American Institute of Mathematical Sciences

Advanced
March  2020, 28(1): 1-13. doi: 10.3934/era.2020001

Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence

Yang Yang , Yun-Rui Yang , and  Xin-Jun Jiao

School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

* Corresponding author. Supported by the NSF of China (11761046)

Received  September 2019 Revised  December 2019 Published  March 2020

In this paper, the existence and non-existence of traveling wave solutions are established for a nonlocal dispersal SIR model equipped delay and generalized incidence. In addition, the existence and asymptotic behaviors of traveling waves under critical wave speed are also contained. Especially, the boundedness of traveling waves is obtained completely without imposing additional conditions on the nonlinear incidence.

Keywords: Nonlocal dispersal, traveling waves, delay.
Mathematics Subject Classification: 35K57, 35R20, 92D25.
Citation: Yang Yang, Yun-Rui Yang, Xin-Jun Jiao. Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence. Electronic Research Archive, 2020, 28 (1) : 1-13. doi: 10.3934/era.2020001
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combusion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 446 (1975), 5-49.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

Z. G. Bai and S. L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Applied Mathematics and Computation, 263 (2015), 221-232.  doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[4]

J. Coville and L. Dupaigne, Propagation speed of traveling fronts in nonlocal reaction-diffusion equation, Nonl. Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[5]

A. Ducrot, P. Magal and S. G. Ruan, Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ratinal Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.  Google Scholar

[6]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A Math., 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.  Google Scholar

[7]

P. C. Fife and J. B. Mcleod, The approach of solutions nonlinear diffusion equations to traveling front solutions, Arch. Ratinal Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[8]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Can. Appl. Math. Q., 10 (2002), 473-499.   Google Scholar

[9]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. Ser. B, 115 (1927), 700-721.   Google Scholar

[10]

W. T. Li, J. B. Wang and X. Q Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, Journal of Nonlinear Science, 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2.  Google Scholar

[11]

W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, Journal of Integral Equations and Applications, 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.  Google Scholar

[12]

J. B. Wang, W. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Communications in Nonlinear Science and Numerical Simulation, 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.  Google Scholar

[13]

X. S. Wang, H. Y. Wang and J. H. Wu, Travelling waves of diffusive predator-pery systems: Disease outbreak propagation, Discrete Cont. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[14]

Z. C. Wang and J. H. Wu, Travelling waves of a diffiusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[15] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ, 1941.   Google Scholar
[16]

S. L. Wu and S. G. Ruan, Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case, J. Differential Equations, 258 (2015), 2435-2470.  doi: 10.1016/j.jde.2014.12.013.  Google Scholar

[17]

F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Cont. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[18]

F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.  Google Scholar

[19]

S. P. Zhang, Y. R. Yang and Y. H. Zhang, Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence, J. Math. Phys., 59 (2018), 011513, 15pp. doi: 10.1063/1.5021761.  Google Scholar

[20]

G. B. Zhang, W. T. Li and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.  Google Scholar

[21]

X. Zou and S. L. Wu, Traveling waves in a nonlocal dispersal SIR epidemic model with delay and nonlinear incidence, Acta Mathematica Scientia, 38 (2018), 496-513.   Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combusion, and nerve pulse propagation, Partial Differential Equations and Related Topics, 446 (1975), 5-49.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

Z. G. Bai and S. L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Applied Mathematics and Computation, 263 (2015), 221-232.  doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[4]

J. Coville and L. Dupaigne, Propagation speed of traveling fronts in nonlocal reaction-diffusion equation, Nonl. Anal., 60 (2005), 797-819.  doi: 10.1016/j.na.2003.10.030.  Google Scholar

[5]

A. Ducrot, P. Magal and S. G. Ruan, Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ratinal Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.  Google Scholar

[6]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A Math., 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.  Google Scholar

[7]

P. C. Fife and J. B. Mcleod, The approach of solutions nonlinear diffusion equations to traveling front solutions, Arch. Ratinal Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar

[8]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Can. Appl. Math. Q., 10 (2002), 473-499.   Google Scholar

[9]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. Ser. B, 115 (1927), 700-721.   Google Scholar

[10]

W. T. Li, J. B. Wang and X. Q Zhao, Spatial dynamics of a nonlocal dispersal population model in a shifting environment, Journal of Nonlinear Science, 28 (2018), 1189-1219.  doi: 10.1007/s00332-018-9445-2.  Google Scholar

[11]

W. T. Li and F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, Journal of Integral Equations and Applications, 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.  Google Scholar

[12]

J. B. Wang, W. T. Li and F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Communications in Nonlinear Science and Numerical Simulation, 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.  Google Scholar

[13]

X. S. Wang, H. Y. Wang and J. H. Wu, Travelling waves of diffusive predator-pery systems: Disease outbreak propagation, Discrete Cont. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[14]

Z. C. Wang and J. H. Wu, Travelling waves of a diffiusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[15] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, NJ, 1941.   Google Scholar
[16]

S. L. Wu and S. G. Ruan, Entire solutions for nonlocal dispersal equations with spatio-temporal delay: Monostable case, J. Differential Equations, 258 (2015), 2435-2470.  doi: 10.1016/j.jde.2014.12.013.  Google Scholar

[17]

F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Cont. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[18]

F. Y. Yang and W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.  Google Scholar

[19]

S. P. Zhang, Y. R. Yang and Y. H. Zhang, Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence, J. Math. Phys., 59 (2018), 011513, 15pp. doi: 10.1063/1.5021761.  Google Scholar

[20]

G. B. Zhang, W. T. Li and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.  Google Scholar

[21]

X. Zou and S. L. Wu, Traveling waves in a nonlocal dispersal SIR epidemic model with delay and nonlinear incidence, Acta Mathematica Scientia, 38 (2018), 496-513.   Google Scholar

[1]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[2]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[3]

Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969
[4]

Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
[5]

Fang-Di Dong, Wan-Tong Li, Jia-Bing Wang. Asymptotic behavior of traveling waves for a three-component system with nonlocal dispersal and its application. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6291-6318. doi: 10.3934/dcds.2017272
[6]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure & Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019
[7]

Guo-Bao Zhang, Fang-Di Dong, Wan-Tong Li. Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1511-1541. doi: 10.3934/dcdsb.2018218
[8]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093
[9]

Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107
[10]

Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161
[11]

Kun Li, Jianhua Huang, Xiong Li. Asymptotic behavior and uniqueness of traveling wave fronts in a delayed nonlocal dispersal competitive system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 131-150. doi: 10.3934/cpaa.2017006
[12]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227
[13]

Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
[14]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659
[15]

Albert Erkip, Abba I. Ramadan. Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2125-2132. doi: 10.3934/cpaa.2017105
[16]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[17]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621
[18]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[19]

Alexander Pankov. Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2097-2113. doi: 10.3934/dcdss.2019135
[20]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028

2018 Impact Factor: 0.263

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences