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Dynamic treatment regimes: the mathematics of unstable switched systems

In this paper we begin the study of dynamic treatment regimes. We use switched systems that are neither stable for all switching patterns nor unstable for all switching patterns. We ﬁnd conditions on the dynamics of the systems for which we can contain the orbits between two concentric balls. A patient with a chronic condition cannot be cured but on the other hand it is necessary to keep the condition from becoming more serious. In the treatment of chronic conditions such as high blood pressure, Type II diabetes, obesity, addictions and many others a “cure” is not really possible and so the goal of treatments is to maintain the patient at an acceptable level. It is well understood that a single treatment is seldom successful over a long time periods and therefore, treatments are adapted to the patient. Medicines are changed or modiﬁed and in many such treatments counseling is adapted to the patient along with pharmaceuticals. All of these cases are compatible with switched systems. While there is a signiﬁcant amount of work being done on modeling the bodys response to treatment there has been hardly any eﬀort to model multiple treatments. This paper is a step in that direction.