Some mathematical models of cold pill concentration in the body

The motivation for this discussion comes from a set of exercises in Borelli and Coleman's Ordinary Differential Equations: A Modeling Approach. In this particular set of exercises the focus is to model the amount of cold medication (antihistamines and decongestants) in the body. We will consider a two-compartment model with the two components being the GI tract and the bloodstream, producing a two-dimensional system of ordinary differential equations. Initially we assume a simple linear model, then we increase the complexity by including a saturation effect in the amount of the drug clearing the bloodstream. We also examine three ways of doing the medication, administering just one dose, continuous dosing, and periodic dosing. Closed-form solutions are determined for the one dose, continuous dosing and non-saturation periodic cases, but only numerical solutions are given for the nonlinear, periodic cases. While we do not find closed-form solutions for the nonlinear, periodic cases we do explore their existence and stability. A software package called ODE Architect is used to generate the numerical solutions to all the models in this thesis.