Mean-field limits for multi-type birth and death process with a view to applications in phylodynamicsProbability
|Speaker:||Steven N Evans, UC Berkeley|
|Start time:||Fri, Sep 29 2023, 1:10PM|
Germinal centers (GC) are micro-anatomical structures which transiently form in lymph nodes during an adaptive immune response. In a GC, B-cells—the cells that make antibodies—diversify and compete based on the ability of the antibodies they express to recognize a foreign antigen molecule. As GC B-cells proliferate, they undergo targeted mutations in the genomic locus encoding the antibody protein that can modify its antigen binding affinity (they undergo type transitions). Via signaling from other GC cell types, the GC is able to monitor the binding phenotype of the B-cell population it contains, and provide survival signals to B-cells with the highest-affinity antibodies (i.e., birth and death rates depend on type). Motivated by this mechanism, we develop a develop a mean-field model that couples the birth and death rates in a focal multi-type birth and death process (MTBDP) with d types to the empirical distribution of states—i.e., the mean-
field —over an exchangeable system of N replica MTBDPs. Using propagation of chaos theory we show that the empirical distribution process of the N replicas converges to a deterministic probability measure-valued flow as N goes to infinity. In the limit, the focal process evolves via McKean-Vlasov-type dynamics governed by the probability measure-valued flow which is in turn the flow of one-dimensional marginal distributions of the focal process.