Delay differential systems for tick population dynamics

Guihong Fan, Horst Thieme, Huaiping Zhu

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


Ticks play a critical role as vectors in the transmission and spread of Lyme disease, an emerging infectious disease which can cause severe illness in humans or animals. To understand the transmission dynamics of Lyme disease and other tick-borne diseases, it is necessary to investigate the population dynamics of ticks. Here, we formulate a system of delay differential equations which models the stage structure of the tick population. Temperature can alter the length of time delays in each developmental stage, and so the time delays can vary geographically (and seasonally which we do not consider). We define the basic reproduction number R0 of stage structured tick populations. The tick population is uniformly persistent if R0>1 and dies out if R0<1. We present sufficient conditions under which the unique positive equilibrium point is globally asymptotically stable. In general, the positive equilibrium can be unstable and the system show oscillatory behavior. These oscillations are primarily due to negative feedback within the tick system, but can be enhanced by the time delays of the different developmental stages.

Original languageEnglish (US)
Pages (from-to)1017-1048
Number of pages32
JournalJournal Of Mathematical Biology
Issue number5
StatePublished - Nov 1 2015


  • Basic reproduction number
  • Delay differential systems
  • Global stability
  • Integral equations
  • Local stability
  • Persistence
  • Stage structure
  • Tick populations

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics


Dive into the research topics of 'Delay differential systems for tick population dynamics'. Together they form a unique fingerprint.

Cite this