Traveling waves for a diffusive SIR model with delay and nonlinear incidence

Volume 11, Issue 12, pp 1313--1330 http://dx.doi.org/10.22436/jnsa.011.12.03

Publication Date: September 13, 2018 Submission Date: December 12, 2017 Revision Date: August 04, 2018 Accteptance Date: August 23, 2018

Download PDF

Download XML

1770 Downloads
3717 Views

Authors

Yanmei Wang - School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China. - School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030006, China. Guirong Liu - School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China. Aimin Zhao - School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China.

Abstract

This paper is concerned with the existence and non-existence of traveling wave solutions for a diffusive SIR model with delay and nonlinear incidence. First, we construct a pair of upper and lower solutions and a bounded cone. Then we prove the existence of traveling wave by using Schauder's fixed point theorem and constructing a suitable Lyapunov functional. The nonexistence of traveling wave is obtained by two-sided Laplace transform. Moreover, numerical simulations support the theoretical results. Finally, we also obtain that the minimal wave speed is decreasing with respect to the latent period and increasing with respect to the diffusion rate of infected individuals.

Share and Cite

ISRP Style

Yanmei Wang, Guirong Liu, Aimin Zhao, Traveling waves for a diffusive SIR model with delay and nonlinear incidence, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 12, 1313--1330

AMA Style

Wang Yanmei, Liu Guirong, Zhao Aimin, Traveling waves for a diffusive SIR model with delay and nonlinear incidence. J. Nonlinear Sci. Appl. (2018); 11(12):1313--1330

Chicago/Turabian Style

Wang, Yanmei, Liu, Guirong, Zhao, Aimin. "Traveling waves for a diffusive SIR model with delay and nonlinear incidence." Journal of Nonlinear Sciences and Applications, 11, no. 12 (2018): 1313--1330

Keywords

SIR model
traveling wave
time delay
nonlinear incidence

MSC

35K57
35C07
92D30

References

[1] Z. G. Bai, S.-L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221–232.

[2] E. Beretta, T. Hara, W. Ma, Y. Takeuchi , Global asymptotic stability of an SIR epidemic model with distributed time delay , Nonlinear Anal., 47 (2001), 4107–4115.

[3] J. Fang, J. J. Wei, X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction diffusion system, J. Differential Equations, 245 (2008), 2749–2770.

[4] S.-C. Fu , Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20–37.

[5] Y. Hosono, B. Ilyas, Traveling waves for a simple diffusive epidemic model , Math. Models Methods Appl. Sci., 5 (1995), 935–966.

[6] G. Huang, Y. Takeuchi, W. Ma, D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207.

[7] W. O. Kermack, A. G. Mckendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London Ser. A, 115 (1927), 700–721.
- Google Scholar

[8] A. Korobeinikov , Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871–1886.

[9] A. Korobeinikov , Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615–626.

[10] A. Korobeinikov, P. K. Maini , Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113–128.
- View Article
- Google Scholar

[11] W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54 (2002), 581–591.

[12] S. G. Ruan, W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135–163.

[13] M. A. Safi, A. B. Gumel, The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay, Nonlinear Anal. Real World Appl., 12 (2011), 215–235.

[14] Y. Takeuchi, W. Ma, E. Beretta , Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal. Ser. A: Theory Methods, 42 (2000), 931–947.

[15] P. X. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270–296.

[16] C. F. Wu, P. X. Weng, Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 867–892.

[17] Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Anal., 111 (2014), 66–81.

[18] R. Xu, Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. Real World Appl., 10 (2009), 3175–3189.