## Taiwanese Journal of Mathematics

### Traveling Waves for a Spatial SIRI Epidemic Model

#### 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

https://projecteuclid.org/euclid.twjm/1545361214

Digital Object Identifier
doi:10.11650/tjm/181205

#### 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

#### 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.