  The adaptive immune system engages in an arms race with evolving viruses,
trying to generate new responses to viral strains that continually move away
from the set of variants that have already elicited a functional immune
response. In previous work, it has been argued that this dynamical process can
lead to a propagating pulse of ever-changing viral population and concomitant
immune response. Here, we introduce a new stochastic model of viral-host
coevolution and a deterministic approximation thereof with which to study these
dynamics. We show that there is indeed a possible pulse solution, but for a
large host population size, the pulse becomes unstable to the generation of new
infections in its wake, leading to an extended endemic infection pattern. This
time-dependent endemic pattern eventually reaches a fluctuating steady-state
when the infected population size reaches a large enough fraction of the total
number of hosts.
