Journal of Mathematical Biology

, Volume 46, Issue 5, pp 425–444 | Cite as

A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

  • Rebecca V. Culshaw
  • Shigui Ruan
  • Glenn Webb


 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1).


Infected Cell Hopf Bifurcation Leukaemia Virus Healthy Cell Bifurcation Parameter 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rebecca V. Culshaw
    • 1
  • Shigui Ruan
    • 2
  • Glenn Webb
    • 3
  1. 1.Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5.CA
  2. 2.Department of Mathematics, University of Miami, P. O. Box 249085, Coral Gables, FL 33124-4250, USA. e-mail:
  3. 3.Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240, USAUS

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