  Lymphocyte selection is a fundamental operation of adaptive immunity. In
order to produce B-lymphocytes with a desired antigenic profile, a process of
mutation-selection occurs in the germinal center, which is part of the lymph
nodes. We introduce in this article a simplified mathematical model of this
process, taking into account the main mechanisms of division, mutation and
selection. This model is written as a non-linear, non-local, inhomogeneous
second order partial differential equation, for which we develop a mathematical
analysis in the case of piecewise-constant coefficients. We assess,
mathematically and numerically, the performance of the biological function by
evaluating the duration of this production process as a function of several
parameters such as the mutation rate or the selection profile, in various
asymptotic regimes.
