Journal of Dynamics and Differential Equations

, Volume 31, Issue 3, pp 1247–1278 | Cite as

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

  • Xing Liang
  • Lei Zhang
  • Xiao-Qiang ZhaoEmail author


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 \(\mathcal {R}_0\).


Basic reproduction ratio Abstract functional differential system Periodic solution Lyme disease Threshold dynamics 

AMS Subject Classification

34K20 35K57 37B55 92D30 



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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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. JohnsCanada

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