SpringerLink
Log in

Traveling wave of a reaction–diffusion vector-borne disease model with nonlocal effects and distributed delay

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

This paper is devoted to investigate the existence and nonexistence of traveling wave solution for a diffusive vector-borne disease model with nonlocal reaction and distributed delays. We demonstrate that the basic reproduction number \({\mathcal {R}}_0\) of the corresponding ordinary differential equation system as a threshold determines whether the model admits traveling waves or not and there exists a critical wave speed \(c_m^*>0\) when \({\mathcal {R}}_0>1\). Specifically, (i) As \({\mathcal {R}}_0>1\) and the wave speed \(c>c_m^*\), the existence of traveling waves for the system is established with the aid of a perturbed system; (ii) As \({\mathcal {R}}_0>1\) and \(0<c<c_m^*\), the nonexistence of traveling waves is proved via the two-sided Laplace transform; (iii) As \({\mathcal {R}}_0\le 1\) and \(c>0\), the nonexistence is obtained by utilizing the comparison principle. The theoretical results are applied to dengue fever epidemics. We study the effects of geographical movement, nonlocal interaction, incubation period and \({\mathcal {R}}_0\) on the threshold speed \(c_m^*\) for dengue fever.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Canada)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Al-Omari, J.F.M., Gourley, S.A.: A nonlocal reaction–diffusion model for a single species with stage structure and distributed maturation delay. Eur. J. Appl. Math. 16, 37–51 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Becker, N., Petric, D., Zgomba, M., et al.: Mosquitoes and Their Control, 2nd edn. Springer, New York (2010)

    Google Scholar 

  3. Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Berestycki, H., Hamel, F.: Generalized transition waves and their properties. Commun. Pure Appl. Math. 65, 592–648 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  6. Brady, O.J., Gething, P.W., Bhatt, S., et al.: Refining the global spatial limits of dengue virus transmission by evidence-based consensus. PLoS Negl. Trop. Dis. 6, e1760e1760 (2012)

    Google Scholar 

  7. Cai, L., Li, X., Fang, B., et al.: Global properties of vector-host disease models with time delays. J. Math. Biol. 74, 1397–1423 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Carr, J., Chmaj, A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Chapwanya, M., Dumont, Y.: On crop vector-borne disease. Impact of virus lifespan and contact rate on the traveling-wave speed of infective fronts. Ecol Complex. 34, 119–133 (2018)

    Google Scholar 

  10. 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)

    Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured model with diffusion. Proc. R. Soc. Edinb. 139, 459–482 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Dunbar, S.R.: Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits. SIAM J. Appl. Math. 46, 1057–1078 (1986)

    MathSciNet  MATH  Google Scholar 

  14. Dunbar, S.R.: Travelling wave solutions of diffusive Lotka–Volterra equations. J. Math. Biol. 17, 11–32 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Dunbar, S.R.: Traveling wave solutions of diffusive Lotka–Volterra equations: a heteroclinic connection in \(R^4\). Trans. Am. Math. Soc. 286, 557–594 (1984)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Fang, J., Zhao, X.-Q.: Monotone wavefronts for partially degenerate reaction–diffusion systems. J. Dyn. Differ. Equ. 21, 663–680 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Halstead, S.B.: Dengue. Lancet 370, 1644–1652 (2007)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. Hsu, C.H., Yang, T.S.: Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models. Nonlinearity 26, 121–139 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Huang, W.Z.: A geometric approach in the study of traveling waves for some classes of non-monotone reaction–diffusion systems. J. Differ. Equ. 260, 2190–2224 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Kendall, D.G.: Discussion of Measles periodicity and community size. J. R. Stat. Soc. A. 120, 64–76 (1959)

    Google Scholar 

  23. Kermack, W.O., McKendrik, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A. 115, 700–721 (1927)

    MATH  Google Scholar 

  24. Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Lou, Y., Zhao, X.-Q.: A reaction–diffusion malaria model with incubation period in the vector population. J. Math. Boil. 62, 543–568 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Ma, S.W.: Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Maidana, N.A., Yang, H.: Spatial spreading of West Nile virus described by traveling waves. J. Theoret. Biol. 258, 403–417 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Mirski, T., Bartoszcze, M., Bielawska-Drozd, A.: Impact of climate change on infectious diseases. Pol. J. Environ. Stud. 21, 525–532 (2012)

    Google Scholar 

  29. Ruan, S.: Spatial-Temporal Dynamics in Nonlocal Epidemiological Models. In: Mathematics for Life Science and Medicine. Springer, Berlin, pp. 97–122 (2007)

