  An interesting inference drawn by some Covid-19 epidemiological models is
that there exists a proportion of the population who are not susceptible to
infection -- even at the start of the current pandemic. This paper introduces a
model of the immune response to a virus. This is based upon the same sort of
mean-field dynamics as used in epidemiology. However, in place of the location,
clinical status, and other attributes of people in an epidemiological model, we
consider the state of a virus, B and T-lymphocytes, and the antibodies they
generate. Our aim is to formalise some key hypotheses as to the mechanism of
resistance. We present a series of simple simulations illustrating changes to
the dynamics of the immune response under these hypotheses. These include
attenuated viral cell entry, pre-existing cross-reactive humoral
(antibody-mediated) immunity, and enhanced T-cell dependent immunity. Finally,
we illustrate the potential application of this sort of model by illustrating
variational inversion (using simulated data) of this model to illustrate its
use in testing hypotheses. In principle, this furnishes a fast and efficient
immunological assay--based on sequential serology--that provides a (i)
quantitative measure of latent immunological responses and (ii) a Bayes optimal
classification of the different kinds of immunological response (c.f., glucose
tolerance tests used to test for insulin resistance). This may be especially
useful in assessing SARS-CoV-2 vaccines.
