Designing good policies for medical applications
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This thesis examines two topics at the intersection of mathematical decision-making and healthcare. The first part addresses a problem of designing optimal policy in ranking and selection. The second part critiques the standard mathematical measure of outbreak severity in epidemiology, proposes a more accurate measure, and discusses how to design an effective prevention policy. Chapter 2 of this thesis deals with the multinomial selection problem (MSP), a problem in ranking and selection. MSPs arise when designing a protocol to select the most effective drug or treatment from among multiple alternatives. The objective of MSP is to find a stopping policy for repeated independent trials, each of which reports a winner among competing alternatives, that has low expected cost and high probability of correct selection (PCS) of the best alternative. In 1959, Bechhofer, Elmaghraby and Morse formulated the problem as minimizing the worst-case expected number of trials, subject to a lower bound on PCS and upper bound on the maximum number of trials, over all probability vectors outside an indifference zone. For the case of two alternatives, we prove that if one employs a particular probability vector known as the slippage configuration, then a linear program always finds an optimal stopping policy. Chapters 3 and 4 discuss the basic reproduction number, a standard measure of potential disease spread. A proxy for the computationally intractable expected fraction of the population to be infected, it is intended to be less than one if the outbreak will die out, and to exceed one if the outbreak will become pandemic. It has long been used to predict the urgency and efficacy of proposed interventions by public health organizations which must determine the best use of limited resources. However, traditional homogeneous contact models have been largely replaced by more accurate heterogeneous contact network models. We prove that in shifting to heterogeneous contact models, the reproduction number loses its crucial theoretical properties. It cannot be used to approximate the scale of the epidemic, does not provide a threshold, and lacks a fundamental monotonicity property. Its worst-case inaccuracy is infinite. We propose to replace the reproduction number by an approximation of the expected fraction of population infected. We prove that accurate approximation is computationally feasible. We conduct a case study of a fine-grained spatial network model of the ongoing cholera outbreak in Yemen. We find that the reproduction number neither aids in assessing severity, nor identifies the important factors that affect the outbreak. One of the main motivations behind studying the spread of diseases is to mitigate their impact. Resource scarcity makes developing good prevention strategies a challenging optimization problem. In Chapter 5, we tackle the problem of minimizing disease spread on a contact network subject to a limited immunization budget. We present a stochastic programming formulation to design an intervention and discuss its computational complexity and approximability. On a more practical side, we evaluate the performance of such intervention against alternatives studied in the literature. We conduct this evaluation on three real contact networks and find that our approach significantly outperforms the alternatives. Moreover, the discrepancy is especially pronounced in the most dangerous cases of highly infectious diseases and tight intervention budgets.