92D25, 91A10, 37N25.

Previous modelling studies have formalized the "Tragedy of the Commons" that can occur under a voluntary vaccination policy, when there is a significant payoff not to vaccinate under conditions where high vaccine coverage affords indirect protection to nonvaccinators through herd immunity effects. Most of these previous studies have considered only a homogeneous population. However, in real populations, vaccine uptake can vary enormously across different social groups, often leading to localized outbreaks. In this paper, we consider a population under a voluntary vaccination policy consisting of distinct social groups. Unlike previous work on vaccination game theory in heterogeneous populations, these social groups differ both in the perceived vaccine risk as well as the perceived probability of becoming infected. Using game theory, projected dynamical systems theory, and variational inequality theory, we characterize the Nash equilibria of the system and analyze the game dynamics. The approach allows us to predict, in principle, the vaccine coverage in various social groups with distinct perceived vaccine and infection risks, where individuals are attempting to minimize health risks. We find that, under a wide range of parameter values, the vaccine coverage in a multi-group population can be higher than the vaccine coverage in the corresponding homogeneous population with the same average perceived relative risk of vaccination. This paper generalizes previous work by Cojocaru et al. [10] on applications of PDS and VI in vaccine game theory.