一类带治疗项的非局部扩散SIR传染病模型的行波解

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 72-102.

一类带治疗项的非局部扩散SIR传染病模型的行波解

邓栋¹(),李燕^2,*()

¹ 重庆工商大学数学与统计学院重庆 400000
² 西安电子科技大学数学与统计学院西安 710126

收稿日期:2018-09-13 出版日期:2020-02-26 发布日期:2020-04-08
通讯作者: 李燕 E-mail:dd0328a@163.com;yanli@xidian.edu.cn
作者简介:邓栋, E-mail:dd0328a@163.com
基金资助:
陕西省科技厅资助(2017JQ1024)

Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Treatment

Dong Deng¹(),Yan Li^2,*()

¹ College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400000
² School of Mathematics and Statistics, Xidian University, Xi'an 710126

Received:2018-09-13 Online:2020-02-26 Published:2020-04-08
Contact: Yan Li E-mail:dd0328a@163.com;yanli@xidian.edu.cn
Supported by:
陕西省科技厅资助(2017JQ1024)

摘要/Abstract

摘要：

该文主要考虑一类非局部扩散传染病模型的行波解的存在性与不存在性.首先，利用Schauder不动点定理和取极限的方法，得到了行波解的存在性.其次，利用双边拉普拉斯变换和Fubini定理，证明了行波解的不存在性.上述结果表明，最小波速是预测疾病是否传播且以多大速度传播的重要阈值.

关键词: 行波解, 非局部扩散, 最小波速, 双边拉普拉斯变换

Abstract:

This paper is concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal epidemic model with treatment. The existence of traveling wave solutions is established by Schauder's fixed point theorem as well as a limiting argument, while the nonexistence of traveling wave solutions is proved by two-sided Laplace transform and Fubini's theorem. From the results, we conclude that the minimal wave speed is an important threshold to predict how fast the disease invades.

Key words: Traveling waves, Nonlocal dispersal, Minimal wave speed, Two-sided Laplace transform

中图分类号:

邓栋,李燕. 一类带治疗项的非局部扩散SIR传染病模型的行波解[J]. 数学物理学报, 2020, 40(1): 72-102.

Dong Deng,Yan Li. Traveling Waves in a Nonlocal Dispersal SIR Epidemic Model with Treatment[J]. Acta mathematica scientia,Series A, 2020, 40(1): 72-102.

