Antigenic waves of virus-immune coevolution

Abstract

The evolution of many microbes and pathogens, including circulating viruses such as seasonal influenza, is driven by immune pressure from the host population. In turn, the immune systems of infected populations get updated, chasing viruses even farther away. Quantitatively understanding how these dynamics result in observed patterns of rapid pathogen and immune adaptation is instrumental to epidemiological and evolutionary forecasting. Here we present a mathematical theory of coevolution between immune systems and viruses in a finite-dimensional antigenic space, which describes the cross-reactivity of viral strains and immune systems primed by previous infections. We show the emergence of an antigenic wave that is pushed forward and canalized by cross-reactivity. We obtain analytical results for shape, speed, and angular diffusion of the wave. In particular, we show that viral-immune coevolution generates an emergent timescale, the persistence time of the wave's direction in antigenic space, which can be much longer than the coalescence time of the viral population. We compare these dynamics to the observed antigenic turnover of influenza strains, and we discuss how the dimensionality of antigenic space impacts the predictability of the evolutionary dynamics. Our results provide a concrete and tractable framework to describe pathogen-host coevolution.

Keywords: coevolution; fitness wave; host–pathogen dynamics; viral evolution.

Fig. 1.

A simple model of viral–host coevolution predicts the emergence of an antigenic wave. (A) Schematic of the coevolution model. Viruses proliferate while effectively diffusing in antigenic space (here in two dimensions) through mutations, with coefficient $D$. Past virus positions are replaced by immune protections (light blue). Immune protections create a fitness gradient for the viruses (green gradient), favoring strains at the front. Both populations of viruses and immune populations are coarse grained into densities in antigenic space. (B) Snapshot of a numerical simulation of Eqs. 2 and 3 showing the existence of a wave solution. The blue colormap represents the density of immune protections $h\left(\mathbf{x},t\right)$ left behind by past viral strains. The current virus density $n\left(x\right)$ is shown in red. (C) Close-up onto the viral population, showing fitness isolines. The wave moves in the direction of the fitness gradient (arrow) through the enhanced growth of stains at the edge of the wave (black dots). (D) Distribution of fitness across the viral population (corresponding to the projection of B along the fitness gradient). Parameters for B–D: $D/r^2=3\cdot 10^{-9}$, $N_h=10^8$, $\ln R_0=3$, $M=1$.

Fig. 2.

Analytical prediction of wave properties. Shown are the numerical versus analytical predictions for the wave’s population size $N$ (A), speed $v$ (B), width $\sigma$ along the wave’s direction of motion (C), and width ${\sigma }_{\perp }$ in the direction perpendicular to motion (D), with $d=2$ dimensions. Lengths are in units of the cross-reactivity range (so that $r=1$, with no loss of generality). Parameters: $N_h=10^8$ (squares), $10^{10}$ (circles), or $10^{12}$ (triangles); $\ln R_0=1$ (solid symbols) or 3 (open symbols); $M=1$ (small symbols) or 5 (large symbols).

Fig. 3.

Stochastic behavior of the wave: diffusive motion, splits, and extinctions. (A) The wave moves forward in antigenic space but is driven by its nose tip, which undergoes antigenic drift (diffusion) in directions perpendicular to its direction of motion. These fluctuations deviate that direction, resulting in effective angular diffusion. (B) When antigenic drift is large, the wave may randomly split into subpopulations, creating independent waves going in different directions. Each wave can also go extinct as size fluctuations bring it to 0. (C) Cartoon illustrating the wave’s angular diffusion. Selection and drift combine to create an inertial random walk of persistence time $t_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}}$. (D) Analytical prediction (Eq. 17) for the persistence time, versus estimates from simulations. Symbols and colors are the same as in Fig. 2.

Fig. 4.

Rate of speciation. (A) Rescaled rate of splitting events, defined as the emergence of two substrains at distance $\mathrm{\Delta }{x}_0=0.1r$ from each other in antigenic space, meaning that they are becoming antigenically independent. The predicted scaling, $k_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}\sim \left(v^2/D\right)e^{-\mathcal{L}}$, as well as the definition of the collective variable $\mathcal{L}$ as a function of the model parameters, is given by Eq. 20. The line shows a linear fit of the logarithm of the ordinate. (B) Predicted rate of splitting as a function of the dimension $d$, for $R_0=2$, $M=1$, $N_h=10^9$, and $D/r^2=3\cdot 10^{-6}$, with $\mathrm{\Delta }{x}_0=r$. Symbols and colors are the same as in Fig. 2.