Skip to main content
SpringerLink
Log in

Wave Propagation for a Discrete Diffusive Mosquito-Borne Epidemic Model

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper is concerned with the existence and nonexistence of traveling wave solutions for a discrete diffusive mosquito-borne epidemic model with general incidence rate and constant recruitment. It is observed that whether the traveling wave solutions exist or not depend on the so-called basic reproduction ratio \(R_0\) of the corresponding kinetic system and the critical wave speed \(c^*\). More precisely, when \( R_0 >1\) and \(c\ge c^*\), the system admits a nontrivial traveling wave solution by constructing an invariant cone in a bounded domain with initial functions being defined on, and employing the method of upper and lower solution, Schauder’s fixed point theorem and a limiting approach. Moreover, the asymptotic behavior of traveling wave solutions at positive infinity is obtained by constructing a suitable Lyapunov functional. When \(0<c<c^*\) or \( R_0 \le 1\), the system has no nontrivial traveling wave solution by using a contradictory approach and two-sided Laplace transforms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bhatt, S., Gething, P.W., Brady, O.J., et al.: The global distribution and burden of dengue. Nature 496, 504–507 (2013)

    Article  Google Scholar 

  2. Capasso, V., Serio, G.: A generalization of the Kermack–McKendrick deterministic epidemic model. Math. Biosci. 42, 43–61 (1978)

    Article  MathSciNet  Google Scholar 

  3. Chaves, L.S.M., Fry, J., Malik, A., et al.: Global consumption and international trade in deforestation-associated commodities could influence malaria risk. Nat. Commun. 11, 1258 (2020)

    Article  Google Scholar 

  4. Chen, Y.-Y., Guo, J.-S., Hamel, F.: Traveling waves for a lattice dynamical system arising in a diffusive endemic model. Nonlinearity 30, 2334–2359 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chow, S.-N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149, 248–291 (1998)

    Article  MathSciNet  Google Scholar 

  6. Deng, D., Zhang, D.P.: Traveling waves for a discrete diffusive SIR epidemic model with treatment. Nonlinear Anal. Real World Appl. 61, 103325 (2021)

    Article  MathSciNet  Google Scholar 

  7. Denu, D., Ngoma, S., Salako, R.B.: Existence of traveling wave solutions of a deterministic vector-host epidemic model with direct transmission. J. Math. Anal. Appl. 487, 123995 (2020)

    Article  MathSciNet  Google Scholar 

  8. Erneux, T., Nicolis, G.: Propagating waves in discrete bistable reaction diffusion systems. Physica D 67, 237–244 (1993)

    Article  MathSciNet  Google Scholar 

  9. Esteva, L., Vargas, C.: Analysis of a dengue disease transmission model. Math. Biosci. 150, 131–151 (1998)

    Article  Google Scholar 

  10. Fang, J., Lai, X., Wang, F.-B.: Spatial dynamics of a dengue transmission model in time-space periodic environment. J. Differ. Equ. 269, 149–175 (2020)

    Article  MathSciNet  Google Scholar 

  11. Fu, S.-C., Guo, J.-S., Wu, C.-C.: Traveling wave solutions for a discrete diffusive epidemic model. J. Nonlinear Convex Anal. 17, 1739–1751 (2016)

    MathSciNet  Google Scholar 

  12. Hu, C.-B., Li, B.: Spatial dynamics for lattice differential equations with a shifting habitat. J. Differ. Equ. 259, 1967–1989 (2015)

    Article  MathSciNet  Google Scholar 

  13. Kapral, R.: Discrete models for chemically reacting systems. J. Math. Chem. 6, 113–163 (1991)

    Article  MathSciNet  Google Scholar 

  14. Lewis, M., Renclawowicz, J., Van den Driessche, P.: Traveling waves and spread rates for a West Nile virus model. Bull. Math. Biol. 68, 3–23 (2006)

    Article  MathSciNet  Google Scholar 

  15. Lin, Z., Zhu, H.: Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381–1409 (2017)

    Article  MathSciNet  Google Scholar 

  16. Liu, T., Zhang, G.-B.: Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electron. Res. Arch. 29, 2599–2618 (2021)

    Article  MathSciNet  Google Scholar 

  17. Lou, Y., Liu, K., He, D., Gao, D., Ruan, S.: Modelling diapause in mosquito population growth. J. Math. Biol. 78, 2259–2288 (2019)

    Article  MathSciNet  Google Scholar 

  18. Pang, L.-Y., Wu, S.-L.: Propagation dynamics for lattice differential equations in a time-periodic shifting habitat. Z. Angew. Math. Phys. 72, 93 (2021)

    Article  MathSciNet  Google Scholar 

  19. Ran, X., Hu, L., Nie, L.-F., Teng, Z.: Effects of stochastic perturbation and vaccinated age on a vector-borne epidemic model with saturation incidence rate. Appl. Math. Comput. 394, 125798 (2021)

    MathSciNet  Google Scholar 

  20. San, X., Wang, Z.: Traveling waves for a two-group epidemic model with latent period in a patchy environment. J. Math. Anal. Appl. 475, 1502–1531 (2019)

    Article  MathSciNet  Google Scholar 

  21. Su, T., Zhang, G.-B.: Invasion traveling waves for a discrete diffusive ratio-dependent predator-prey model. Acta Math. Sci. Ser. B 40, 1459–1476 (2020)

