Time-dependent heterogeneity leads to transient suppression of the COVID-19 epidemic, not herd immunity

Abstract

Epidemics generally spread through a succession of waves that reflect factors on multiple timescales. On short timescales, superspreading events lead to burstiness and overdispersion, whereas long-term persistent heterogeneity in susceptibility is expected to lead to a reduction in both the infection peak and the herd immunity threshold (HIT). Here, we develop a general approach to encompass both timescales, including time variations in individual social activity, and demonstrate how to incorporate them phenomenologically into a wide class of epidemiological models through reparameterization. We derive a nonlinear dependence of the effective reproduction number [Formula: see text] on the susceptible population fraction S. We show that a state of transient collective immunity (TCI) emerges well below the HIT during early, high-paced stages of the epidemic. However, this is a fragile state that wanes over time due to changing levels of social activity, and so the infection peak is not an indication of long-lasting herd immunity: Subsequent waves may emerge due to behavioral changes in the population, driven by, for example, seasonal factors. Transient and long-term levels of heterogeneity are estimated using empirical data from the COVID-19 epidemic and from real-life face-to-face contact networks. These results suggest that the hardest hit areas, such as New York City, have achieved TCI following the first wave of the epidemic, but likely remain below the long-term HIT. Thus, in contrast to some previous claims, these regions can still experience subsequent waves.

Keywords: COVID-19; epidemic theory; heterogeneity; overdispersion.

Fig. 1.

$R_e/R$ vs. $S$ dependence for gamma-distributed susceptibility with $\lambda =3\pm 1$ (light blue area). The dashed line shows the classical homogeneous result, $R_e=R_{0}S$. The dark blue region corresponds to the range $2<R_0<3.5$ representative for COVID-19.

Fig. 2.

TCI threshold (dot-dashed), long-term heterogeneous HIT (solid), and homogeneous HIT (dotted) for various values of $R_0$ (x axis). HIT (solid line) is determined by persistent heterogeneity. The corresponding immunity factor ${\lambda }_{\infty }\approx 2$ was estimated from Eq. 20 assuming strong short-term overdispersion (${\chi }^{*}\ll 1$) and the exponential distribution of $\alpha$ ($\eta =1$). For transient behavior, ${\lambda }_{\mathrm{e}\mathrm{ff}}\approx 4$ is assumed based on analysis of empirical data for COVID-19 epidemic in select locations.

Fig. 3.

Correlation between the relative reduction in the effective reproduction number $R_e\left(t\right)/R_0$ (y axis) with the susceptible population $S\left(t\right)$. (A) The progression of these two quantities for NYC and Chicago, as given by the epidemiological model described in ref. . (B) The scatter plot of $R_e\left(t_0\right)/R_0$ and $S\left(t_0\right)$ in individual states of the United States, evaluated in ref. ($t_0$ is the latest date covered in that study).

Fig. 4.

Projections of daily deaths under the hypothetical scenario in which any mitigation is completely eliminated as of June 15, 2020, for (A) NYC and (B) Chicago. Different curves correspond to different values of the transient immunity factor ${\lambda }_{\mathrm{e}\mathrm{ff}}=1$ (blue), 3 (red), 4 (green), and 5 (black lines). The model described in ref. was fully calibrated on daily deaths (circles), ICU occupancy, and hospitalization data up to the end of May. See SI Appendix for additional details, including CIs.

Fig. 5.

Effect of social rewiring on the epidemic dynamics. The time course of an epidemic in a heterogeneous SIR model with $R_0=2.5$ and $\lambda =3$. During the first 100 d, a mitigation factor $\mu =0.7$ is applied. Social networks gradually rewire with a time constant ${\tau }_s=150$ d. The figure shows multiple waves. (Inset) $R_e\left(t\right)$ plotted as a function of $S\left(t\right)$. Solid black line shows the homogeneous limit reached after multiple waves.