What the reproductive number R0 can and cannot tell us about COVID-19 dynamics

Abstract

The reproductive number R (or R₀, the initial reproductive number in an immune-naïve population) has long been successfully used to predict the likelihood of pathogen invasion, to gauge the potential severity of an epidemic, and to set policy around interventions. However, often ignored complexities have generated confusion around use of the metric. This is particularly apparent with the emergent pandemic virus SARS-CoV-2, the causative agent of COVID-19. We address some misconceptions about the predictive ability of the reproductive number, focusing on how it changes over time, varies over space, and relates to epidemic size by referencing the mathematical definition of R and examples from the current pandemic. We hope that a better appreciation of the uses, nuances, and limitations of R and R₀ facilitates a better understanding of epidemic spread, epidemic severity, and the effects of interventions in the context of SARS-CoV-2.

Keywords: COVID-19; Epidemic size; Heterogeneity; Interventions; SARS-CoV-2.

Fig. 1

Estimates of the ${\mathcal{R}}_0$ of SARS-CoV-2 vary substantially between locations (Binny et al., 2020, Chen et al., 2020, Choi and Ki, 2020, Deb and Majumdar, 2020, Giordano et al., 2020, Johndrow et al., 2020, Ke et al., 2020, Korolev, 2021, Lewnard et al., 2020, Li et al., 2020b, Majumder and Mandl, 2020, Mizumoto et al., 2020, Peirlinck et al., 2020, Pitzer et al., 2020, Ranjan, 2020, Read et al., 2020, Riou and Althaus, 2020, Sanche et al., 2020, Senapati et al., 2020, Shim et al., 2020, Singh and Adhikari, 2020, Tang et al., 2020, Wu et al., 2020, Xiao et al., 2020, Yuan et al., 2020, Zhao et al., 2020). Each point represents a literature-compiled average ${\mathcal{R}}_0$ estimate for a different geographic area (sample size noted alongside means, error bars show plus or minus 1 standard error). For individual studies that provided multiple estimates for a single geographic area, the median estimate was used to avoid pseudo-replication. An analysis (not shown) confirmed that ${\mathcal{R}}_0$ estimation method (transmission model, exponential growth model, or stochastic simulation method) did not drive the pattern of variation in ${\mathcal{R}}_0$ by location. Recent meta-analyses of ${\mathcal{R}}_0$ values for SARS-CoV-2 consider the effects of estimation methods in more detail (Alimohamadi et al., 2020, Barber et al., 2020).

Fig. 2

Simulation of transmission model of SARS-CoV-2 presented in Eqs. (2.1)–(2.6) for (left) ${\mathcal{R}}_0=2$ and (right) ${\mathcal{R}}_0=3$. We achieved this change in ${\mathcal{R}}_0$ by altering the parameter $\lambda$, which is the per day detection rate of symptomatic cases, thus illustrating the potential value of rapid case detection. Note that a larger ${\mathcal{R}}_0$ leads to a faster epidemic and a greater fraction of the population becoming infected. The horizontal, dotted gray line in each panel indicates the herd immunity threshold (i.e. the value of $S$ that corresponds to $\mathcal{R}=1$). Note that infection declines, but does not immediately disappear, after crossing the herd immunity threshold. Parameter values: ${\beta }_A=0.12,{\beta }_P=0.4,{\beta }_C=0.4,\phi =0.50,\nu =0.25,{\gamma }_{\mathrm{A}}=0.1,{\gamma }_{\mathrm{C}}=0.1,{\gamma }_{\mathrm{Q}}=0.1,{\alpha }_{\mathrm{C}}=0.001,{\alpha }_Q=0.001$,(left) $\lambda =0.2323$, (right) $\lambda =0.024$. Initial conditions: $S\left(0\right)=1,I_A\left(0\right)=0.0001,I_P\left(0\right)=0.0001,I_C\left(0\right)=0,Q\left(0\right)=0,R\left(0\right)=0$.

Fig. 3

Epidemic size contours and shading show that when ${\mathcal{R}}_0$ is close to 1, the epidemic is more strongly influenced by a reduction of $I_0$ than by a reduction of ${\mathcal{R}}_0$. For instance, if ${\mathcal{R}}_0=0.95$ (dashed line), the epidemic could infect from less than 0.1% to greater than 16% of the population as $I_0$ ranges from just above 0% to 3% of the population. Epidemic size was calculated using the final size equation, $Z=S_0(1-e^{-{\mathcal{R}}_0\left(Z-I_0\right)})$, where $S_0=1$. Shading indicates the cube root of epidemic size with lighter colors corresponding to smaller outbreaks.