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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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