### Mathematical Biosciences and Engineering

2019, Issue 3: 1654-1682. doi: 10.3934/mbe.2019079
Research article Special Issues

# Traveling waves for SVIR epidemic model with nonlocal dispersal

• Received: 09 December 2018 Accepted: 30 January 2019 Published: 27 February 2019
• In this paper, we studied an SVIR epidemic model with nonlocal dispersal and delay, and we find that the existence of traveling wave is determined by the basic reproduction number $\Re_0$ and minimal wave speed $c^*$. By applying Schauder's fixed point theorem and Lyapunov functional, the existence and boundary asymptotic behaviour of traveling wave solutions is investigated for $\Re_0 \gt 1$ and $c \gt c^*$. The existence of traveling waves is obtained for $\Re_0 \gt 1$ and $c = c^*$ by employing a limiting argument. We also show that the nonexistence of traveling wave solutions by Laplace transform. Our results imply that (ⅰ) the diffusion and infection ability of infected individuals can accelerate the wave speed; (ⅱ) the latent period and successful rate of vaccination can slow down the wave speed.

Citation: Ran Zhang, Shengqiang Liu. Traveling waves for SVIR epidemic model with nonlocal dispersal[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1654-1682. doi: 10.3934/mbe.2019079

### Related Papers:

• In this paper, we studied an SVIR epidemic model with nonlocal dispersal and delay, and we find that the existence of traveling wave is determined by the basic reproduction number $\Re_0$ and minimal wave speed $c^*$. By applying Schauder's fixed point theorem and Lyapunov functional, the existence and boundary asymptotic behaviour of traveling wave solutions is investigated for $\Re_0 \gt 1$ and $c \gt c^*$. The existence of traveling waves is obtained for $\Re_0 \gt 1$ and $c = c^*$ by employing a limiting argument. We also show that the nonexistence of traveling wave solutions by Laplace transform. Our results imply that (ⅰ) the diffusion and infection ability of infected individuals can accelerate the wave speed; (ⅱ) the latent period and successful rate of vaccination can slow down the wave speed.

 [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 multi-group SVIR epidemic model, Nonlinear Anal.-Real World Appl., 14 (2013), 1135–1143. [4] J. Xu and Y. Zhou, Global stability of a multi-group 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 reaction-diffusion 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 reaction-diffusion model, Nonlinear Anal.-Theory Methods Appl., 71 (2008), 239–247. [12] Y. Lou and X. Q. Zhao, A reaction-diffusion 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 infection-age 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 two-group 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, Non-local 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 Kermack-Mckendrick 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 non-cooperative reaction-diffusion 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, Springer-Verlag, 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 reaction-diffusion 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.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142