Glycaemic stability of the diabetic patient and therapeutic adjustments

In previous papers, we provided a modeling of the behavior "insulin delivery/glycaemia" of the diabetic patient under continuous insulin infusion, continuous glucose monitoring and we provided a method of regulation of his glycaemia. This behavioral model is bilinear, predicting the behavior on an interval of 15 minutes, with an average error of 15%. And consequently, the model is adjusted for every 15 minutes. The aim of this paper is to study the Bounded-Input-Bounded-Output (BIBO) stability of the bilinear model in order to point out that the patient is entering in a period of stable/unstable equilibrium. In case of stable equilibrium, the prediction will be valid for a longer time interval, when in case of unstable equilibrium, it will leads one to reduce the time intervals and to pilot closely the variations of insulin delivery. The BIBO stability is studied by computing the generating series G of the model. This series, generalization of the transfer function, presents an useful tool for analyzing the stability of bilinear systems. It is a rational power series in noncommutative variables and by evaluating it, a formal expression of the output in form of iterated integrals is provided. Three cases arise: firstly, the output can be explicitly computed; secondly, the output can be bounded/unbounded if the input is bounded; thirdly, no conclusion seems available about the BIBO stability by using G. In this case, we propose a stabilizing constant input eta by studying the univariate series Geta.