Interplay between evolutionary and epidemic time scales challenges the outcome of control policies

Abstract

The SIR model is the cornerstone model for mathematical epidemiology, explaining key epidemic features such as the second-order transition between disease-free and epidemic states, the initial exponential growth of outbreaks or the short-term benefits of control measures. Nonetheless, the classical SIR model assumes that pathogen traits remain fixed, thus neglecting viral evolution. Here we propose a minimal extension of the SIR model, allowing infectiousness to evolve. We show that such evolution can cause superexponential early growth of outbreaks, create abrupt epidemic transitions, and undermine the effectiveness of control policies, as lifting interventions too early can lead to worse epidemic scenarios than no action. We derive analytical expressions for the critical mutation rate and intervention time governing this behavior, and identify a strong asymmetry between control strategies: while shortening the infectious period hinders transmission without suppressing viral evolution, lowering transmission both reduces cases and slows down viral evolution.

Figures (4)

• Figure 1: Effect of infectivity evolution on epidemic trajectories.a Schematic illustration of the epidemic model with evolving infectivity. b-c Temporal evolution of the epidemic prevalence $i(t)$ (b) and basic reproduction number $\mathcal{R}_0(t)$ (c). d Epidemic peak $i_{\max}$ as a function of the initial basic reproduction number $R_0$ for different values of $\mathcal{D}$, revealing an abrupt epidemic diagram induced by infectivity evolution. In panels b and c, dashed lines show the theoretical predictions from Eqs. (\ref{['eq:initial']})-(\ref{['eq:superexp']}), and we set $R_0=2$. In panels b, c and d, we set $i_0=10^{-3}$, $\mu=1/7$, $k=10$ and $\Delta\lambda=10^{-3}$, and stop simulations if $i(t)<10^{-4}$.
• Figure 1: Robustness of the nonmonotonic epidemic impact. Epidemic peak $i_{\max}$ as a function of the intervention duration $\tau$. Columns show the dependence on: a the initial basic reproduction number $R_0$, b the intervention strength $\varepsilon$, and c the activation threshold $i_{\text{thr}}$. Unless otherwise varied, parameters are $R_0=2$, $\varepsilon=0.6$, $i_{\text{thr}}=0$, $i_0=10^{-3}$, $\mu=1/7$, $k=10$, $\mathcal{D}=10^{-6}$ and $\Delta\lambda=10^{-3}$.
• Figure 2: Effect of infectivity evolution on epidemic control. a Epidemic peak $i_{\max}$ under $k-$ or $\lambda-$control as a function of the intervention time $\tau$ for different values of the effective diffusion rate $\mathcal{D}$. b Same as in a for $\mu-$control. In both panels, inset represents the theoretical (red dashed line) prediction obtained with Eq. (\ref{['eq:tau_star']}) and the numerical (blue dots) values for the peak of the critical intervention time yielding the highest $i_{\max}$ value. In both panels we set $i_0=10^{-3}$, $R_0=2$, $\mu=1/7$, $k=10$ and $\varepsilon=0.6$.
• Figure 3: Asymmetric effect of control policies.a Difference in epidemic peak $\Delta i_{\max}$ between the $k,\lambda-$control and $\mu-$control as a function of $\tau$ and $\mathcal{D}$. Dashed curves indicate the theoretical predictions for the critical intervention durations $\tau^{\star}$ associated with each strategy. b-e Temporal evolution of the epidemic prevalence $i(t)$ (top row) and the corresponding time-varying basic reproduction number $\mathcal{R}_0(t)$ (bottom row) for (b) $\tau=30$ and $\mathcal{D}=10^{-6}$, (c) $\tau=75$ and $\mathcal{D}=10^{-6}$, (d) $\tau=200$ and $\mathcal{D}=10^{-8}$ and (e) $\tau=200$ and $\mathcal{D}=10^{-7}$. Panels b-e include the theoretical prediction for the early-time dynamics (red dashed line) and, in the inset, the evolution of the susceptible population $s(t)$. Moreover, we stop simulations if $i(t)<10^{-4}$. In all panels, we set $i_0=10^{-3}$, $R_0=2$, $\mu=1/7$, $k=10$ and $\varepsilon=0.6$. Colors distinguish control strategies, as indicated in panel c, and vertical lines indicate lifting times. Symbols in panel a correspond to the parameter choices shown in panels b-e.