图/表 1

参考文献 30

1	Arino J , Brauer F , Driessche P , et al. A model for influenza with vaccination and antiviral treatment. J Theoret Biol, 2008, 253, 118- 130 doi: 10.1016/j.jtbi.2008.02.026
2	Alexander M E , Bowman C , Moghzdas S , et al. A vaccination model for transmission dynamics of influenza. SIAM J Appl Dyn Syst, 2004, 3 (4): 503- 524 doi: 10.1137/030600370
3	Chen X , Guo J S . Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math Ann, 2003, 326, 123- 146 doi: 10.1007/s00208-003-0414-0
4	Diekmann O , Heesterbeek J A . Mathematical Epidemiology of Infective Diseases:Model Building, Analysis and Interpretation. New York: Wiley, 2000
5	Ducrot A , Magal P . Travelling wave solution for an infection-age strutured model with diffusion. Proc Roy Soc Edinburgy Sec A, 2009, 139, 459- 482 doi: 10.1017/S0308210507000455
6	Ducrot A , Magal P , Ruan S . Travelling wave solutions in Mltigroup Age-Structured Epidemic Models. Arch Rational Mech Anal, 2010, 195, 311- 331 doi: 10.1007/s00205-008-0203-8
7	Ducrot A , Magal P . Travelling wave solution for an infection-age strutured epidemic model with external supplies. Nonliearity, 2011, 24, 2891- 2911 doi: 10.1088/0951-7715/24/10/012
8	Fu S C , Guo J S . Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math Ann, 2003, 326, 123- 146 doi: 10.1007/s00208-003-0414-0
9	Ferguson N M , Cummings D , Fraser C , et al. Strategies for mitigating an influenza pandemic. Nature, 2006, 442, 448- 452 doi: 10.1038/nature04795
10	Hosono Y , IIyas B . Traveling waves of a simple diffusive epidemic model. Math Model Meth Appl Sci, 1995, 5, 935- 966 doi: 10.1142/S0218202595000504
11	Lin J , Andreasen V , Casagrandi R , Levin S A . Traveling waves in a model of influenza a drift. J Theoret Biol, 2003, 222, 437- 445 doi: 10.1016/S0022-5193(03)00056-0
12	Lipsitch M , Cohen T , Muary M , Levin B R . Antiviral resistantce and the control of pandemic influenza. PLoS Med, 2007, 4, 0111- 0120 doi: 10.1371/journal.pmed.0040111
13	Li Y , Li W T , Yang F Y . Traveling waves for a nonlocal dispersal SIR model with delay and external supplies. Appl Math Compute, 2014, 247, 723- 740
14	Merler S , Ajelli M . The role of population heterogeneity and human mobility in the spread of pandemic influenza. Proc R Soc B, 2010, 277, 557- 565 doi: 10.1098/rspb.2009.1605
15	Murray J D . Mathematical Biology. Berlin: Springer, 2002
16	Nuno M , Feng Z , Martcheva M , Castillo-Chavez C . Dynamics of two-strain influenza with isolation and partial crossimmunity. SIAM J Appl Math, 2005, 65, 964- 982 doi: 10.1137/S003613990343882X
17	Ruan S G. Spatial-temporal Dynamics in Nonlocal Epidemiological Models//Takeuchi Y, et al. Mathematics for Life Science and Medicine. New York: Springer, 2007: 97-122
18	Ruan S, Wu J. Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts//Cantrell S, et al. Spatial Ecology. New York: Chapman and Hall/CRC, 2009: 293-316
19	Thieme H R . Asymptotic estimates of the the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J Renie Angew Math, 1979, 306, 94- 121
20	Thieme H R , Zhao X Q . Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J Differential Equations, 2003, 195, 430- 470 doi: 10.1016/S0022-0396(03)00175-X
21	Wu C C . Existence of traveling waves with the critical speed for a discrete diffusive epidemic model. J Differential Equations, 2017, 262, 272- 282 doi: 10.1016/j.jde.2016.09.022
22	Wang J B , Li W T , Yang F Y . Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission. Commun Nonlinear Sci Numer Simulat, 2015, 27, 136- 152 doi: 10.1016/j.cnsns.2015.03.005
23	Wang X S , Wang H , Wu J . Traveling waves of diffusive predator-prey systems:disease outbreak propagation. Discrete Contin Dyn Syst B, 2002, 32, 3303- 3324
24	Wang Z C , Wu J , Liu R . Traveling waves of the spread avian influenza. Proc Amer Math Soc, 2012, 140, 3931- 3946 doi: 10.1090/S0002-9939-2012-11246-8
25	Xu D , Zhao X Q . Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete Contin Dyn Syst (Ser B), 2005, 5 (4): 1043- 1056 doi: 10.3934/dcdsb.2005.5.1043
26	Yang F Y , Li Y , Li W T , Wang Z C . Travelling waves in a nonlocal dispersal Kermack-Mckendrick epidemic model. Discrete Contin Dyn Syst (Ser B), 2013, 18, 1969- 1993 doi: 10.3934/dcdsb.2013.18.1969
27	Yang F Y , Li W T , Wang Z C . Traveling waves in a nonlocal dispersal SIR epidemic model. Nonlinear Anal, 2015, 23, 129- 147 doi: 10.1016/j.nonrwa.2014.12.001
28	Yang F Y. Li W T . Traveling waves in a nonlocal dispersal SIR epidemic with critical wave speed. J Math Anal Appl, 2018, 458, 1131- 1146 doi: 10.1016/j.jmaa.2017.10.016
29	Zhang G B , Li W T , Wang Z C . Spreading speed and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity. J Differential Equations, 2012, 252, 5096- 5124 doi: 10.1016/j.jde.2012.01.014
30	Zhang T R , Wang W D . Existence of traveling wave solutions for influenza model with treatment. J Math Anal Appl, 2014, 419, 469- 495 doi: 10.1016/j.jmaa.2014.04.068

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