<p>Hybrid hospitals combine on-site hospitalization with remote care via telemedicine, requiring new operational policies to balance costs, efficiency, and patient well-being across both care modalities. We address two key questions: (i) how to assign patients to remote or on-site care based on individual characteristics and proximity and (ii) how to optimally allocate shared medical resources between care modes and patient types. We develop a stochastic model using Brownian motion to capture the randomness in recovery and travel-related risk during remote and on-site care. While the optimal call-in threshold is shaped by a cost-minimization objective, its behavior—specifically, its non-monotonicity in travel time and the narrowing of the effective distance range for more severe cases—is also driven by clinical constraints on allowable delays before hospital admission. These constraints, motivated by medical guidelines, limit the threshold and lead to cases where remote hospitalization becomes infeasible for very distant or severely ill patients. Under resource constraints, the optimal solution mirrors a simultaneous increase in remote and on-site costs relative to the abundant-resource case. For multiple patient types, we characterize how optimal thresholds shift with resource availability. Our findings indicate that distant patients may at times be better served by on-site care. This outcome arises not purely from economic trade-offs, but from the interplay between clinical constraints (e.g., safe limits on call-in delays), operational considerations, and treatment costs. These insights can help inform healthcare decision-makers and policymakers in designing hybrid care systems.</p>