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Rapidly evolving pathogens like influenza viruses can persist by accumulating antigenic novelty fast enough to evade the adaptive immunity of the host population, yet without continuous accumulation of genetic diversity. This dynamical state is often compared to the Red Queen evolving as fast as it can just to maintain its foothold in the host population: Accumulation of antigenic novelty is balanced by the build-up of host immunity. Such Red Queen States (RQS) of continuous adaptation in large rapidly mutating populations are well understood in terms of Traveling Wave (TW) theories of population genetics. Here we shall make explicit the mapping of the established Multi-strain Susceptible-Infected-Recovered (SIR) model onto the TW theory and demonstrate that a pathogen can persist in RQS if cross-immunity is long-ranged and its population size is large populations allowing for rapid adaptation. We then investigate the stability of this state focusing on the rate of extinction and the rate of speciation defined as antigenic divergence of viral strains beyond the range of cross-inhibition. RQS states are transient, but in a certain range of evolutionary parameters can exist for the time long compared to the typical time to the most recent common ancestor. In this range the steady TW is unstable and the antigenic advance of the lead strains relative to the typical co-circulating viruses tends to oscillate. This results in large fluctuations in prevalence that facilitate extinction. We shall demonstrate that the rate of TW fission into antigenically uncoupled viral populations is related to fluctuations of diversity and construct a phase diagram identifying different regimes of viral phylodynamics as a function of evolutionary parameters.

• the directory simulation_data contains raw data in zip files of extinction and speciation time from the simulation of FluEpiTreeNM.m and extinction time from the simulation of mfdist.m. Data_antid_Sp.zip, Data_antid_Ext.zip, Data_popN.zip, and Data_beta.zip are from the multi-strain SIR simulation FluEpiTreeNM.m, where _antid implies the crossimmunity range d is varied, _popN implies the host population N is varied, and _beta implies the effect of mutation in infectivity is tuned in the set of data. Data_mf.zip is from the stochastic fitness class simulation mfdist.m. In the unzipped folders, Extinct files record the time between successive extinction events. WaitTime files record the time interval to the most recent common ancester (MRCA), time to the MRCA (TMRCA), generations to the MRCA, and total number of strains in the clone at each TMRCA advance event. The file names record the input parameters in the following format: Extinct\WaitTime_birth rate gamma_mutation rate m_selection coefficient s_log10 of population N_replica number.dat. The average of the extinction and speciation time of the corresponding data are recorded in the subdirectory ExtSpTime.
• the MATLAB file mfdist.m implements the stochastic fitness class simulation of a large population in a one dimensional landscape. It the was used to produce simulation data on extinction times.
• the file pop2tw.m is an analogous fitness class simulation of two coupled populations to investigate the speciation behavior.
• the file FluEpiTreeNM.m implements a tree structured viral population use to simulate the SIR model.