The COVID-19 pandemic, which spread rapidly in late 2019, has revealed that the use of computing and communication technologies provides significant aid in preventing, controlling, and combating infectious diseases. With the ongoing research in next-generation networking (NGN), the use of secure and reliable communication and networking is of utmost importance when dealing with users' health records and other sensitive information. Through the adaptation of artificial-intelligence-enabled NGN, the shape of healthcare systems can be altered to achieve smart and secure healthcare capable of coping with epidemics that may emerge at any given moment. In this article, we envision a cooperative and distributed healthcare framework that relies on state-of-the-art computing, communication, and intelligence capabilities, namely, federated learning, mobile edge computing, and blockchain, to enable epidemic (or suspicious infectious disease) discovery, remote monitoring, and fast health authority response. The introduced framework can also enable secure medical data exchange at the edge and between different health entities. This technique, coupled with the low latency and high bandwidth functionality of 5G and beyond networks, would enable mass surveillance, monitoring, and analysis to occur at the edge. Challenges, issues, and design guidelines are also discussed in this article with highlights on some trending solutions.