    Article  MathSciNet  Google Scholar 

  22. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  23. Wang, C., Wang, J.: Analysis of a malaria epidemic model with age structure and spatial diffusion. Z. Angew. Math. Phys. 72, 74 (2021)

    Article  MathSciNet  Google Scholar 

  24. Wang, J., Wang, J.: Analysis of a reaction–diffusion cholera model with distinct dispersal rates in the human population. J. Dyn. Differ. Equ. 33, 549–575 (2021)

    Article  MathSciNet  Google Scholar 

  25. Wang, K., Zhao, H., Wang, H.: Traveling waves for a diffusive mosquito-borne epidemic model with general incidence. Z. Angew. Math. Phys. 73, 31 (2022)

    Article  MathSciNet  Google Scholar 

  26. Wang, W., Zhao, X.-Q.: A nonlocal and time-delayed reaction–diffusion model of dengue transmission. SIAM J. Appl. Math. 71, 147–168 (2011)

    Article  MathSciNet  Google Scholar 

  27. Wang, W., Zhao, X.-Q.: Basic reproduction numbers for reaction–diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11, 1652–1673 (2012)

    Article  MathSciNet  Google Scholar 

  28. Wu, C.C.: Existence of traveling waves with the critical speed for a discrete diffusive epidemic model. J. Differ. Equ. 262, 272–282 (2017)

    Article  MathSciNet  Google Scholar 

  29. Wu, R., Zhao, X.-Q.: A reaction-diffusion model of vector-borne disease with periodic delays. J. Nonlinear Sci. 29, 29–64 (2019)

    Article  MathSciNet  Google Scholar 

  30. Yang, Z.-X., Zhang, G.-B.: Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. Sci China Math. 61, 1789–1806 (2018)

    Article  MathSciNet  Google Scholar 

  31. Yang,X.-X., Zhang,G.-B., Hao,Y.-C.: Existence and stability of traveling wavefronts for a discrete diffusion system with nonlocal delay effects. Discrete Contin. Dyn. Syst. Ser. B (2023) (in press). https://doi.org/10.3934/dcdsb.2023160

  32. Zhang, Q., Wu, S.-L.: Wave propagation of a discrete SIR epidemic model with a saturated incidence rate. Int. J. Biomath. 12, 1950029 (2019)

    Article  MathSciNet  Google Scholar 

  33. Zhang, R., Liu, S.-Q.: Wave propagation for a discrete diffusive vaccination epidemic model with bilinear incidence. J. Appl. Anal. Comput. 13, 715–733 (2023)

    MathSciNet  Google Scholar 

  34. Zhang, R., Wang, J.-L., Liu, S.-Q.: Traveling wave solutions for a class of discrete diffusive SIR epidemic model. J. Nonlinear Sci. 31, 10 (2021)

    Article  MathSciNet  Google Scholar 

  35. Zhang, T.: Minimal wave speed for a class of non-cooperative reaction–diffusion systems of three equations. J. Differ. Equ. 262, 4724–4770 (2017)

    Article  MathSciNet  Google Scholar 

  36. Zhao, L., Wang, Z.-C., Ruan, S.: Traveling wave solutions in a two-group SIR epidemic model with constant recruitment. J. Math. Biol. 1, 1–45 (2018)

    MathSciNet  Google Scholar 

  37. Zhao, X.-Q.: Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. 29, 67–82 (2017)

    Article  MathSciNet  Google Scholar 

  38. Zhou, J., Song, L.-Y., Wei, J.-D.: Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay. J. Differ. Equ. 268, 4491–4524 (2020)

    Article  MathSciNet  Google Scholar 

  39. Zhou, J., Xu, J., Wei, J.-D., Xu, H.: Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate. Nonlinear Anal. RWA. 41, 204–231 (2018)

    Article  MathSciNet  Google Scholar 

  40. Zhou, J., Yang, Y., Hsu, C.-H.: Traveling waves of a discrete diffusive waterborne pathogen model with general incidence. Commun. Nonlinear Sci. Numer. Simul. 126, 107431 (2023)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for several valuable comments and suggestions which helped to improve this paper. The second author is supported by NSF of China (12261081) and NSF of Gansu Province (21JR7RA121). The third author is supported by NSF of Gansu Province (22JR5RA172), Northwest Normal University: Starting Fund for Doctoral Research (202103101204) and the Foundation for Young Teacher of Northwest Normal University (NWNU-LKQN2022-01).

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, 730070, China

    Jiao Dang, Guo-Bao Zhang & Ge Tian

Authors
  1. Jiao Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guo-Bao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ge Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

JD: Writing-original draft. G-BZ: Conceptualization, Supervision, Funding acquisition, Writing review and editing. GT: Made modifications and corrections. All authors reviewed the manuscript.

Corresponding author

Correspondence to Guo-Bao Zhang.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dang, J., Zhang, GB. & Tian, G. Wave Propagation for a Discrete Diffusive Mosquito-Borne Epidemic Model. Qual. Theory Dyn. Syst. 23, 104 (2024). https://doi.org/10.1007/s12346-024-00964-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-024-00964-7

Keywords

Mathematics Subject Classification

Search

Navigation