April  2020, 25(4): 1469-1495. doi: 10.3934/dcdsb.2019236

Traveling waves for a nonlocal dispersal vaccination model with general incidence

1. 

Department of Mathematics, Zhejiang International Studies University, Hangzhou 310023, China

2. 

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

3. 

Department of Mathematics, National Central University, Zhongli District, Taoyuan City 32001, Taiwan

* Corresponding author: Yu Yang

Received  November 2018 Revised  March 2019 Published  April 2020 Early access  November 2019

Fund Project: The third author was partially supported by the MOST (Grant No. 107-2115-M-008-009-MY3) and NCTS of Taiwan.

This paper is concerned with the existence and asymptotic behavior of traveling wave solutions for a nonlocal dispersal vaccination model with general incidence. We first apply the Schauder's fixed point theorem to prove the existence of traveling wave solutions when the wave speed is greater than a critical speed $ c^* $. Then we investigate the boundary asymptotic behaviour of traveling wave solutions at $ +\infty $ by using an appropriate Lyapunov function. Applying the method of two-sided Laplace transform, we further prove the non-existence of traveling wave solutions when the wave speed is smaller than $ c^* $. From our work, one can see that the diffusion rate and nonlocal dispersal distance of the infected individuals can increase the critical speed $ c^* $, while vaccination reduces the critical speed $ c^* $. In addition, two specific examples are provided to verify the validity of our theoretical results, which cover and improve some known results.

Citation: Jinling Zhou, Yu Yang, Cheng-Hsiung Hsu. Traveling waves for a nonlocal dispersal vaccination model with general incidence. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1469-1495. doi: 10.3934/dcdsb.2019236
References:
[1]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[2]

J. W. CahnJ. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Applied Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703.

[3]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.

[4]

H. M. Cheng and R. Yuan, Traveling wave solutions for a nonlocal dispersal KermackMcKendrick epidemic model with spatio-temporal delay (in Chinese), Sci. China Math., 45 (2015), 765-788.

[5]

H. M. Cheng and R. Yuan, Traveling waves of a nonlocal dispersal Kermack-McKendrick epidemic model with delayed transmission, J. Evol. Equ., 17 (2017), 979-1002.  doi: 10.1007/s00028-016-0362-2.

[6]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.

[7]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.

[8]

Z. L. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: Fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.

[9]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8.

[10]

H. B. GuoM. Y. Li and Z. S. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261-279.  doi: 10.1137/110827028.

[11]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.  doi: 10.1088/0951-7715/26/1/121.

[12]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.  doi: 10.1007/s00285-010-0368-2.

[13]

L. Ignat and J. D. Rossi, A nonlocal convolution-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013.

[14]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626.  doi: 10.1007/s11538-005-9037-9.

[15]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[16]

C. T. Lee and et al., Non-local concepts in models in biology, J. Theor. Biol., 210 (2001), 201-219.

[17]

M. A. LewisB. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.

[18]

W.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Intergral Equ. Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.

[19]

W.-T. LiW. B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2483-2512.  doi: 10.3934/dcds.2017107.

[20]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.

[21]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.

[22]

X. N. LiuY. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253 (2008), 1-11.  doi: 10.1016/j.jtbi.2007.10.014.

[23]

Y. P. Liu and J.-A. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.

[24]

Z. H. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, Int. J. Biomath., 10 (2017), 1750071, 23 pp. doi: 10.1142/S1793524517500711.

[25]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, 1822 (2003), 231–298. doi: 10.1007/978-3-540-45204-1_4.

[26]

M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61. Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[27]

J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[28]

Y.-J. SunL. ZhangW.-T. Li and Z.-C. Wang, Entire solutions in nonlocal monostable equations: Asymmetric case, Comm. Pure Appl. Anal., 18 (2019), 1049-1072.  doi: 10.3934/cpaa.2019051.

[29]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801.  doi: 10.1016/j.jde.2011.01.007.

[30]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994.

[31]

J.-B. WangW. T. Li and F.-Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.

[32]

W. Wang and W. B. Ma, Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model, J. Math. Anal. Appl., 457 (2018), 868-889.  doi: 10.1016/j.jmaa.2017.08.024.

[33]

C. F. WuY. YangQ. Y. ZhaoY. L. Tian and Z. T. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068.

[34]

Z. Q. Xu and D. M. Xiao, Minimal wave speed and uniqueness of traveling waves for a nonlocal diffusion population model with spatio-temporal delays, Differ. Integral Equ., 27 (2014), 1073-1106. 

[35]

Z. T. XuY. Q. Xu and Y. H. Huang, Stability and traveling waves of a vaccination model with nonlinear incidence, Comput. Math. Appl., 75 (2018), 561-581.  doi: 10.1016/j.camwa.2017.09.042.

[36]

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

[37]

F.-Y. YangW.-T. Li and 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.

[38]

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

[39]

J. B. ZhouJ. XuJ. D. Wei and H. M. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal. RWA, 41 (2018), 204-231.  doi: 10.1016/j.nonrwa.2017.10.016.

[40]

J. L. Zhou and Y. Yang, Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1719-1741.  doi: 10.3934/dcdsb.2017082.

[41]

C.-C. ZhuW.-T. Li and F.-Y. Yang, Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73 (2017), 1707-1723.  doi: 10.1016/j.camwa.2017.02.014.

show all references

References:
[1]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[2]

J. W. CahnJ. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Applied Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703.

