Vaccination has become an integral part of public health, since an increase in overall vaccination in a given population contributes to a decline in infectious diseases and mortality. Vaccination also contributes to a lower rate of infection even for nonvaccinators due to herd immunity ((Brisson and Edmunds, 2002)). In this work we model human decision-making (with respect to a vaccination program in a single-payer health care provider country) using a leader-follower game framework. We then extend our model to a discrete dynamic game, where time passing is modelled by risk perception changes among population groups considering whether or not to vaccinate. The risk perception changes are encapsulated by probability transition matrices. We assume that the single-payer provider has a given fixed budget which would not be sufficient to cover 100% of a new vaccine for the entire population. To increase the potential coverage, we propose the introduction of a partial vaccine adoption policy, whereby an individual would pay a portion of the vaccine price and the single payer would support the rest for the entire population. We show how this policy, together with changes in risk perceptions regarding vaccination, impact the strategic decisions of individuals in each group, the policy cost under budgetary constraints and, ultimately, how it impacts the overall uptake of the vaccine in the entire population.