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Basic Reproduction Ratios for Periodic Abstract Functional Differential Equations (with Application to a Spatial Model for Lyme Disease)

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Abstract

In this paper, we develop the theory of basic reproduction ratios \(\mathcal {R}_0\) for abstract functional differential systems in a time-periodic environment. It is proved that \(\mathcal {R}_0-1\) has the same sign as the exponential growth bound of an associated linear system. Then we apply it to a time-periodic Lyme disease model with time-delay and obtain a threshold type result on its global dynamics in terms of \(\mathcal {R}_0\).

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Acknowledgements

Liang’s research is supported by the the National Natural Science Foundation of China (11571334) and the Fundamental Research Funds for the Central Universities; Zhang’s research is supported by the China Scholarship Council under a joint-training program at Memorial University of Newfoundland; and Zhao’s research is supported in part by the NSERC of Canada. We are also grateful to the anonymous referee for careful reading and valuable comments which led to improvements of our original manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

    Xing Liang & Lei Zhang

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. Johns, NL, A1C 5S7, Canada

    Lei Zhang & Xiao-Qiang Zhao

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  1. Xing Liang
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  2. Lei Zhang
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  3. Xiao-Qiang Zhao
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Correspondence to Xiao-Qiang Zhao.

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In Memory of Professor George Sell

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Liang, X., Zhang, L. & Zhao, XQ. Basic Reproduction Ratios for Periodic Abstract Functional Differential Equations (with Application to a Spatial Model for Lyme Disease). J Dyn Diff Equat 31, 1247–1278 (2019). https://doi.org/10.1007/s10884-017-9601-7

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  • DOI: https://doi.org/10.1007/s10884-017-9601-7

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