  Historically, infectious diseases caused considerable damage to human
societies, and they continue to do so today. To help reduce their impact,
mathematical models of disease transmission have been studied to help
understand disease dynamics and inform prevention strategies. Vaccination - one
of the most important preventive measures of modern times - is of great
interest both theoretically and empirically. And in contrast to traditional
approaches, recent research increasingly explores the pivotal implications of
individual behavior and heterogeneous contact patterns in populations. Our
report reviews the developmental arc of theoretical epidemiology with emphasis
on vaccination, as it led from classical models assuming homogeneously mixing
(mean-field) populations and ignoring human behavior, to recent models that
account for behavioral feedback and/or population spatial/social structure.
Many of the methods used originated in statistical physics, such as lattice and
network models, and their associated analytical frameworks. Similarly, the
feedback loop between vaccinating behavior and disease propagation forms a
coupled nonlinear system with analogs in physics. We also review the new
paradigm of digital epidemiology, wherein sources of digital data such as
online social media are mined for high-resolution information on
epidemiologically relevant individual behavior. Armed with the tools and
concepts of statistical physics, and further assisted by new sources of digital
data, models that capture nonlinear interactions between behavior and disease
dynamics offer a novel way of modeling real-world phenomena, and can help
improve health outcomes. We conclude the review by discussing open problems in
the field and promising directions for future research.
