  The evolution of many microbes and pathogens, including circulating viruses
such as seasonal influenza, is driven by immune pressure from the host
population. In turn, the immune systems of infected populations get updated,
chasing viruses even further away. Quantitatively understanding how these
dynamics result in observed patterns of rapid pathogen and immune adaptation is
instrumental to epidemiological and evolutionary forecasting. Here we present a
mathematical theory of co-evolution between immune systems and viruses in a
finite-dimensional antigenic space, which describes the cross-reactivity of
viral strains and immune systems primed by previous infections. We show the
emergence of an antigenic wave that is pushed forward and canalized by
cross-reactivity. We obtain analytical results for shape, speed, and angular
diffusion of the wave. In particular, we show that viral-immune co-evolution
generates a new emergent timescale, the persistence time of the wave's
direction in antigenic space, which can be much longer than the coalescence
time of the viral population. We compare these dynamics to the observed
antigenic turnover of influenza strains, and we discuss how the dimensionality
of antigenic space impacts on the predictability of the evolutionary dynamics.
Our results provide a rigorous and tractable framework to describe
pathogen-host co-evolution.
