Eco-evolutionary constraints for the endemicity of rapidly evolving viruses

TL;DR

It is found that the evolution in both traits during the first epidemic wave plays a critical role in determining long-term viral persistence, and the results prove that the long-term behavior of epidemic trajectories hinges on the complex interplay between both evolutionary pathways and the underlying contagion dynamics.

Abstract

Antigenic escape constitutes the main mechanism allowing rapidly evolving viruses to achieve endemicity. Beyond granting immune escape, empirical evidence also suggests that mutations of viruses might increase their inter-host infectiousness. While both mechanisms are well-studied individually, their combined effects on viral endemicity remain to be explored. Here we propose a minimal eco-evolutionary framework to simulate epidemic outbreaks generated by pathogens evolving both their infectiousness and immune escape. Our results reveal that the main driver of viral evolution shifts over time: from intrinsic selection for infectiousness at early stages of the outbreak to antigenic diversification in the transition to the endemic phase. We find that the evolution in both traits during the first epidemic wave plays a critical role in determining long-term viral persistence. Evolution in infectiousness enhances the endemicity of viruses, especially in viruses with lower baseline infectiousness due to the longer duration of their first epidemic wave. Likewise, control policies flattening epidemic curves might increase viral endemicity as a result of the greater antigenic diversity generated in the prolonged epidemic waves. Our results thus prove that the long-term behavior of epidemic trajectories hinges on the complex interplay between both evolutionary pathways and the underlying contagion dynamics.

Figures (10)

