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Rapidly evolving pathogens like influenza viruses can persist by accumulatingantigenic novelty fast enough to evade the adaptive immunity of the hostpopulation, yet without continuous accumulation of genetic diversity. Thisdynamical state is often compared to the Red Queen evolving as fast as it canjust to maintain its foothold in the host population: Accumulation of antigenicnovelty is balanced by the build-up of host immunity. Such Red Queen States(RQS) of continuous adaptation in large rapidly mutating populations are wellunderstood in terms of Traveling Wave (TW) theories of population genetics.Here we shall make explicit the mapping of the established Multi-strainSusceptible-Infected-Recovered (SIR) model onto the TW theory and demonstratethat a pathogen can persist in RQS if cross-immunity is long-ranged and itspopulation size is large populations allowing for rapid adaptation. We theninvestigate the stability of this state focusing on the rate of extinction andthe rate of "speciation" defined as antigenic divergence of viral strainsbeyond the range of cross-inhibition. RQS states are transient, but in acertain range of evolutionary parameters can exist for the time long comparedto the typical time to the most recent common ancestor ($T_{MRCA}$). In thisrange the steady TW is unstable and the antigenic advance of the lead strainsrelative to the typical co-circulating viruses tends to oscillate. This resultsin large fluctuations in prevalence that facilitate extinction. We shalldemonstrate that the rate of TW fission into antigenically uncoupled viralpopulations is related to fluctuations of $T_{MRCA}$ and construct a "phasediagram" identifying different regimes of viral phylodynamics as a function ofevolutionary parameters.

Phylodynamics of rapidly adapting pathogens: extinction and speciation of a Red Queen