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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

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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

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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

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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

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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

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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

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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

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