Optimal strategies in immunology. I. B-cell differentiation and proliferation

The optimal strategy available to the immune system for responding to a non-replicating thymus-independent antigen is examined. By applying Pontryagin's maximum principle to a set of mathematical models of lymphocyte populations and their antibody production, it is found that the optimal strategy of bang-bang control appears robust. In a variety of structurely related biological models, similar behaviour is observed. The models that we consider assume that antigen triggers a population of B-lymphocytes. These triggered lymphocytes can either proliferate and secrete modest amounts of antibody or differentiate into nondividing plasma cells which secrete large amounts of antibody. For biologically reasonable parameter values it is found that for low doses of antigen, immediate differentiation into plasma cells is optimal, while for high antigen doses a proliferative state followed by differentiation is the best strategy.