Dynamic resource allocation for prevention, screening, and treatment interventions in population disease management has received much attention in recent years due to excessive healthcare costs. In this paper, our goal is to design a model and an efficient algorithm to optimize sequential intervention policies under resource constraints to improve population health outcomes. We consider a discrete-time finite-horizon budget allocation problem with disease progression within a closed birth-cohort population. To address the computational challenges associated with large-state and multiple-period dynamic decision-making problems, we propose a low-fidelity approximation that preserves the population dynamics under a stationary policy. To improve the healthcare interventions in terms of population health outcomes, we then embed the low-fidelity approximation into a high-fidelity optimization model to efficiently identify a good non-stationary sequential intervention policy. Our approach is illustrated by a numerical example of screening and treatment policy implementation for chronic hepatitis C virus (HCV) infection over a budget planning period. We numerically compare our Multi-Fidelity Rollout Algorithm (MF-RA) to a grid search approach and demonstrate the similarity of sequential policy trends and closeness of overall health outcomes measured by quality-adjusted life-years (QALYs) and the total number of individuals that undergo screening and treatment for different annual budgets and birth-cohorts. We also show how our approach scales well to problems with high dimensionality due to many decision periods by studying time to elimination of HCV.