  30. Sadilek, A., Hswen, Y., Bavadekar, S., et al.: Lymelight: forecasting Lyme disease risk using web search data. nej Digit. Med. 16(3) (2020)

  31. Shuai, Z., van den Driessche, P.: Global dynamics of cholera models with differential infectivity. Math. Biosci. 234, 118–126 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Tian, B., Yuan, R.: Traveling waves for a diffusive SEIR epidemic model with non-local reaction. Appl. Math. Model. 50, 432–449 (2017)

    MathSciNet  MATH  Google Scholar 

  33. 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)

    MathSciNet  MATH  Google Scholar 

  34. Volpert, A.I., Volpert, V.A., Volpert, V.A.: Travelling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, vol. 140. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  35. Wang, K., Wang, W.: Propagation of HBV with spatial dependence. Math. Biosci. 210, 78–95 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Wang, J.L., Chen, Y.M.: Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias. Appl. Math. Lett. 100, 106052 (2020)

    MathSciNet  MATH  Google Scholar 

  37. Wang, S.M., Feng, Z., Wang, Z.-C., et al.: Periodic traveling wave of a time periodic and diffusive epidemic model with nonlocal delayed transmission. Nonlinear Anal. RWA 55, 103117103117 (2020)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. Wang, Z., Wang, H.: Bistable traveling waves in impulsive reaction–advection–diffusion models. J. Differ. Equ. 285, 17–39 (2021)

    MathSciNet  MATH  Google Scholar 

  41. Wang, Z.-C., Li, W.-T., Ruan, S.: Travelling fronts in monostable equations with nonlocal delayed effects. J. Dyn. Differ. Equ. 20, 573–607 (2008)

    MATH  Google Scholar 

  42. Wang, Z.-C., Li, W.-T., Ruan, S.: Travelling wave fronts in reaction–diffusion systems with spatio-temporal delays. J. Differ. Equ. 222, 185–232 (2006)

    MathSciNet  MATH  Google Scholar 

  43. Wang, Z.-C., Wu, J.-H.: 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, 237–261 (2010)

    MathSciNet  MATH  Google Scholar 

  44. Widder, D.: The Laplace Transform. Princeton University Press, Princeton (1941)

    MATH  Google Scholar 

  45. Wu, J.-H., Zou, X.: Traveling wave fronts of reaction–diffusion systems with delay. J. Dyn. Differ. Equ. 13, 651–687 (2001)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Zeidler, E.: Nonlinear Functional Analysis and Its Applications I. Springer, New York (1986)

    MATH  Google Scholar 

  48. 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)

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

    MathSciNet  MATH  Google Scholar 

  50. Zhang, T., Wang, W., Wang, K.: Minimal wave speed for a class of non-cooperative diffusion–reaction system. J. Differ. Equ. 260, 2763–2791 (2016)

    MathSciNet  MATH  Google Scholar 

  51. 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  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  53. Zhao, X.-Q.: Global dynamics of a reaction and diffusion model for Lyme disease. J. Math. Biol. 65, 787–808 (2012)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  55. Zhu, M., Lin, Z.G., Zhang, L.: Spatial-temporal risk index and transmission of a nonlocal dengue model. Nonlinear Anal. RWA 53, 103076 (2020)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for helpful suggestions which improved the manuscript. The research is supported by Natural Science Foundation of China (No. 11971013); an NSERC Discovery Grant; Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0169); China Postdoctoral Science Foundation (No. 2021M691577).

Author information

Authors and Affiliations

  1. Department of Mathematics, Nan**g University of Aeronautics and Astronautics, Nan**g, 211106, China

    Kai Wang, Hongyong Zhao & Ran Zhang

  2. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nan**g, 211106, China

    Kai Wang, Hongyong Zhao & Ran Zhang

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Hao Wang

Authors
  1. Kai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongyong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ran Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongyong Zhao.

Additional information

Publisher's Note

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

This work is supported by Natural Science Foundation of China (No. 11971013); an NSERC Discovery Grant; Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX20_0169); China Postdoctoral Science Foundation (No. 2021M691577)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, K., Zhao, H., Wang, H. et al. Traveling wave of a reaction–diffusion vector-borne disease model with nonlocal effects and distributed delay. J Dyn Diff Equat 35, 3149–3185 (2023). https://doi.org/10.1007/s10884-021-10062-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-10062-w

Keywords

Mathematics Subject Classification

Search

Navigation