Taiwanese Journal of Mathematics

Traveling Waves for a Spatial SIRI Epidemic Model

Zhiting Xu, Yixin Xu, and Yehui Huang

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

The aim of this paper is to study the traveling waves in a spatial SIRI epidemic model arising from herpes viral infection. We obtain the complete information about the existence and non-existence of traveling waves in the model. Namely, we prove that when the basic reproduction number $\mathcal{R}_0 \gt 1$, there exists a critical wave speed $c^* \gt 0$ such that for each $c \gt c^*$, the model admits positive traveling waves; and for $c \lt c^*$, the model has no non-negative and bounded traveling wave. We also give some numerical simulations to illustrate our analytic results.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 26 pages.

Dates
First available in Project Euclid: 21 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1545361214

Digital Object Identifier
doi:10.11650/tjm/181205

Subjects
Primary: 92D30: Epidemiology 35K57: Reaction-diffusion equations 35C07: Traveling wave solutions

Keywords
traveling wave solutions SIRI epidemic model basic reproduction number critical wave speed

Citation

Xu, Zhiting; Xu, Yixin; Huang, Yehui. Traveling Waves for a Spatial SIRI Epidemic Model. Taiwanese J. Math., advance publication, 21 December 2018. doi:10.11650/tjm/181205. https://projecteuclid.org/euclid.twjm/1545361214


Export citation

References

  • S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dynam. Differential Equations 26 (2014), no. 1, 143–164.
  • Z. Bai and S.-L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput. 263 (2015), 221–232.
  • V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci. 42 (1978), no. 1-2, 43–61.
  • J. Chin, Control of Communicable Diseases Manual, American Public Health Association, Washington, 1999.
  • A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commun. Pure Appl. Anal. 11 (2012), no. 1, 97–113.
  • 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.
  • J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations. 21 (2009), no. 4, 663–680.
  • S.-C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl. 435 (2016), no. 1, 20–37.
  • P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math. Comput. 219 (2013), no. 16, 8496–8507.
  • W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math. 38 (1985), no. 6, 733–753.
  • C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity 26 (2013), no. 1, 121–139.
  • W.-T. Li, G. Lin, C. Ma and F.-Y. Yang, Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 2, 467–484.
  • S. W. Martin, Livestock Disease Eradication: Evaluation of the Cooperative State-Federal Bovine Tuberculosis Eradication Program, National Academy Press, Washington, D.C., 1994.
  • H. N. Moreira and Y. Wang, Global stability in an $S \to I \to R \to I$ model, SIAM Rev. 39 (1997), no. 3, 496–502.
  • J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, third edition, Interdisciplinary Applied Mathematics 18, Springer-Verlag, New York, 2002.
  • W. Pei, Q. Yang and Z. Xu, Traveling waves of a delayed epidemic model with spatial diffusion, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), no. 82, 19 pp.
  • L. Perko, Differential Equations and Dynamical Systems, third edition, Texts in Applied Mathematics 7, Springer-Verlag, New York, 2001.
  • G. P. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, J. Math. Anal. Appl. 421 (2015), no. 2, 1651–1672.
  • D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev. 32 (1990), no. 1, 136–1139.
  • H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci. 21 (2011), no. 5, 747–783.
  • H. Wang and X.-S. Wang, Traveling wave phenomena in a Kermack-McKendrick SIR model, J. Dynam. Differential Equations 28 (2016), no. 1, 143–166.
  • X.-S. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst. 32 (2012), no. 9, 3303–3324.
  • 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.
  • ––––, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl. 385 (2012), no. 2, 683–692.
  • P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations 229 (2006), no. 1, 270–296.
  • C. Wu, Y. Yong, Q. Zhao, Y. Tian and Z. Xu, Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal, Appl. Math. Comput. 313 (2017), 122–143.
  • Z. Xu, Traveling waves in a Kermack-McKendrick epidemic model with diffusion and latent period, Nonlinear Anal. 111 (2014), 66–81.
  • ––––, Traveling waves for a diffusive SEIR epidemic model, Commun. Pure Appl. Anal. 15 (2016), no. 3, 871–892.
  • ––––, Wave propagation in an infectious disease model, J. Math. Anal. Appl. 449 (2017), no. 1, 853–871.
  • Z. Xu and D. Chen, An SIS epidemic model with diffusion, Appl. Math. J. Chinese Univ. Ser. B 32 (2017), no. 2, 127–146.
  • Z. Xu, Y. Xu and Y. Huang, Stability and traveling waves of a vaccination model with nonlinear incidence, Comput. Math. Appl. 75 (2018), no. 2, 561–581.
  • Q. Ye and Z. Li, Introduction to Reaction-diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.
  • Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M. S. Wang and X. A. Yang, Theory of Limit Cycles, Translations of Methematical Monographs 66, American Mathematical Society, Providence, RI, 1986.
  • T. Zhang, Minimal wave speed for a class of non-cooperative reaction-diffusion systems of three equations, J. Differential Equations 262 (2017), no. 9, 4724–4770.
  • T. Zhang and W. Wang, Existence of traveling wave solutions for influenza model with treatment, J. Math. Anal. Appl. 419 (2014), no. 1, 469–495.
  • Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Methematical Monographs 101, American Mathematical Society, Providence, RI, 1992.