Skip to main content

The asymptotic analysis of a vector–host epidemic model with finite growing domain

Zeitschrift für angewandte Mathematik und Physik volume 73, Article number: 112 (2022) Cite this article

Abstract

To study the impact of vector’s habitat expansion and mobility of hosts on the geographic spread of diseases, we propose a diffusive vector–host epidemic model with a finite growing domain and mainly focus on its asymptotic profile. In a situation that the domain grows uniformly and isotropically with growth ratio \(\rho \), we employ the Lagrangian transformations to transform the model in the growing domain into the one in a fixed domain, along with dilution terms and time-dependent diffusion coefficients. We analyze the well posedness of the model and define the basic reproduction number \(\Re _0^\rho \). Our results indicate that if \(\Re _0^\rho < 1\), the disease-free equilibrium is globally asymptotically stable without any extra conditions, while the infected populations will eventually tend to the set formulated by the maximum and minimum solutions of the associated steady-state problem if \(\Re _0^\rho >1\). The analysis is carried out by using the comparison principle, the theory of quasimonotone nondecreasing elliptic and parabolic system, and convergence of abstract asymptotic autonomous system. Comparing the results with those in a fixed domain, we confirm that the growth of domain brings negative influences on disease control.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Chen, J., Huang, J., Beier, J.C., Cantrell, R.S., Cosner, C., Fuller, D.O., Zhang, G., Ruan, S.: Modeling and control of local outbreaks of West nile virus in the United States. Discrete Contin. Dyn. Syst. Ser B. 21, 2423–2449 (2016)

    MathSciNet  Article  Google Scholar 

  2. Bowman, C., Gumel, A.B., van den Driessche, P., Wu, J., Zhu, H.: A mathematical model for assessing control strategies against West Nile virus. Bull. Math. Biol. 67, 1107–1133 (2005)

    MathSciNet  Article  Google Scholar 

  3. Ross, R.: An application of the theory of probabilities to the study of a priori pathometry. Proc. R. Soc. Lond A. 92, 204–230 (1916)

    Article  Google Scholar 

  4. Macdonald, G.: The analysis of equilibrium in malaria. Trop. Dis. Bull. 49, 813–829 (1952)

    Google Scholar 

  5. Macdonald, G.: The Epidemiology and Control of Malaria. Oxford University Press, London (1957)

  6. Chamchod, F., Britton, N.F.: Analysis of a vector-bias model on malaria transmission. Bull. Math. Biol. 73, 639–657 (2011)

    MathSciNet  Article  Google Scholar 

  7. Fitzgibbon, W.E., Morgan, J.J., Webb, G.F.: An outbreak vector-host epidemic model with spatial structure: the 2015–2016 Zika outbreak in Rio De Janeiro. Theor. Biol. Med. Modell 14, 7 (2017)

    Article  Google Scholar 

  8. Wang, X., Zhao, X.Q.: A periodic vector-bias malaria model with incubation period. SIAM J. Appl Math. 77, 181–201 (2017)

    MathSciNet  Article  Google Scholar 

  9. Magal, P., Webb, G.F., Wu, Y.: On a vector-host epidemic model with spatial structure. Nonlinearity 31, 5589–5614 (2018)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  11. Bailey, N.T.J.: The Mathematical Theory of Epidemics. Charles Griffin & Company Limited, London (1957)

  12. Dietz, K.: Mathematical models for transmission and control of malaria. In: Wernsdorfer, W., McGregor, I. (eds.) Malaria: Principles and Practice of Malariology. Churchill Livingstone, Edinburgh (1988)

  13. Busenberg, S., Vargas, C.: Modeling Chagas’ disease: variable population size and demographic implications. In Ovide Arino, David E. Axelrod, and Marek Kimmel, editors, Mathematical Population Dynamics, Lecture Notes Pure and Applied Mathematics. Boca Raton:CRC Press; (1991)

