In a rare life-threatening disease setting the number of patients in the trial is a high proportion of all patients with the condition (if not all of them). Further, this number is usually not enough to guarantee the required statistical power to detect a treatment effect of a meaningful size. In such a context, the idea of prioritizing patient benefit over hypothesis testing as the goal of the trial can lead to a trial design that produces useful information to guide treatment, even if it does not do so with the standard levels of statistical confidence. The idealised model to consider such an optimal design of a clinical trial is known as a multi-armed bandit problem with a finite patient horizon and a patient benefit objective function. Such a design maximises patient benefit by balancing the learning and earning goals as data accumulates and given the patient horizon. On the other hand, optimally solving such a model has a very high computational cost (many times prohibitive) and more importantly, a cumbersome implementation, even for populations as small as a hundred patients. Several computationally feasible heuristic rules to address this problem have been proposed over the last 40 years in the literature. In this article we study a novel heuristic approach to solve it based on the reformulation of the problem as a Restless bandit problem and the derivation of its corresponding Whittle index rule. Such rule was recently proposed in the context of a clinical trial in Villar et al (2015). We perform extensive computational studies to compare through both exact value calculations and simulated values the performance of this rule, other index rules and simpler heuristics previously proposed in the literature. Our results suggest that for the two and three-armed case and a patient horizon less or equal than a hundred patients, all index rules are a priori practically identical in terms of the expected proportion of success attained when all arms start with a uniform prior. However, we find that a posteriori, for specific values of the parameters of interest, the index policies outperform the simpler rules in every instance and specially so in the case of many arms and a larger, though still relatively small, total number of patients with the diseases. The very good performance of bandit rules in terms of patient benefit (i.e. expected number of successes and mean number of patients allocated to the best arm, if it exists) makes them very appealing in context of the challenge posed by drug development for rare life threatening diseases.

Motivated by a class of Partially Observable Markov Decision Processes with application in surveillance systems in which a set of imperfectly observed state processes is to be inferred from a subset of available observations through a Bayesian approach, we formulate and analyze a special family of multi-armed restless bandit problems. We consider the problem of finding an optimal policy for observing the processes that maximizes the total expected net rewards over an infinite time horizon subject to the resource availability. From the Lagrangian relaxation of the original problem, an index policy can be derived, as long as the existence of the Whittle index is ensured. We demonstrate that such a class of bandits in which the projects' state deteriorates while active and resets to its initial state when passive until its completion possesses the structural property of indexability and we further show how to compute the index in closed form. In general, the Whittle index rule for restless bandit problems does not achieve optimality. However, we show that the proposed Whittle index rule is optimal for the problem under study in the case of stochastically heterogenous arms under the expected total criterion, and it is further recovered by a simple tractable rule referred to as the rule. Moreover, we illustrate the significant suboptimality of other widely used heuristic: the Myopic index rule, by computing in closed form its suboptimality gap. We present numerical studies which illustrate for the more general instances the performance advantages of the Whittle index rule over other simple heuristics.