Currently available medication for treating many chronic diseases is often effective only for a subgroup of patients, and biomarkers accurately assessing whether an individual belongs to this subgroup typically do not exist. In such settings, physicians learn about the effectiveness of a drug primarily through experimentation, i.e., by initiating treatment and monitoring the patient's response. Precise guidelines for discontinuing treatment are often lacking or left entirely to the physician's discretion. We introduce a framework for developing adaptive, personalized treatments for such chronic diseases. Our model is based on a continuous-time, multi-armed bandit setting where drug effectiveness is assessed by aggregating information from several channels: by continuously monitoring the state of the patient, but also by (not) observing the occurrence of particular infrequent health events, such as relapses or disease flare-ups. Recognizing that the timing and severity of such events provides critical information for treatment decisions is a key point of departure in our framework compared with typical (bandit) models used in healthcare. We show that the model can be analyzed in closed form for several settings of interest, resulting in optimal policies that are intuitive and may have practical appeal. We illustrate the effectiveness of the methodology by developing a set of efficient treatment policies for multiple sclerosis, which we then use to benchmark several existing treatment guidelines.