Journal of Mathematical Biology

, Volume 78, Issue 5, pp 1331–1364 | Cite as

How ticks keep ticking in the adversity of host immune reactions

  • Rachel Jennings
  • Yang Kuang
  • Horst R. ThiemeEmail author
  • Jianhong Wu
  • Xiaotian Wu


Ixodid ticks are acknowledged as one of the most important hematophagous arthropods because of their ability in transmitting a variety of tick-borne diseases. Mathematical models have been developed, based on emerging knowledge about tick ecology, pathogen epidemiology and their interface, to understand tick population dynamics and tick-borne diseases spread patterns. However, no serious effort has been made to model and assess the impact of host immunity triggered by tick feeding on the distribution of the tick population according to tick stages and on tick population extinction and persistence. Here, we construct a novel mathematical model taking into account the effect of host immunity status on tick population dynamics, and analyze the long-term behaviours of the model solutions. Two threshold values, \({\mathcal {R}}_{11}\) and \({\mathcal {R}}_{22}\), are introduced to measure the reproduction ratios for the tick-host interaction in the absence and presence of host immunity. We then show that these two thresholds (sometimes under additional conditions) can be used to predict whether the tick population goes extinct (\({\mathcal {R}}_{11}<1\)) and the tick population grows without bound (\({\mathcal {R}}_{22}>1\)). We also prove tick permanence (persistence and boundedness of the tick population) and the existence of a tick persistence equilibrium if \({\mathcal {R}}_{22}<1<{\mathcal {R}}_{11}\). As the host species adjust their immunity to tick infestation levels, they form for the tick population an environment with a carrying capacity very much like that in logistic growth. Numerical results show that the host immune reactions decrease the size of the tick population at equilibrium and apparently reduce the tick-borne infection risk.


Host resistance Persistence Extinction Basic reproduction ratios Quasi-steady-state approximation Global stability 

Mathematics Subject Classification

92D40 34C11 34C60 34D23 93D30 



This project was initiated during the workshop on “Mathematics inspired by immunoepidemiology” held at the American Institute of Mathematics (August 24-28, 2015), and the authors and all rodents and deer plagued by ticks gratefully acknowledge the AIM’s support. This research was also financially supported by the National Natural Science Foundation of China (No.11501358, held by XW), the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs program (JW), and by NSF grant DMS-1615879 (YK). The authors thank an anonymous referee for helpful comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Rachel Jennings
    • 1
  • Yang Kuang
    • 2
  • Horst R. Thieme
    • 2
    Email author
  • Jianhong Wu
    • 3
  • Xiaotian Wu
    • 4
  1. 1.SavvysherpaMinnetonkaUSA
  2. 2.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  3. 3.Department of Mathematics and StatisticsYork UniversityTorontoCanada
  4. 4.Department of MathematicsShanghai Maritime UniversityShanghaiChina

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