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


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.

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


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)

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Correspondence to Jinliang Wang.

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Ji, D., Wang, J. The asymptotic analysis of a vector–host epidemic model with finite growing domain. Z. Angew. Math. Phys. 73, 112 (2022).

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


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