The emergence of SARS-CoV-2 (COVID-19) has demonstrated the severe impact of infectious diseases on global society, politics, and economies. To mitigate future pandemics, preemptive measures for effectively managing infection outbreaks are essential. In this context, Societal Digital Twin (SDT) technology offers a promising solution. To the best of our knowledge, this survey is the premier to conceptualize an SDT framework for infection containment under a novel systematic taxonomy. The framework categorizes infection management into five stages, namely infection initiation, spread, control, combat, and recovery. It provides an overview of SDT approaches within each category, discussing their validation strategies, generalizability, and limitations. Additionally, the survey examines applications, data-driven design issues, key components, and limitations of DT technology in healthcare. Finally, it explores key challenges, open research directions, and emerging paradigms to advance DT applications in the healthcare domain, highlighting smart service paradigms such as SDT as a Smart Service (SDTaaSS) and Healthcare Metaverse as a Smart Service (HMaaSS).