Digital Contact Tracing (DCT) is a great example of information systems assisting societal problems. However, privacy concerns lead to reduced DCT adoption rates. When dealing with critical societal issues, policymakers seek to use various strategies, such as interventions and financial subsidies, to steer the behavors of individuals. This causes individuals to face a sophisticated decision-making process when coping with the public health crisis and the adoption of DCT, i.e., giving up privacy and freedom to gain information to remain healthy. In this paper, we consider a scenario, where policymakers allocate rewards to individuals to motivate their compliance with the interventions; And the individuals decide the optimal compliance effort based on their health state, privacy loss, interaction with their neighbors, and rewards. To tackle the trade-off between a number of individuals and policymakers, in this paper, we propose a Leader-Followers Mean-Field Game model to analyze this time-dependent, dynamic, and large-scale decision-making problem. The numerical results demonstrate that by allocating appropriate rewards, policymakers can play a role in guiding the behavior of individuals in various scenarios.