A dynamical model of the adaptive immune system: Effects of cells promiscuity, antigens and B-B interactions

We analyse a minimal model for the immune response in the adaptive immune system comprising three different players: antigens, T and B cells. We assume B-T interactions to be diluted and sampled locally from heterogeneous degree distributions, which mimic B cells receptors' promiscuity. We derive dynamical equations for the order parameters quantifying the B cells activation and study the nature and stability of the stationary solutions using linear stability analysis and Monte Carlo simulations. The system's behaviour is studied in different scaling regimes of the number of B cells, dilution in the interactions and number of antigens. Our analysis shows that: (i) B cells activation depends on the number of receptors in such a way that cells with an insufficient number of triggered receptors cannot be activated; (ii) idiotypic (i.e. B-B) interactions enhance parallel activation of multiple clones, improving the system's ability to fight different pathogens in parallel; (iii) the higher the fraction of antigens within the host the harder is for the system to sustain parallel signalling to B cells, crucial for the homeostatic control of cell numbers.