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A persistent public health challenge is identifying immunization schemes that are effective against 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 determines the selection forces that steer affinity maturation to generate antibodies. We focus on identifying the optimal selection forces exerted by a generic time-dependent vaccination protocol to maximize the production of broadly neutralizing antibodies (bnAbs) that can protect against a broad spectrum of pathogen strains. The model utilizes a path integral representation and operator approximations within a mean-field limit and provides 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 we discuss their similarities and differences.