[3]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.

[4]

H. M. Cheng and R. Yuan, Traveling wave solutions for a nonlocal dispersal KermackMcKendrick epidemic model with spatio-temporal delay (in Chinese), Sci. China Math., 45 (2015), 765-788.

[5]

H. M. Cheng and R. Yuan, Traveling waves of a nonlocal dispersal Kermack-McKendrick epidemic model with delayed transmission, J. Evol. Equ., 17 (2017), 979-1002.  doi: 10.1007/s00028-016-0362-2.

[6]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with external supplies, Nonlinearity, 24 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.

[7]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153–191.

[8]

Z. L. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: Fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.

[9]

R. A. Gardner, Existence and stability of traveling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8.

[10]

H. B. GuoM. Y. Li and Z. S. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261-279.  doi: 10.1137/110827028.

[11]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139.  doi: 10.1088/0951-7715/26/1/121.

[12]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.  doi: 10.1007/s00285-010-0368-2.

[13]

L. Ignat and J. D. Rossi, A nonlocal convolution-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013.

[14]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626.  doi: 10.1007/s11538-005-9037-9.

[15]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.

[16]

C. T. Lee and et al., Non-local concepts in models in biology, J. Theor. Biol., 210 (2001), 201-219.

[17]

M. A. LewisB. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.

[18]

W.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Intergral Equ. Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.

[19]

W.-T. LiW. B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2483-2512.  doi: 10.3934/dcds.2017107.

[20]

Y. LiW.-T. Li and F.-Y. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.

[21]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.

[22]

X. N. LiuY. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253 (2008), 1-11.  doi: 10.1016/j.jtbi.2007.10.014.

[23]

Y. P. Liu and J.-A. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.

[24]

Z. H. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, Int. J. Biomath., 10 (2017), 1750071, 23 pp. doi: 10.1142/S1793524517500711.

[25]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, 1822 (2003), 231–298. doi: 10.1007/978-3-540-45204-1_4.

[26]

M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61. Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.

[27]

J. D. Murray, Mathematical Biology, Biomathematics, 19. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.

[28]

Y.-J. SunL. ZhangW.-T. Li and Z.-C. Wang, Entire solutions in nonlocal monostable equations: Asymmetric case, Comm. Pure Appl. Anal., 18 (2019), 1049-1072.  doi: 10.3934/cpaa.2019051.

[29]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801.  doi: 10.1016/j.jde.2011.01.007.

[30]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994.

[31]

J.-B. WangW. T. Li and F.-Y. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.

[32]

W. Wang and W. B. Ma, Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model, J. Math. Anal. Appl., 457 (2018), 868-889.  doi: 10.1016/j.jmaa.2017.08.024.

[33]

C. F. WuY. YangQ. Y. ZhaoY. L. Tian and Z. T. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput., 313 (2017), 122-143.  doi: 10.1016/j.amc.2017.05.068.

[34]

Z. Q. Xu and D. M. Xiao, Minimal wave speed and uniqueness of traveling waves for a nonlocal diffusion population model with spatio-temporal delays, Differ. Integral Equ., 27 (2014), 1073-1106. 

[35]

Z. T. XuY. Q. Xu and Y. H. Huang, Stability and traveling waves of a vaccination model with nonlinear incidence, Comput. Math. Appl., 75 (2018), 561-581.  doi: 10.1016/j.camwa.2017.09.042.

[36]

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

[37]

F.-Y. YangW.-T. Li and 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.

[38]

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

[39]

J. B. ZhouJ. XuJ. D. Wei and H. M. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal. RWA, 41 (2018), 204-231.  doi: 10.1016/j.nonrwa.2017.10.016.

[40]

J. L. Zhou and Y. Yang, Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1719-1741.  doi: 10.3934/dcdsb.2017082.

[41]

C.-C. ZhuW.-T. Li and F.-Y. Yang, Traveling waves in a nonlocal dispersal SIRH model with relapse, Comput. Math. Appl., 73 (2017), 1707-1723.  doi: 10.1016/j.camwa.2017.02.014.

Figure 1.  Graphs of $ ( \overline{S}(\xi), \overline{V}(\xi),\overline{I}(\xi)) $ and $ ( \underline{S}(\xi), \underline{V}(\xi),\underline{I}(\xi)) $
[1]

Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1209-1225. doi: 10.3934/dcdsb.2021087

[2]

Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339

[3]

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

[4]

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

[5]

Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817

[6]

Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389

[7]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040

[8]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022045

[9]

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

[10]

Kai Li, Yan Li, Jing Liu, Nenggui Zhao. Two-sided vertical competition considering product quality in a manufacturing-remanufacturing system. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021187

[11]

Evgeny Korotyaev, Natalia Saburova. Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1691-1714. doi: 10.3934/cpaa.2022042

[12]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[13]

William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020

[14]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

[15]

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

[16]

Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467

[17]

Aijun Zhang. Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations. Conference Publications, 2013, 2013 (special) : 815-824. doi: 10.3934/proc.2013.2013.815

[18]

Ronald Mickens, Kale Oyedeji. Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's. Evolution Equations and Control Theory, 2019, 8 (1) : 139-147. doi: 10.3934/eect.2019008

[19]

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

[20]

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

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (357)
  • HTML views (160)
  • Cited by (1)

Other articles
by authors

[Back to Top]