Journal of Mathematical Biology

, Volume 78, Issue 6, pp 1637–1679 | Cite as

Response of an oscillatory differential delay equation to a periodic stimulus

  • Daniel C. De SouzaEmail author
  • Michael C. Mackey


Periodic hematological diseases such as cyclical neutropenia or cyclical thrombocytopenia, with their characteristic oscillations of circulating neutrophils or platelets, may pose grave problems for patients. Likewise, periodically administered chemotherapy has the unintended side effect of establishing periodic fluctuations in circulating white cells, red cell precursors and/or platelets. These fluctuations, either spontaneous or induced, often have serious consequences for the patient (e.g. neutropenia, anemia, or thrombocytopenia respectively) which exogenously administered cytokines can partially correct. The question of when and how to administer these drugs is a difficult one for clinicians and not easily answered. In this paper we use a simple model consisting of a delay differential equation with a piecewise linear nonlinearity, that has a periodic solution, to model the effect of a periodic disease or periodic chemotherapy. We then examine the response of this toy model to both single and periodic perturbations, meant to mimic the drug administration, as a function of the drug dose and the duration and frequency of its administration to best determine how to avoid side effects.


Delay differential equation Periodic perturbation Delayed negative feedback Cycle length map Resetting time Blood cells Dynamical disease Cyclical neutropenia Cyclical thrombocytopenia 

Mathematics Subject Classification

92B25 92C45 34K27 34K13 



MCM would like to thank the Institut für Theoretische Neurophysik, Universität Bremen for their hospitality during the time in which much of the writing of this paper took place. We are very grateful to Tony Humphries for fruitful discussions and suggestions.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Departments of Physiology, Physics and MathematicsMcGill UniversityMontrealCanada
  3. 3.Institute of Immunology and Infection ResearchUniversity of Edinburgh, Ashworth LabsEdinburghScotland

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