Periodic versus Intermittent Adaptive Cycles in Quasispecies Coevolution

We study an abstract model for the coevolution between mutating viruses and the adaptive immune system. In sequence space, these two populations are localized around transiently dominant strains. Delocalization or error thresholds exhibit a novel interdependence because immune response is conditional on the viral attack. An evolutionary chase is induced by stochastic fluctuations and can occur via periodic or intermittent cycles. Using simulations and stochastic analysis, we show how the transition between these two dynamic regimes depends on mutation rate, immune response, and population size.

Figure 1

(a) Schematic model for the coevolutionary dynamics of virus (V) and immune system (IS). The two populations are subject to mutation and selection (left), but also to ecological interactions (right). (b) Exemplary trajectories of the relative frequencies of virulent strains $x_i$ (top) and corresponding antibodies $y_i$ (bottom). Regular oscillatory dynamics involving three strains turn into simpler two-strain oscillations at $T_{3\to 2}$ and finally transition into intermittency at $T_{\mathrm{int}}$. Genetic variability within the populations is calculated from the average pairwise Hamming distance and is indicated in gray dashed lines. (c) Sketch of the dynamics of the full population distributions in sequence space as described in the text.

Figure 2

(a) Regimes of coevolution. High mutation rates ${\mu }_x$ of the virus lead to population delocalization, while for lower mutation rates a regime of coexistence emerges. Intermediate values lead to a degenerate localization regime for the virus (see text). (c) Steady-state values for relative frequencies $x_p=x_q$ and $y_p=y_q$ as a function of ${\mu }_y$ with ${\mu }_x={\mu }_y$ (above) or ${\mu }_x=0.05$ (below). Solid lines are solutions of Eq. (2) and dots are simulation results. Panels (a) and (c) are for $\gamma =\alpha$, while (b) and (d) show analogous result for $\gamma =10\alpha$, where $L=8$, $n=2$, and $\alpha =10$.

Figure 3

Mean time until transition from regular oscillations to intermittency. Dashed lines are from the analytical result (5). (a) Rescaled simulation data for $L=8$, $\alpha =\gamma =10$, and different choices of ${\mu }_{x/y}$ collapse onto a universal curve (unscaled data shown in the inset). (b) Transition times decrease for increasing $\gamma$ [parameters otherwise as in (a)]. (c) For $n=3$ virulent strains, the transition time $T_{3\to 2}$ until one strain is lost is much shorter than $T_{\mathrm{int}}$, especially for large populations.