A Minimal Model for Affinity Maturation and Vaccination Protocols

The adaptive immune system's ability to recognize previously unseen pathogens through affinity maturation is a remarkable biological inspiration. In light of this, we develop a minimal model focusing on B cells inhabiting a trait space that quantifies their affinity towards various strains of a mutating pathogen. The objective of an effective vaccination protocol is to guide and accelerate affinity maturation in order to optimize specific characteristics of the resulting B cell population, e.g., their neutralizing breadth. We explore this question using a simple linear stochastic birth-death-mutation model with varying levels of precision. We start with an analytical investigation of the mean-field model and utilize operator techniques and a path integral formulation to obtain qualitative features of the vaccination protocols that enhance the desired changes of the population within the trait space. Subsequently, we assess the validity of our mean-field predictions through stochastic simulations.