Sepsis prediction models in intensive care units often degrade over time due to changes in clinical practice, patient populations, and data recording processes, a phenomenon known as model drift that can compromise patient safety. Traditional federated learning approaches are not well-suited to these evolving conditions, as they assume static data distributions and typically require costly retraining that risks forgetting previously learned knowledge, while also being constrained by privacy limitations that prevent central data pooling. To address these challenges, this paper proposes a federated continual learning framework that enables ongoing, privacy-preserving model adaptation across multiple hospitals without catastrophic forgetting. The framework integrates local continual learning methods (such as elastic weight consolidation or memory replay) with federated aggregation and importance-weighted parameter updates to support continuous learning from new clinical data while preserving prior knowledge. This design allows each institution to adapt models to local data shifts while collaboratively improving a shared global model without sharing patient-level data. Overall, the proposed approach offers a scalable solution for maintaining robust, adaptive sepsis prediction systems in dynamic healthcare environments, reducing the need for repeated full retraining and supporting long-term clinical deployment.