzufe.edu.cn
Supported by:
the Science and Technology Planed Projects of Gansu Province(18JR3RA217);the Research Project of Lanzhou University of Finance and Economics(Lzufe2019B-006)

摘要/Abstract

摘要：

研究了一类具有周期性潜伏期的常微分SEIR传染病模型.首先借助于染病年龄分布函数导出了模型.紧接着定义了模型的基本再生数$\mathcal R_0$并利用耗散动力系统的相关理论证明$\mathcal R_0$是决定疾病是否继续流行的阈值.最后，利用数值方法进一步验证了结论，并分析了忽略潜伏期的周期性对估计疾病传播能力的影响.

关键词: 周期性潜伏期, SEIR模型, 基本再生数, 持久性

Abstract:

A SEIR ordinary differential epidemic model with time-periodic latent period is studied. Firstly, the model is derived by means of the distribution function of infected ages. Next, the basic reproduction ratio $\mathcal R_0$ is introduced, and it is shown that $\mathcal R_0$ is a threshold index for determining whether the epidemic will go extinction or become endemic using the theory of dissipative dynamic systems. Finally, numerical methods are carried out to validate the analytical results and further to invetigate the effects on evaluating the propagation of disease owning to the neglect of the periodicity of the incubation period.

Key words: Periodic latent period, SEIR model, Basic Reproduction ratios, Persistence

中图分类号:

O175

王双明,樊馨蔓,张明军,梁俊荣. 具周期性潜伏期的SEIR传染病模型的动力学[J]. 数学物理学报, 2020, 40(2): 527-539.

Shuangming Wang,Xingman Fan,Mingjun Zhang,Junrong Liang. The Dynamics of an SEIR Epidemic Model with Time-Periodic Latent Period[J]. Acta mathematica scientia,Series A, 2020, 40(2): 527-539.

图/表 3

参考文献 18

1	马知恩, 周义仓, 王稳地, 等. 传染病动力学的数学建模与研究. 北京: 科学出版社, 2004: 42- 56
	Ma Z E , Zhou Y C , Wang W D , et al. Mathematics Modeling and Research of Infectious Disease Dynamics. Beijing: Science Press, 2004: 42- 56
2	Brauer F , Castillo-Chavez C . Mathematical Models in Population Biology and Epidemiology. New York: Springer, 2012: 1- 60
3	陈兰荪, 孟新柱, 焦建军. 生物动力学. 北京: 科学出版社, 2009: 150- 440
	Chen L S , Meng X Z , Jiao J J . Biodynamics. Beijing: Science Press, 2009: 150- 440
4	Zhao X Q . Dynamical Systems in Population Biology. New York: Springer, 2017: 1- 116
5	王宾国, 邵昶, 李海萍. 仓室传染病模型基本再生数的发展简介. 兰州大学学报(自然科学版), 2016, 52 (3): 380- 384
	Wang B G , Shao C , Li H P . Basic repoduction numbers for compartmetal epidemic models. Journal of Lanzhou University (Natural Sciences), 2016, 52 (3): 380- 384
6	Liang X , Zhang L , Zhao X Q . Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J Dynam Differential Equations, 2019, 31, 1247- 1278 doi: 10.1007/s10884-017-9601-7
7	Zhao X Q . Basic reproduction ratios for periodic compartmental models with time delay. J Dynam Differential Equations, 2017, 29, 67- 82 doi: 10.1007/s10884-015-9425-2
8	Bai Z G , Peng R , Zhao X Q . A reaction-diffusion malaria model with seasonality and incubation period. J Math Biol, 2018, 77, 201- 228 doi: 10.1007/s00285-017-1193-7
9	Lou Y J , Zhao X Q . A theoretical aproach to understanding population dynamics with seasonal developmental durations. J Nonlinear Sci, 2017, 27, 573- 603 doi: 10.1007/s00332-016-9344-3
10	刘胜强, 陈兰荪. 阶段结构种群生物模型与研究. 北京: 科学出版社, 2010: 8- 15
	Liu S Q , Chen L S . Population Biological Model with Stage Structure Population and Research. Beijing: Science Press, 2010: 8- 15
11	Zhang L , Wang Z C , Zhao X Q . Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period. J Differential Equations, 2015, 258 (9): 3011- 3036 doi: 10.1016/j.jde.2014.12.032
12	王双明, 张明军, 樊馨蔓. 一类具时滞的周期logistic传染病模型空间动力学研究. 应用数学和力学, 2018, 39 (2): 226- 238
	Wang S M , Zhang M J , Fan X M . Spatial dynamics of periodic reaction-diffusion epidemic models with delay and logistic growth. Applied Mathematics and Mechanics (Chinese Edition), 2018, 39 (2): 226- 238
13	Hay S I , Graham A , Rogers D J . Global Mapping of Infectious Diseases:Methods, Examples and Emerging Applications. London: Academic Press, 2006
14	Paaijmans K P , Read A F , Thomas M B . Understanding the link between malaria risk and climate. Proc Nat Acad Sci USA, 2009, 106 (33): 13844- 13849 doi: 10.1073/pnas.0903423106
15	王智诚, 王双明. 一类时间周期的时滞反应扩散模型的空间动力学研究. 兰州大学学报:自然科学版, 2013, (4): 535- 540
	Wang Z C , Wang S M . Spatial dynamics of a class of delayed nonlocal reaction-diffusion models with a time period. Journal of Lanzhou University (Natural Sciences), 2013, (4): 535- 540
16	Magal P , Zhao X Q . Global attractors and steady states for uniformly persistent dynamical systems. SIAM J Math Anal, 2005, 37 (1): 251- 275 doi: 10.1137/S0036141003439173
17	Lou Y J , Zhao X Q . Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete Contin Dyn Syst Ser B, 2009, 12, 169- 186
18	Posny D , Wang J . Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments. Appl Math Comput, 2014, 242, 473- 490

[1]	靖晓洁, 赵爱民, 刘桂荣. 考虑部分免疫和环境传播的麻疹传染病模型的全局稳定性[J]. 数学物理学报, 2019, 39(4): 909-917.
[2]	李华,刘三红,方奕乐,张兴安. 儿童手足口病的数学建模和计算机模拟[J]. 数学物理学报, 2018, 38(5): 1032-1040.
[3]	张丽萍, 赵瑜, 原三领. 一类具有非线性发生率的随机SIS传染病模型阈值动力学行为研究[J]. 数学物理学报, 2018, 38(1): 197-208.
[4]	韦爱举, 张新建, 王俊义, 李科赞. 一类埃博拉传染病模型的动力学分析[J]. 数学物理学报, 2017, 37(3): 577-592.
[5]	梁志清;陈兰荪. 离散Leslie捕食与被捕食系统周期解的稳定性[J]. 数学物理学报, 2006, 26(4): 634-640.
[6]	娄洁, 马知恩. 娄洁等：一类有被动免疫的流行病模型研究[J]. 数学物理学报, 2003, 23(3): 357-368.