Pulse Instabilities Can Shape Virus-Immune Co-evolution

Adaptive immune systems engage in an arms race with evolving viruses, trying to generate new responses to viral strains that continually move away from the set of genetically-varying strains that have already elicited a functional immune response. It has been argued that this dynamical process can lead to a propagating pulse of an ever-changing viral population and concomitant immune response. Here, we introduce a new stochastic model of viral-host co-evolution, taking into account finite-sized host populations and varying processes of immune "forgetting". Using both stochastic and determinstic calculations, we show that there is indeed a possible pulse solution, but for a large host population size and for finite memory capacity, the pulse becomes unstable to the generation of new infections in its wake. This instability leads to an extended endemic infection pattern, demonstrating that the population-level behavior of virus infections can exhibit a wider range of behavior than had been previously realized.