Stability and persistence in a model for bluetongue dynamics

Stephen A. Gourley, Horst Thieme, P. Van Den Driessche

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


A model for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors, is formulated as a delay differential equation system of six equations. Midges are assumed to have a preadult stage of constant duration and a general incubation period for bluetongue. A linear stability analysis leads to identification of a basic reproduction number that determines if the disease introduced at a low level dies out or is uniformly weakly persistent in the midges. Stronger conditions sufficient for global stability of the disease-free equilibrium are derived. The control reproduction numbers, which guide control strategies for midges, cattle, or sheep, are determined in the special case in which the incubation period for midges is exponentially distributed. The possibility of backward bifurcation is briefly discussed as is an equilibrium situation in which the disease wipes out sheep populations that are introduced in small numbers.

Original languageEnglish (US)
Pages (from-to)1280-1306
Number of pages27
JournalSIAM Journal on Applied Mathematics
Issue number4
StatePublished - 2011


  • Bluetongue
  • Delay
  • Disease persistence
  • Stability
  • Type reproduction number

ASJC Scopus subject areas

  • Applied Mathematics


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