Effect of population heterogeneity on herd immunity and on vaccination decision making process

We study the influence of population heterogeneity on herd immunity level and on individual's vaccination decision making process. We first formulate the mathematical model in a population with two subgroups, based on different activity levels or different susceptibilities. The herd immunity threshold is derived and discussed. It is calculated that the required vaccine coverage level for herd immunity in a heterogeneous mixing population can be varied significantly. The required vaccine coverage level is lower than the classical herd immunity level, if the vaccine coverage level in the more active group or more susceptible group is higher than the other subgroup. It is suggested that the classical herd immunity levels can be misleading in the process of planning mass vaccination programs. The analysis is further extended to study the population with more subgroups. We then study the formal vaccination games to simulate the process of vaccination decision making, in either homogeneous or heterogeneous mixing populations. It is proved that the Nash equilibrium in the vaccination game is not unique if population heterogeneity is considered. Moreover, herd immunity is not achieved if individuals are solely driven by self-interests.