Response of an oscillatory differential delay equation to a periodic stimulus

• Published in: Journal of mathematical biology Vol. 78; no. 6; pp. 1637 - 1679
• Main Authors: De Souza, Daniel C., Mackey, Michael C.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-05-2019; Springer Nature B.V
• Subjects: Anemia; Applications of Mathematics; Chemotherapy; Cytokines; Delay; Differential equations; Fluctuations; Hematological diseases; Hematology; Leukocytes (neutrophilic); Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Neutropenia; Nonlinear systems; Oscillations; Periodic variations; Platelets; Side effects; Thrombocytopenia; Resetting time; Cyclical neutropenia; Blood cells; Cyclical thrombocytopenia; 34K27; Cycle length map; Delayed negative feedback; 34K13; Delay differential equation; Periodic perturbation; 92C45; 92B25; Dynamical disease
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-018-1322-y

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.