Contact tracing is a very effective way to control the COVID-19-like pandemics. It aims to identify individuals who closely contacted an infected person during the incubation period of the virus and notify them to quarantine. However, the existing systems suffer from privacy, security, and efficiency issues. To address these limitations, in this article, we propose an efficient and privacy-preserving Blockchain-based infection control system. Instead of depending on a single authority to run the system, a group of health authorities, that form a consortium Blockchain, run our system. Using Blockchain technology not only secures our system against single point of failure and denial of service attacks, but also brings transparency because all transactions can be validated by different parties. Although contact tracing is important, it is not enough to effectively control an infection. Thus, unlike most of the existing systems that focus only on contact tracing, our system consists of three integrated subsystems, including contact tracing, public places access control, and safe-places recommendation. The access control subsystem prevents infected people from visiting public places to prevent spreading the virus, and the recommendation subsystem categorizes zones based on the infection level so that people can avoid visiting contaminated zones. Our analysis demonstrates that our system is secure and preserves the privacy of the users against identification, social graph disclosure, and tracking attacks, while thwarting false reporting (or panic) attacks. Moreover, our extensive performance evaluations demonstrate the scalability of our system (which is desirable in pandemics) due to its low communication, computation, and storage overheads.