# American Institute of Mathematical Sciences

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. Cahn, J. 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. Guo, M. 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. Lewis, B. 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. Li, W. 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. Li, W.-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. Liu, S. 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. Liu, Y. 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. Sun, L. Zhang, W.-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. Wang, W. 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. Wu, Y. Yang, Q. Y. Zhao, Y. 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. Xu, Y. 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. Yang, Y. Li, W.-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. Yang, W.-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. Zhou, J. Xu, J. 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. Zhu, W.-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.

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##### 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. Cahn, J. 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. Guo, M. 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. Lewis, B. 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. Li, W. 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. Li, W.-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. Liu, S. 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. Liu, Y. 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. Sun, L. Zhang, W.-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. Wang, W. 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. Wu, Y. Yang, Q. Y. Zhao, Y. 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. Xu, Y. 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. Yang, Y. Li, W.-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. Yang, W.-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. Zhou, J. Xu, J. 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. Zhu, W.-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.
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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