Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed

Shiqiang Feng; Dapeng Gao; Shiqiang Feng; Dapeng Gao

doi:10.3934/mbe.2021460

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 9357-9380. doi: 10.3934/mbe.2021460

Previous Article Next Article

Research article Special Issues

Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed

Shiqiang Feng ^1,2,
Dapeng Gao ^{1,2
,
,}

1.
School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637009, China
2.
Internet of Things Perception and Big Data Analysis Key Laboratory of Nanchong, Nanchong, Sichuan 637009, China

Received: 11 August 2021 Accepted: 17 October 2021 Published: 27 October 2021

This paper is about the existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Because of the introduction of nonlocal dispersal and the generality of incidence function, it is difficult to investigate the existence of critical traveling waves. To this end, we construct an auxiliary system and show the existence of traveling waves for the auxiliary system. Employing the results for the auxiliary system, we obtain the existence of traveling waves for the delayed nonlocal dispersal SIR epidemic model with the critical wave speed under mild conditions.
- delayed SIR model,
- nonlocal dispersal,
- nonlinear incidence,
- minimal wave speed,
- traveling waves
Citation: Shiqiang Feng, Dapeng Gao. Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 9357-9380. doi: 10.3934/mbe.2021460

Related Papers:

Abstract

This paper is about the existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Because of the introduction of nonlocal dispersal and the generality of incidence function, it is difficult to investigate the existence of critical traveling waves. To this end, we construct an auxiliary system and show the existence of traveling waves for the auxiliary system. Employing the results for the auxiliary system, we obtain the existence of traveling waves for the delayed nonlocal dispersal SIR epidemic model with the critical wave speed under mild conditions.

References

[1]	J. H. Wu, X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equations, 13 (2001), 651–687. doi: 10.1023/A:1016690424892
[2]	J. B. Wang, W. T. Li, F. Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Nonlinear Sci. Numer. Simul., 27 (2015), 136–152. doi: 10.1016/j.cnsns.2015.03.005
[3]	G. B. Zhang, Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate nonlinearity, Nonlinear Anal. TMA., 74 (2011), 6518–6529. doi: 10.1016/j.na.2011.06.035
[4]	G. B. Zhang, W. T. Li, Z. C. Wang, Speading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equations, 252 (2012), 5096–5124. doi: 10.1016/j.jde.2012.01.014
[5]	R. Zhang, L. L. Liu, X. M. Feng, Z. Jin, Existence of traveling wave solutions for a diffusive tuberculosis model with fast and slow progression, Appl. Math. Lett., 112 (2021), 106848. doi: 10.1016/j.aml.2020.106848
[6]	W. Pei, Q. S. Yang, Z. T. Xu, Traveling waves of a delayed epidemic model with spatial diffusion, Electron. J. Qual. Theory Differ. Equations, 82 (2017), 1–19.
[7]	O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109–130. doi: 10.1007/BF02450783
[8]	W. T. Li, F. Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equ. Appl., 26 (2014), 243–273.
[9]	J. F. He, J. C. Tsai, Traveling waves in the Kermark-Mckendrick epidemic model with latent period, Z. Angew. Math. Phys., 70 (2019), 2722. doi:10.1007/s00033-018-1072-0. doi: 10.1007/s00033-018-1072-0
[10]	Y. Hosono, B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Model Meth. Appl. Sci., 5 (1995), 935–966. doi: 10.1142/S0218202595000504
[11]	Z. C. Wang, J. H. Wu, Traveling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. Roy. Soc. Lond. Ser. A, 466 (2010), 237–261.
[12]	Z. C. Wang, J. H. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683–692. doi: 10.1016/j.jmaa.2011.06.084
[13]	J. V. Noble, Geographic and temporal development of plagues, Nature, 250 (1974), 726–729. doi: 10.1038/250726a0
[14]	G. Saccomandi, The spatial diffusion of diseases, Math. Comput. Model., 25 (1997), 83–95.
[15]	Z. G. Bai, S. L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221–232.
[16]	F. Y. Yang, Y. Li, W. T. Li, Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969–1993.
[17]	H. M. Cheng, R. Yuan, Traveling waves of a nonlocal dispersal Kermack-McKendrick epidemic model with delayed transmission, J. Evol. Equations, 17 (2017), 979–1002. doi: 10.1007/s00028-016-0362-2
[18]	S. P. Zhang, Y. R. Yang, Y. H. Zhou, Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence, J. Math. Phys., 59 (2018), 011513. DOI:10.1063/1.5021761. doi: 10.1063/1.5021761
[19]	J. B. Zhou, J. Xu, J. D. Wei, H. M. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nolinear Anal. RWA, 41 (2018), 204–231. doi: 10.1016/j.nonrwa.2017.10.016
[20]	Z. G. Bai, S. L. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370–1381. doi: 10.1016/j.cnsns.2014.07.005
[21]	Y. L. Cheng, D. C. Lu, Wave propagation in a infectious disease model with non-local diffusion, Adv. Differ. Equ., 1 (2019). doi:10.1186/s13662-019-2057-9. doi: 10.1186/s13662-019-2057-9
[22]	F. Y. Yang, W. T. Li, Z. C. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. RWA, 23 (2015), 129–147. doi: 10.1016/j.nonrwa.2014.12.001
[23]	Y. Li, W. T. Li, F. Y. Yang, Traveling waves for a nonlocal dispesal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723–740.
[24]	F. Y. Yang, W. T. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 485 (2017), 1131–1146.
[25]	Y. Y. Chen, J. S. Guo, F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2016), doi:10.1088/1361-6544/aa6b0a. doi: 10.1088/1361-6544/aa6b0a
[26]	C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equations, 262 (2017), 272–282. doi: 10.1016/j.jde.2016.09.022
[27]	R. Zhang, J. L. Wang, S. Q. Liu, Traveling wave solutions for a class of discrete diffusive SIR epidemic model, J. Nonlinear Sci., 31 (2021), 10. doi:10.1007/s00332-020-09656-3. doi: 10.1007/s00332-020-09656-3
[28]	Y. Li, W. T. Li, G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal., 14 (2015), 1001–1022. doi: 10.3934/cpaa.2015.14.1001
[29]	Q. Zhang, G. S. Chen, Existence of critical traveling waves for nonlocal diepersal SIR models with delay and nonlinear incidence, Appl. Math. Mech., 40(2019), 713–727.
[30]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. doi: 10.1137/S0036144500371907
[31]	R. Temam, Infinite Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1998.
[32]	X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)