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Response of an oscillatory differential delay equation to a periodic stimulus

Authors
  • De Souza, Daniel C.1, 2
  • Mackey, Michael C.3
  • 1 McGill University, Department of Mathematics and Statistics, Montreal, QC, H3A 0B9, Canada , Montreal (Canada)
  • 2 University of Edinburgh, Ashworth Labs, Institute of Immunology and Infection Research, Edinburgh, EH9 3FL, Scotland , Edinburgh (United Kingdom)
  • 3 McGill University, Departments of Physiology, Physics and Mathematics, 3655 Promenade Sir William Osler, Montreal, QC, H3G 1Y6, Canada , Montreal (Canada)
Type
Published Article
Journal
Journal of Mathematical Biology
Publisher
Springer-Verlag
Publication Date
Jan 12, 2019
Volume
78
Issue
6
Pages
1637–1679
Identifiers
DOI: 10.1007/s00285-018-1322-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

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.

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