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.

参考文献

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
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
3	Chen X , Guo J S . Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math Ann, 2003, 326, 123- 146
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
6	Ducrot A , Magal P , Ruan S . Travelling wave solutions in Mltigroup Age-Structured Epidemic Models. Arch Rational Mech Anal, 2010, 195, 311- 331
7	Ducrot A , Magal P . Travelling wave solution for an infection-age strutured epidemic model with external supplies. Nonliearity, 2011, 24, 2891- 2911
8	Fu S C , Guo J S . Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. Math Ann, 2003, 326, 123- 146
9	Ferguson N M , Cummings D , Fraser C , et al. Strategies for mitigating an influenza pandemic. Nature, 2006, 442, 448- 452
10	Hosono Y , IIyas B . Traveling waves of a simple diffusive epidemic model. Math Model Meth Appl Sci, 1995, 5, 935- 966
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
12	Lipsitch M , Cohen T , Muary M , Levin B R . Antiviral resistantce and the control of pandemic influenza. PLoS Med, 2007, 4, 0111- 0120
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
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
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
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
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
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
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
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
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
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
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
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

[1]	武文斌, 任雪, 张冉. 具有时滞的离散扩散疫苗接种模型的行波解[J]. 数学物理学报, 2025, 45(3): 858-874.
[2]	吴鹏, 张帅, 方诚. 一类具有非局部扩散和空间异质性年龄-空间结构松材线虫病模型动力学分析[J]. 数学物理学报, 2025, 45(3): 946-959.
[3]	刘佳, 包雄雄. 非局部时滞扩散方程棱锥形波前解的渐近稳定性[J]. 数学物理学报, 2025, 45(1): 44-53.
[4]	杨咏丽, 杨赟瑞. 非局部扩散的时空时滞霍乱传染病系统的行波解[J]. 数学物理学报, 2025, 45(1): 110-135.
[5]	吴鹏, 方诚. 具有非局部扩散和空间异质性的年龄-空间结构HIV潜伏感染模型的动力学分析[J]. 数学物理学报, 2025, 45(1): 279-294.
[6]	张广新, 杨赟瑞, 宋雪. 具有非局部效应的时滞SEIR系统的周期行波解[J]. 数学物理学报, 2024, 44(1): 140-159.
[7]	吴文菊,范伯全. 广义地表准地转方程的非对称行波涡对解[J]. 数学物理学报, 2023, 43(4): 1037-1061.
[8]	邵春晖,孙吉江,马世旺. 带有超二次位势无限格点上的基态行波解[J]. 数学物理学报, 2022, 42(5): 1451-1461.
[9]	杨瑜. 一类非局部扩散的SIR模型的行波解[J]. 数学物理学报, 2022, 42(5): 1409-1415.
[10]	王凯,赵洪涌. 一类具有时滞的反应扩散登革热传染病模型的行波解[J]. 数学物理学报, 2022, 42(4): 1209-1226.
[11]	程红梅,袁荣. 移动环境下带有非局部扩散项和时滞的反应扩散方程的强迫行波解[J]. 数学物理学报, 2022, 42(2): 491-501.
[12]	吴鑫,袁荣,马兆海. 非局部扩散Holling-Tanner捕食者-食饵系统的临界与非临界行波解分析[J]. 数学物理学报, 2021, 41(6): 1705-1717.
[13]	张雪,孙峪怀. 广义的(3+1)维Kadomtsev-Petviashvili方程的动力分析及其行波解[J]. 数学物理学报, 2019, 39(3): 501-509.
[14]	邹霞, 吴事良. 一类具有非线性发生率与时滞的非局部扩散SIR模型的行波解[J]. 数学物理学报, 2018, 38(3): 496-513.
[15]	郭庭光, 徐志庭. 带有扩散项和接种的传染病模型的行波解[J]. 数学物理学报, 2017, 37(6): 1129-1147.