Figure 3. Extinction, speciation, and oscillations in multi-strain SIR models.

(A) Typical trajectories of infection prevalence in the parameter regimes corresponding to extinction (red), traveling wave RQS (yellow) and oscillatory RQS (green). Panels B and C schematically show parameter regimes corresponding to these qualitatively different behaviors. As explained in the text the boundaries $q_{ex},q_{sp}$ of the RQS regime depend explicitly on the time scale considered (see also Figure 7). Simulation results supporting this diagram are shown in Figure 3—figure supplements 1 and 2. Panel (D) is a schematic illustrating how a novel pandemic strain (red) can settle into an endemic RQS state. As the cumulative number of infected individuals increases, the susceptible fraction decreases, and survival of the strain depends on the emergence of antigenic escape mutations (gray). The top part of the panel illustrates the population composition at a particular time point. Rare pioneering variants are $q$ mutations ahead of the dominant variant and grow with rate $x_n$. (Note, that boundaries of the ‘extinction’ regime in (B,C) correspond to $q$ value close to one.) Different lineages are related via their phylogenetic tree embedded in the fitness distribution in the population.

Figure 3—figure supplement 1. ‘Phase diagram’ in the clone-based simulation.

The data points are assigned to phases using the following criteria. The extinction phase (red): ${\tau }_{\mathrm{ext}}<10{\tau }_{\mathrm{sw}}$; The speciation phase (blue): ${\tau }_{\mathrm{sp}}<10{\tau }_{\mathrm{sw}}$; The transient RQS regime (yellow and green): otherwise. Red and blue lines correspond to the extinction boundary $q_{ext}$ and speciation boundary $q_{sp}$ respectively, defined for $\tau =10{\tau }_{\mathrm{sw}}$. The black solid line corresponds to the onset of spontaneous oscillation $\log (N_h\overline{I})=8.3$. Color in the transient phase labels the amplitude of prevalence $\mathrm{var}(\log I_{tot})$, increasing from yellow to green. Simulation set $d=50$.

Figure 3—figure supplement 2. Value of $q$ across the ‘Phase diagram’.

This figure shows the same data as one with each data point colored by the corresponding value of $q$, which is constant along the straight of different angle.