  14. Inaba, H., Sekine, H.: A mathematical model for Chagas disease with infection-age-dependent infectivity. Math. Biosci. 190, 39–69 (2004)

    MathSciNet  Article  Google Scholar 

  15. Velasco-Hernández, J.X.H.: An epidemiological model for the dynamics of Chaga’s disease. Biosystems 26, 127–134 (2004)

    Article  Google Scholar 

  16. Xu, Z., Zhao, X.Q.: A vector-bias malaria model with incubation period and diffusion. Discrete Contin. Dyn. Syst. Ser. B. 17, 2615–2634 (2012)

    MathSciNet  MATH  Google Scholar 

  17. 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  Article  Google Scholar 

  18. Bai, Z., Peng, R., Zhao, X.Q.: A reaction-diffusion malaria model with seasonality and incubation period. J. Math. Biol. 77, 201–228 (2018)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  20. Allen, L.J.S., Bolker, B.M., Lou, Y., Nevai, A.L.: Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Contin. Dyn. Syst. 21, 1–20 (2008)

    MathSciNet  Article  Google Scholar 

  21. Peng, R.: Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I. J Differ. Equ. 247, 1096–1119 (2009)

    MathSciNet  Article  Google Scholar 

  22. Li, B., Li, H., Tong, T.: Analysis on a diffusive SIS epidemic model with logistic source. Z Angew Math. Phys. 68, 96 (2017)

    MathSciNet  Article  Google Scholar 

  23. Cui, R., Lam, L., Lou, Y.: Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments. J. Differ. Equ. 263, 2343–2373 (2017)

    MathSciNet  Article  Google Scholar 

  24. Han, S., Lei, C., Zhang, X.: Qualitative analysis on a diffusive SIRS epidemic model with standard incidence infection mechanism. Z Angew Math. Phys. 71, 190 (2020)

    MathSciNet  Article  Google Scholar 

  25. Wu, Y., Zou, X.: Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J. Differ. Equ. 261, 4424–4447 (2016)

    MathSciNet  Article  Google Scholar 

  26. Fitzgibbon, W.E., Morgan, J.J., Webb, G.F., Wu, Y.: A vector-host epidemic model with spatial structure and age of infection. Nonlinear Anal. Real. World Appl. 41, 692–705 (2018)

    MathSciNet  Article  Google Scholar 

  27. Jiang, D.H., Wang, Z.C.: The diffusive Logistic equation on periodically evolving domain. J. Math. Anal. Appl. 378, 93–111 (2018)

    MathSciNet  Article  Google Scholar 

  28. Zhang, M., Lin, Z.: The diffusive model for Aedes Aegypti mosquito on a periodically evolving domain. Discrete Contin. Dyn. Syst. Ser B. 24, 4703–4720 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Zhu, M., Xu, Y., Cao, J.: The asymptotic profile of a dengue fever model on a periodically evolving domain. Appl. Math. Comput. 362, 124531 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Sprenger, D., Wuithiranyagool, T.: The discovery and distribution of Aedes albopictus in Harris county. Texas. J Am Mosq Control Assoc. 2, 217–219 (1986)

    Google Scholar 

  31. Yee, D.A., Juliano, S.A., Vamosi, S.M.: Seasonal photoperiods alter developmental time and mass of an invasive mosquito Aedes albopictus (Diptera: Culicidae), across its north-south range in the United States. J. Med. Entomol. 49, 825–832 (2012)

    Article  Google Scholar 

  32. Zhu, M., Lin, Z., Zhang, L.: The asymptotic profile of a dengue model on a growing domain driven by climate change. Appl. Math. Model. 83, 470–486 (2020)

