
Mathematical Biosciences and Engineering, 2019, 16(3): 16541682. doi: 10.3934/mbe.2019079.
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Traveling waves for SVIR epidemic model with nonlocal dispersal
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P.R. China
Received: , Accepted: , Published:
Special Issues: Differential Equations in Mathematical Biology
Keywords: traveling wave solutions; nonlocal dispersal; Schauder’s fixed point theorem; Lyapunov functional; epidemic model
Citation: Ran Zhang, Shengqiang Liu. Traveling waves for SVIR epidemic model with nonlocal dispersal. Mathematical Biosciences and Engineering, 2019, 16(3): 16541682. doi: 10.3934/mbe.2019079
References:
 1. W. Kermack and A. McKendrick, A contribution to mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700–721.
 2. X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253 (2008), 1–11.
 3. T. Kuniya, Global stability of a multigroup SVIR epidemic model, Nonlinear Anal.Real World Appl., 14 (2013), 1135–1143.
 4. J. Xu and Y. Zhou, Global stability of a multigroup model with vaccination age, distributed delay and random perturbation, Math. Biosci. Eng., 12 (2015), 1083–1106.
 5. X. Duan, S. Yuan and X. Li, Global stability of an SVIR model with age of vaccination, Appl. Math. Comput., 226 (2014), 528–540.
 6. J.Wang, R. Zhang and T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321–343.
 7. J.Wang, M. Guo and S. Liu, SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse, IMA J. Appl. Math., 82 (2017), 945–970.
 8. G. F. Webb, A reactiondiffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150–161.
 9. M. Kubo and M. Langlais, Periodic solutions for a population dynamics problem with agedependence and spatial structure, J. Math. Biol., 29 (1991), 393–378.
 10. Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Meth. Appl. Sci., 5 (1995), 935–966.
 11. R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reactiondiffusion model, Nonlinear Anal.Theory Methods Appl., 71 (2008), 239–247.
 12. Y. Lou and X. Q. Zhao, A reactiondiffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543–568.
 13. A. Ducrot and P. Magal, Travelling wave solutions for an infectionage structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891–2911.
 14. Z. Wang and R. Xu, Traveling waves of an epidemic model with vaccination, Int. J. Biomath., 6 (2013), 1350033, 19 pp.
 15. R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equ., 261 (2016), 3305–3343.
 16. B. Tian and R. Yuan, Traveling waves for a diffusive SEIR epidemic model with standard incidences, Sci. China Math., 60 (2017), 813–832.
 17. L. Zhao, Z. C. Wang and S. Ruan, Traveling wave solutions in a twogroup SIR epidemic model with constant recruitment, J. Math. Biol., 77 (2018), 1871–1915.
 18. G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions Part I: the optimal profile problem, Math. Ann., 310 (1998), 527–560.
 19. D. Xu and X. Zhao, Asymptotic speed of spread and traveling waves for a nonlocal epidemic model, Discret. Contin. Dyn. Syst. Ser. B, 5 (2005), 1043–1056.
 20. V. Hutson and M. Grinfeld, Nonlocal dispersal and bistability, Eur. J. Appl. Math., 17 (2006), 221–232.
 21. Z. C.Wang,W. T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dyn. Differ. Equ., 20 (2008), 573–607.
 22. F. Y. Yang, Y. Li, W. T. Li and Z. C. Wang, Traveling waves in a nonlocal dispersal Kermack McKendrick epidemic model, Discret. Contin. Dyn. Syst. Ser. B, 18 (2013), 1969–1993.
 23. 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.
 24. H. Cheng and R. Yuan, Traveling waves of a nonlocal dispersal KermackMckendrick epidemic model with delayed transmission, J. Evol. Equ., 17 (2017), 979–1002.
 25. 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.
 26. W. T. Li, W. B. Xu and L. Zhang, Traveling waves and entire solutions for an epidemic model with asymmetric dispersal, Discret. Contin. Dyn. Syst., 37 (2017), 2483–2512.
 27. T. Kuniya and J. Wang, Global dynamics of an SIR epidemic model with nonlocal diffusion, Nonlinear Anal.Real World Appl., 43 (2018), 262–282.
 28. G. Zhao and S. Ruan, Spatial and temporal dynamics of a nonlocal viral infection model, SIAM J. Appl. Math., 78 (2018), 1954–1980.
 29. S. L. Wu, G. S. Chen and C. H. Hsu, Entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity, J. Differ. Equ., 265 (2018), 5520–5574.
 30. W. Wang and W. Ma, Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model, J. Math. Anal. Appl., 457 (2018), 868–889.
 31. W.Wang andW. Ma, Global dynamics and travelling wave solutions for a class of noncooperative reactiondiffusion systems with nonlocal infections, Discret. Contin. Dyn. Syst. Ser. B, 23 (2018), 3213–3235.
 32. J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences Vol. 119, SpringerVerlag, New York, 1996.
 33. K. C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer Verlag, Berlin, 2005.
 34. 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.
 35. C. C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equ., 262 (2017), 272–282.
 36. Y. Chen, J. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334–2359.
 37. G. B. Zhang,W. T. Li and Z. C.Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differ. Equ., 252 (2012), 5096–5124.
 38. G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207.
 39. K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Philos. Soc., 81 (1977), 431–433.
 40. J. Wu and X. Zou, Traveling wave fronts of reactiondiffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651–687.
 41. D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, 1941.
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