Taiwanese Journal of Mathematics

Traveling Wave Solutions of a Diffusive SEIR Epidemic Model with Nonlinear Incidence Rate

Lin Zhao, Liang Zhang, and Haifeng Huo

Full-text: Open access


This paper is concerned with the existence and nonexistence of traveling wave solutions of a diffusive SEIR epidemic model with nonlinear incidence rate, which are determined by the basic reproduction number $R_0$ and the minimal wave speed $c^*$. Namely, the system admits a nontrivial traveling wave solution if $R_0 \gt 1$ and $c \geq c^*$ and then the non-existence of traveling wave solutions of the system is established if $R_0 \gt 1$ and $0 \lt c \lt c^*$. Especially, using numerical simulation, we give the basic framework of traveling wave solutions of the system.

Article information

Taiwanese J. Math., Volume 23, Number 4 (2019), 951-980.

Received: 5 May 2018
Revised: 20 October 2018
Accepted: 24 October 2018
First available in Project Euclid: 18 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35C07: Traveling wave solutions 35B40: Asymptotic behavior of solutions 35K57: Reaction-diffusion equations 92D30: Epidemiology

SEIR epidemic model nonlinear incidence rate the basic reproduction number the minimal speed traveling wave solutions


Zhao, Lin; Zhang, Liang; Huo, Haifeng. Traveling Wave Solutions of a Diffusive SEIR Epidemic Model with Nonlinear Incidence Rate. Taiwanese J. Math. 23 (2019), no. 4, 951--980. doi:10.11650/tjm/181009. https://projecteuclid.org/euclid.twjm/1563436876

Export citation


  • S. A. Al-Sheikh, Modeling and analysis of an SEIR epidemic model with a limited resource for treatment, Global Journal of Science Frontier Research Mathematics and Decision Sciences 12 (2012), 55–66.
  • R. M. Anderson, The Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol. 53 (1991), no. 1-2, 3–32.
  • R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and control, Oxford University Press, Academic, 1991.
  • R. M. Anderson and W. Trewhella, Population dynamics of the badger (Meles meles) and the epidemiology of bovine tuberculosis (Mycobacterium bovis), Philos. Trans. Roy. Soc. London Ser. B 310 (1985), no. 1145, 327–381.
  • F. Brauer, Compartmental models in epidemiology, in: Mathematical Epidemiology, 19–79, Lecture Notes in Mathematics 1945, Math. Biosci. Subser., Springer, Berlin, 2008.
  • O. Diekmann, Run for your life: A note on the asymptotic speed of propagation of an epidemic, J. Differential Equations 33 (1979), no. 1, 58–73.
  • B. Dubey, A. Patra, P. K. Srivastava and U. S. Dubey, Modeling and analysis of an SEIR model with different types of nonlinear treatment rates, J. Biol. Systems 21 (2013), no. 3, 1350023, 25 pp.
  • A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity 24 (2011), no. 10, 2891–2911.
  • A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 311–331.
  • S.-C. Fu and J.-C. Tsai, Wave propagation in predator-prey systems, Nonlinearity 28 (2015), no. 12, 4389–4423.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
  • H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), no. 4, 599–653.
  • Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci. 5 (1995), no. 7, 935–966.
  • W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations 16 (2004), no. 3, 745–765.
  • 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), no. 5, 1192–1207.
  • A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol. 68 (2006), no. 3, 615–626.
  • ––––, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol. 69 (2007), no. 6, 1871–1886.
  • ––––, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol. 71 (2009), no. 1, 75–83.
  • A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol. 22 (2005), no. 2, 113–128.
  • 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.
  • 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), no. 2, 187–204.
  • M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics 61, Springer, New York, 2015.
  • J. D. Murray, Mathematical Biology, Biomathematics 19, Springer-Verlag, Berlin, 1989.
  • S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in: Mathematics for Life Science and Medicine, 97–122, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007.
  • S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations 188 (2003), no. 1, 135–163.
  • C. Sun, Y. Lin and S. Tang, Global stability for an special SEIR epidemic model with nonlinear incidence rates, Chaos Solitons Fractals 33 (2007), no. 1, 290–297.
  • B. Tian and R. Yuan, Traveling waves for a diffusive SEIR epidemic model with standard incidences, Sci. China Math. 60 (2017), no. 5, 813–832.
  • ––––, Traveling waves for a diffusive SEIR epidemic model with non-local reaction and with standard incidences, Nonliear Anal. Real World Appl. 37 (2017), 162–181.
  • ––––, Traveling waves for a diffusive SEIR epidemic model with non-local reaction, Appl. Math. Model. 50 (2017), 432–449.
  • P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.
  • Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2113, 237–261.
  • Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3931–3946.
  • Z. Wang and R. Xu, Travelling waves of a diffusive epidemic model with latency and relapse, Discrete Dyn. Nat. Soc. 2013 (2013), Art. ID 869603, 13 pp.
  • J. Wu and S. Ruan, Modeling spatial spread of communicable diseases involving animal hosts, in: Spatial Ecology, 293–316, Chapman & Hall/CRC, Boca Raton, FL, 2009.
  • H. Xiang, Y. Wang and H. Huo, Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput. 8 (2018), no. 5, 1535–1554.
  • Z. Xu, Traveling waves for a diffusive SEIR epidemic model, Commun. Pure Appl. Anal. 15 (2016), no. 3, 871–892.
  • ––––, Traveling waves in an SEIR epidemic model with the variable total population, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 10, 3723–3742.
  • Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Discrete Contin. Dyn. Syst. Ser. B 13 (2010), no. 1, 195–211.
  • L. Zhao and Z.-C. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math. 81 (2016), no. 5, 795–823.
  • L. Zhao, Z.-C. Wang and S. Ruan, Traveling wave solutions in a two-group epidemic model with latent period, Nonlinearity 30 (2017), no. 4, 1287–1325.