    MathSciNet  Article  Google Scholar 

  33. Baker, R.E., Maini, P.K.: A mechanism for morphogen-controlled domain growth. J. Math. Biol. 54, 597–622 (2007)

    MathSciNet  Article  Google Scholar 

  34. Crampin, E.J., Gaffney, E.A., Maini, P.K.: Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61, 1093–1120 (1999)

    Article  Google Scholar 

  35. Crampin, E.J., Gaffney, E.A., Maini, P.K.: Mode-doubling and tripling in reaction-diffusion patterns on growing domains: a piecewise linear model. J Math Biol. 200244:107-128

  36. Madzvamuse, A.: Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. J. Comput. Phys. 214, 239–263 (2006)

    MathSciNet  Article  Google Scholar 

  37. World Health Organization. Dengue and Severe Dengue, (2021)

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

    Article  Google Scholar 

  39. Tewa, J.J., Dimi, J.L., Bowong, S.: Lyapunov functions for a dengue disease transmission model. Chaos Soliton Fract. 39, 936–941 (2009)

    MathSciNet  Article  Google Scholar 

  40. Villela, D.A.M., Bastos, L.S., D.E. Carvalho, L.M., Cruz, O.G., Gomes, M.F.C., Durovni, B., Lemos, M.C., Saraceni, V., Coelho, F.C., CodeçSo, C.T.: Zika in Rio de Janeiro: Assessment of basic reproduction number and comparison with dengue outbreaks. Epidemiol Infect. 2017;145:1649-1657

  41. Killilea, M.E., Swei, A., Lane, R.S., Briggs, C.J., Ostfeld, R.S.: Spatial dynamics of Lyme disease: a review. EcoHealth 5, 167–195 (2008)

    Article  Google Scholar 

  42. Yu, X., Zhao, X.Q.: A nonlocal spatial model for Lyme disease. J. Diff. Equ. 261, 340–372 (2016)

    MathSciNet  Article  Google Scholar 

  43. Elementary, Acheson D., Dynamics, Fluid: Oxford Applied Mathematics and Computing Science Series. Clarendon Press, Oxford (1990)

  44. Baines, M.J.: Moving Finite Element. Clarendon Press, Monographs on Numerical Analysis. Oxford (1994)

  45. Madzvamuse, A., Maini, P.K.: Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains. J. Comp. Phys. 225, 100–119 (2007)

    MathSciNet  Article  Google Scholar 

  46. Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology. West Sussex:John Wiley and Sons Ltd.; (2003)

  47. Alvarez-Caudevilla, P., Du, Y., Peng, R.: Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate enviroment. SIAM J. Math. Anal. 46, 499–531 (2014)

    MathSciNet  Article  Google Scholar 

  48. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Berlin:Springer; (2001)

  49. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

  50. Ladyženskaja OA, Solonnikov UA, Ural’ceva NN. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Providence:American Mathematical Society; (1968)

  51. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Berlin:Springer; (1981)

  52. 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)

    MathSciNet  Article  Google Scholar 

  53. Pu, L., Lin, Z.: Spatial transmission and risk assessment of West Nile virus on a growing domain. Math. Meth. Appl. Sci. 44, 6067–6085 (2021)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for his/her suggestions that have improved this paper.

Funding

This work was supported by National Natural Science Foundation of China (nos. 12071115, 11871179, 11771107), Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, and Fundamental Research Funds for the Heilongjiang Education Department (No. 2021-KYYWF-0034)

Author information

Authors and Affiliations

  1. School of Mathematical Science, Heilongjiang University, Harbin, 150080, People’s Republic of China

    Desheng Ji & Jinliang Wang

Authors
  1. Desheng Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinliang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinliang Wang.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflicts interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ji, D., Wang, J. The asymptotic analysis of a vector–host epidemic model with finite growing domain. Z. Angew. Math. Phys. 73, 112 (2022). https://doi.org/10.1007/s00033-022-01749-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-022-01749-1

Keywords

  • Vector–host epidemic model
  • Growing domain
  • Basic reproduction number

MSC

  • 35K57
  • 35J57
  • 35B40
  • 92D25