Bulletin of Mathematical Biology

, Volume 72, Issue 4, pp 1006–1028 | Cite as

Backward Bifurcation and Optimal Control in Transmission Dynamics of West Nile Virus

  • Kbenesh W. Blayneh
  • Abba B. GumelEmail author
  • Suzanne Lenhart
  • Tim Clayton
Original Article


The paper considers a deterministic model for the transmission dynamics of West Nile virus (WNV) in the mosquito-bird-human zoonotic cycle. The model, which incorporates density-dependent contact rates between the mosquito population and the hosts (birds and humans), is rigorously analyzed using dynamical systems techniques and theories. These analyses reveal the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity) in WNV transmission dynamics. The epidemiological consequence of backward bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for WNV elimination from the population. It is further shown that the model with constant contact rates can also exhibit this phenomenon if the WNV-induced mortality in the avian population is high enough. The model is extended to assess the impact of some anti-WNV control measures, by re-formulating the model as an optimal control problem with density-dependent demographic parameters. This entails the use of two control functions, one for mosquito-reduction strategies and the other for personal (human) protection, and redefining the demographic parameters as density-dependent rates. Appropriate optimal control methods are used to characterize the optimal levels of the two controls. Numerical simulations of the optimal control problem, using a set of reasonable parameter values, suggest that mosquito reduction controls should be emphasized ahead of personal protection measures.

West Nile virus Equilibria Stability Bifurcation Optimal control 


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Copyright information

© Society for Mathematical Biology 2009

Authors and Affiliations

  • Kbenesh W. Blayneh
    • 1
  • Abba B. Gumel
    • 2
    Email author
  • Suzanne Lenhart
    • 3
  • Tim Clayton
    • 4
  1. 1.Department of MathematicsFlorida A & M UniversityTallahasseeUSA
  2. 2.Department of MathematicsUniversity of ManitobaWinnipegCanada
  3. 3.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  4. 4.Mathematics and Natural Science DepartmentTennessee Temple UniversityChattanoogaUSA

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