• Figure 1: Schematic of the eco-evolutionary model here introduced. A) Modified Susceptible-Infected-Recovered (SIR) model to account for the reinfection events driven by the immune escape of the virus. Each agent $j$, when infected, recovers at a rate $\mu$, keeping their associated antigenic position $x_j$. Primary infections caused by an infected agent $j$ occur at a rate $\lambda_j$, whereas the contagion rate for a recovered individual $i$ in contact with that infected individual is $\lambda^\prime_{ij}=\Theta(x_j-x_i)\lambda_j \left(1-e^{-(x_j-x_i)}\right)$, where $\Theta(x)$ represents the Heaviside function. Note that the latter increases with the antigenic distance between variants and that reinfection is only possible when $x_j>x_i$, thus assuming that natural immunity against a variant does not wane over time. B) Evolutionary processes of the model. On the one hand, recovered individuals keep their antigenic position. On the other hand, infected individuals evolve both the infectiousness of the virus and their antigenic position, assuming that changes in trait $m$ ($m \in \left\lbrace x,\lambda \right\rbrace$) are drawn from normal distributions, i.e. $\Delta_m \sim \mathcal{N}(0,D^2_m)$, with $D^2_m$ determining its speed of evolution.
• Figure 2: Epidemic trajectories for viruses not evolving their infectiousness. A) Time dependence of the fraction of population in the infected state $\rho_I$. B): Time evolution of the variance of the distribution of variants in the antigenic space $\sigma_x^2$. In both panels, thick solid (dashed) lines shows the values for those viruses becoming (not becoming) endemic in the population obtained by averaging 200 epidemic outbreaks whereas thin lines represent a sample of 20 individual trajectories in both cases. In these panels, the basic reproduction number of the wild-type variant of the pathogen ${\mathcal{R}_0^{wt}}$ is set to $\mathcal{R}^{wt}_0$=3. C): Fraction of epidemic outbreaks surviving in the population after $t=1000$ days $f_{endemic}$ as a function of ${\mathcal{R}^{wt}_0}$. The results shown in this panel have been obtained by simulating $1000$ epidemic outbreaks for each pair of ($\mathcal{R}^{wt}_0$,$D_x$) values. In all panels, line color represents the value of speed of evolution in antigenic position $D_x$. For the simulations, in all panels we assume $N=10^4$ individuals, $I_0=10$ initially infected agents, and we set the recovery rate to $\mu=1/7$ days$^{-1}$ and the number of daily contacts to $k=10$ interactions.
• Figure 3: Eco-evolutionary dynamics under evolution of antigenic and non-antigenic traits. A)-D): Time evolution of different epidemiological quantities and virus traits in endemic epidemic outbreaks. The quantities shown correspond to: (A) fraction of infected population $\rho_I$, (B) basic reproduction number ${\cal R}_0$, (C) variance of the distribution of strains across the antigenic space $\sigma^2_x$ and (D) an estimation of the case reproduction number $R^{app}_{case}$ (see Appendix \ref{['sec:appendixC']}). In all these panels, line color corresponds to the basic reproduction number of the wild-type variant ${\cal R}_0^{wt}$.The symbol $\langle \cdot \rangle$ denotes that each curve is the result of averaging the individual curves of all endemic realizations observed after simulating $1000$ epidemic outbreaks for each ${\cal R}_0$ value. The speeds of evolution in the infectiousness and antigenic spaces are set to $D_\lambda=0.0003$ and $D_x=0.015$ respectively. E): Endemicity $f_{endemic}$ of the virus as a function of the basic reproduction number of the wild-type variant ${\cal R}_0^{wt}$. Line color here represents the speed of evolution in the antigenic space $D_x$. Solid (dotted) lines represent the values found in presence (absence) of evolution in the infectiousness space by setting $D_\lambda=0.0003$ ($D_\lambda=0$). In all the panels, the rest of epidemiological parameters are the same as in Fig. \ref{['fig:2']}.
• Figure 4: Endemicity of viruses under deterministic evolution. A)-D): Fraction of endemic realizations $f_{endemic}$ as a function of the basic reproduction number of the wild-type variant $\mathcal{R}^{wt}_0$ and the speed of evolution in the antigenic space $\widetilde{D}_x$. The white solid line shows the theoretical estimation of the critical immunity escape value $\widetilde{D}^C_x$ delimiting the region $f_{endemic}=1$. Such quantity is obtained by setting $a=1/12$ and $b=5$ in Eq. \ref{['eq:Dimmtrans']}. The values considered for the speed of evolution in infectiousness are: (A) $\widetilde{D}_\lambda=0$, (B) $\widetilde{D}_\lambda=4\cdot 10^{-5}$, (C) $\widetilde{D}_\lambda=8\cdot 10^{-5}$ and (D) $\widetilde{D}_\lambda=1.2\cdot 10^{-4}$. In all panels, the fraction of endemic realizations is obtained by performing 500 epidemic outbreaks and computing those persisting in the population after $t=1000$ days. The rest of model parameters are the same as in Fig. \ref{['fig:2']}.
• Figure 5: Trade-off between short-term and long-term benefits of control policies. A): Time evolution of the epidemic prevalence $\rho_I$. Line color denotes the duration of the control policies $\tau_{control}$. B): Distribution of epidemic peak as a function of $\tau_{control}$. C): Time evolution of the antigenic diversity $\sigma_x^2$. Line color denotes the value of $\tau_{control}$. D): Distribution of antigenic diversity at the epidemic bottleneck across endemic realizations, $(\sigma^2_x)^{bottleneck}$, as a function of $\tau_{control}$. We define the epidemic bottleneck as the point with minimum incidence following the first epidemic wave. In panels A) and C) thick lines represent average values across endemic realizations whereas thin lines correspond to single realizations. In panels B) and D), the dot shows the mean of the distribution whereas the whiskers denote its IQR. In all these panels, the (constant) speed of antigenic evolution is set to $\widetilde{D}_x = 0.0003$. E): Fraction of endemic realizations as a function of $\tau_{control}$. Line color encodes the value of $\widetilde{D}_x$. The criterion used to classify a realization as endemic is the same as the one used in Fig. \ref{['fig:4']}. In all panels, we do not consider evolution in infectiousness, i.e. $\widetilde{D}_\lambda=0$ and the epidemiological parameters are the same as in Fig. \ref{['fig:2']}.