Linear periodic models of subcutaneous insulin absorption: mathematical analysis

This work proposes a pair of models for subcutaneous (sc) insulin kinetics. Three sc injections are considered over a 24 hour time period. Each injection delivers both fast and slow insulin, so that a third order system is built, whose components are the fast and slow sc masses of insulin and the plasma insulin concentration. The insulin injections are modeled as impulsive inputs. ODE (Ordinary Differential Equation) models can be written when neglecting the delays in the absorption of insulin from the sc injections to the plasma circulation. Two linear, time-varying, periodic models have been presented, both taking into account that di®erent absorption rates are associated with different times of a 24 hour time period. One model consists of a variable structure system, switching among three linear working modes, according to the injection times. The other treats the time-varying absorption rate as an harmonic function. The qualitative behavior of the solutions is investigated in both the cases, showing positivity, boundedness and global stability of solutions. As far as the statistical identi¯ability of the models is concerned, it has also been proven the global identifiability of the switching linear model, according to a suitably defined set of experiments, while a rank condition is given to check local identifiability of the other periodic model.