| Summary: | In this thesis project, we model a sub-part of the adaptive immune system, composed of B and T lymphocytes, which interact to produce a suitable immune response against antigens. From a statistical mechanics perspective, this system can be modelled as a bipartite network with sparse links where the nodes represent B and T cells respectively, signalling via particular proteins called cytokines. Assuming that B lymphocytes evolve on a faster timescale than T cells, we study the dynamics of an effective mono-partite graph of T cells only where the B cells have been integrated out. Interestingly, this system can be mapped into a Hopfield-like associative network, which is able to retrieve and perform multiple immune strategies simultaneously. Using techniques such as Kramers-Moyal expansions for master equations, we carry out a dynamical analysis of the network evolving via Glauber sequential update. We derive equations quantifying the evolution in time of the immune response strength, analysing the nature and the stability of the stationary solutions in different regimes of dilution and network connectivity via linear stability analysis and Monte Carlo simulations. The model has also been extended to include the effect of receptors promiscuity, sampling B-T interactions locally from heterogeneous degree distributions and the effect of the antigens. Finally, we introduce interactions between B lymphocytes, called idiotypic interactions, studying their effect on the system's dynamics. We also analyse the effect of idiotypic interactions in the high storage and finite connectivity regime at equilibrium, using the cavity method to derive equations for the distributions of observables of the system. In particular, we obtain the B clone size distribution, studying its behaviour in different regions of the phase space.
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