  A persistent public health challenge is finding immunization schemes that are
effective in combating highly mutable pathogens such as HIV and influenza
viruses. To address this, we analyze a simplified model of affinity maturation,
the Darwinian evolutionary process B cells undergo during immunization. The
vaccination protocol dictates selection forces that steer affinity maturation
to generate antibodies. We focus on determining the optimal selection forces
exerted by a generic time-dependent vaccination protocol to maximize production
of broadly neutralizing antibodies (bnAbs) that can protect against a broad
spectrum of pathogen strains. The model lends itself to a path integral
representation and operator approximations within a mean-field limit, providing
guiding principles for optimizing time-dependent vaccine-induced selection
forces to enhance bnAb generation. We compare our analytical mean-field results
with the outcomes of stochastic simulations and discuss their similarities and
differences.
