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Analysis of a reaction–diffusion system about West Nile virus with free boundaries in the almost periodic heterogeneous environment

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Abstract

We put forward a reaction–diffusion cooperative system for the West Nile virus in a spatial heterogeneous and time almost periodic environment with free boundaries. The existence, uniqueness and regularity estimates of the global solution for this epidemic model are given. Focused on the effects of spatial heterogeneity and time almost periodicity, we introduce the principal Lyapunov exponent \(\lambda (t)\) dependent on t and get the initial infected critical size \(L^*\) which plays a vital part in analyzing the threshold dynamics and the long-time asymptotic behaviors of the solution. We build the spreading–vanishing dichotomy regimes of this model and obtain several criteria determining the spreading or vanishing. Our analysis result suggests that the solution converges to a positive time almost periodic function \(\left( U^*(x,t),V^*(x,t)\right) \) locally uniformly when the spreading occurs. We discover that the initial disease infected domain and the front expanding rate have momentous impacts on the permanence and extinction of the disease. Moreover, we give the lower and the upper bound estimates about the asymptotic spreading speeds of the double free fronts when the spreading happens.

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Acknowledgements

The authors are supported by the NSF of China (Nos. 11671382, 12031020), CAS Key Project of Frontier Sciences (No. QYZDJ-SSW-JSC003), the Key Lab. of Random Complex Structures and Data Sciences CAS and National Center for Mathematics and Interdisciplinary Sciences CAS.

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Authors and Affiliations

  1. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing, 100875, China

    Chengcheng Cheng

  2. College of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, Hainan, China

    Zuohuan Zheng

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zuohuan Zheng

  4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Zuohuan Zheng

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  1. Chengcheng Cheng
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  2. Zuohuan Zheng
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Correspondence to Chengcheng Cheng.

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Cheng, C., Zheng, Z. Analysis of a reaction–diffusion system about West Nile virus with free boundaries in the almost periodic heterogeneous environment. Z. Angew. Math. Phys. 73, 84 (2022). https://doi.org/10.1007/s00033-022-01729-5

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  • DOI: https://doi.org/10.1007/s00033-022-01729-5

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