Effective Reproduction Number

Effective Reproduction Number Rt in Epidemic Monitoring and Infection Models

The effective reproduction number, written Rt, tracks how many onward infections each case is producing at a given point in an epidemic, once immunity and changes in contact or control are taken into account. A value above 1 means case numbers are growing, a value below 1 means they are shrinking, and the gap from 1 shows how much further transmission has to fall for an outbreak to come under control. In health economics Rt matters because the benefit of a vaccine or a control measure depends on how far it moves transmission relative to that threshold, and because Rt estimated from surveillance data gives a check on whether a dynamic transmission model reproduces what is actually happening. Unlike the basic reproduction number, which assumes everyone is susceptible and stays fixed, Rt changes as an epidemic unfolds and is estimated from surveillance case counts.

Rt, R0 and the share of people still susceptible

The WHO guide for standardisation of economic evaluations of immunisation programmes (WHO/IVB/19.10) defines Rt as the number of secondary cases a single infectious individual causes on average in a population at time t. For a homogeneously mixing population without migration it gives the following expression.

$$R_t = p\,c\,D\,x_t = R_0\,x_t$$

where $p$ is the probability of transmission following exposure, $c$ is the average number of new contacts per unit of time, $D$ is the duration of infectiousness, $x_t$ is the proportion of the population susceptible at time $t$, and $R_0 = pcD$ is the basic reproduction number. The ISPOR-SMDM task force report by Pitman and colleagues defines it equivalently, as R0 multiplied by the susceptible fraction of the population. Gostic and colleagues write the instantaneous version for an SIR or SEIR model as $\beta(t)S(t)D$, with $\beta(t)$ the time-varying transmission rate and $S(t)$ the susceptible fraction. Rt therefore moves for two reasons, because infection and vaccination use up susceptible people and because the transmission rate changes with behaviour and control measures.

The WHO guide reads the value as a direction of travel, with new infections per generation increasing when Rt is above 1, decreasing when it is below 1 and stable when it equals 1. Rt held at 1 for a prolonged period means endemic transmission is sustained. R0, in contrast, stays fixed while the population and the agent are unchanged.

Instantaneous and case reproduction numbers

Two different quantities share the name. Fraser defines the case reproduction number as the average number of people someone infected at time t can expect to infect, and the instantaneous reproduction number as the number they could expect to infect should conditions remain unchanged. Cori and colleagues note that an effective control measure should cause a sudden fall in the instantaneous number and a smoother fall in the case number.

The method of Wallinga and Teunis, developed on the 2003 SARS outbreaks, estimates the case reproduction number by reconstructing who infected whom from the epidemic curve, so it needs data from after time t. Cori and colleagues describe the instantaneous Rt as the only reproduction number easily estimated in real time. Gostic and colleagues report that case estimates lead instantaneous ones by roughly one mean generation interval, and recommend the instantaneous measure for near real-time monitoring, with the case measure suited to retrospective work and individual covariates such as age.

Estimating Rt from incidence with the renewal equation

The common estimator links each day's new cases to the cases of earlier days, weighted by how infectious those earlier cases are now. Cori and colleagues describe an infectivity profile $w_s$, a probability distribution over time since infection, and write expected incidence as follows.

$$E[I_t] = R_t\,\Lambda_t, \qquad \Lambda_t = \sum_{s=1}^{t} I_{t-s}\,w_s$$

where $I_t$ is the number of new cases at time step $t$, $I_{t-s}$ is incidence $s$ steps earlier, $w_s$ is the probability that $s$ steps separate infection of a case and infection of the people it infects, and $\Lambda_t$ is the total infectiousness of earlier cases at time $t$. In practice $w_s$ is approximated by the serial interval, the time between symptom onset in a case and in the people it infects, which is easier to observe in households and other closed settings.

With Poisson case counts, Rt held constant over a window of $\tau$ steps and a gamma distribution prior, the posterior distribution is also gamma. Thompson and colleagues set out the result.

$$R_t \mid \text{data} \sim \text{Gamma}\Big(\text{shape} = a + \sum_{k} I_k,\ \ \text{scale} = \frac{1}{\sum_{k} \Lambda_k + 1/b}\Big)$$

where the sums run over the time steps $k$ in the window ending at $t$, and $a$ and $b$ are the shape and scale of the prior. Thompson and colleagues, following Cori and colleagues, use a prior with mean and standard deviation 5, a conservative choice, so an estimate below 1 reflects the data and not the prior. The posterior mean is $(a + \sum I_k)/(\sum \Lambda_k + 1/b)$, and the credible interval comes from gamma quantiles. Gostic and colleagues recommend this method, released in the R package EpiEstim, for near real-time estimation because it uses only past data and minimal parametric assumptions.

The window length is a real choice. On the 1918 Baltimore influenza data, Cori and colleagues found 1-day windows too variable to show a trend, while longer windows gave smoother estimates but delayed the date on which the median fell below 1, and they judged 1 week a good compromise. Precision depends directly on the number of cases in the window, so early estimates are unreliable.

Data problems that bias an estimate

The estimator is simple, and most of the error comes from its inputs. Gostic and colleagues, writing for COVID-19 practice, describe the main problems.

• The generation interval. An assumed mean that is too high pushes Rt further from 1 than the truth, and one that is too low pulls it towards 1. The bias is small near 1 and grows for values far from it, so it may be largest early in an epidemic.
• Delays to observation. If $d$ is the delay between infection and observation, data at time $t$ inform $R_{t-d}$, so estimates from cases, admissions or deaths lag the true value unless the infection series is reconstructed. Shifting the curve back by the mean delay leaves the blurring from variable delays in place.
• Right truncation. Recent infections are not yet reported, so Rt on the latest dates is underestimated unless adjusted or dropped.
• Ascertainment. A constant reporting fraction leaves point estimates unbiased, but changes in testing are read as changes in Rt, and standard intervals omit uncertainty from incomplete observation.

Cori and colleagues also assumed no imported cases, which Thompson and colleagues relax by separating local and imported cases. For a 2009 influenza outbreak in a Pennsylvania school, Cori and colleagues could not tell from Rt alone whether school closure or depletion of susceptible people caused the fall, and a more detailed analysis by Cauchemez and colleagues favoured depletion.

Worked example with a three-day estimation window

All numbers in this example are illustrative. A respiratory infection has a serial interval of 1, 2, 3 or 4 days with probabilities 0.2, 0.4, 0.3 and 0.1, a mean of 2.3 days. Daily symptom-onset counts on days 4 to 10 are 32, 36, 40, 42, 44, 45 and 46. The prior has mean and standard deviation 5, which is shape $a = 1$ and scale $b = 5$, and the window is days 8 to 10.

1. Total infectiousness. For day 8, $\Lambda_8 = 0.2 \times 42 + 0.4 \times 40 + 0.3 \times 36 + 0.1 \times 32 = 8.4 + 16.0 + 10.8 + 3.2 = 38.4$. In the same way $\Lambda_9 = 8.8 + 16.8 + 12.0 + 3.6 = 41.2$ and $\Lambda_{10} = 9.0 + 17.6 + 12.6 + 4.0 = 43.2$.

2. Daily ratios. Cases over total infectiousness give 44/38.4 = 1.146, 45/41.2 = 1.092 and 46/43.2 = 1.065, above 1 and falling.

3. Posterior over the window. The window holds 44 + 45 + 46 = 135 cases and total infectiousness 38.4 + 41.2 + 43.2 = 122.8, so the posterior is gamma with shape 1 + 135 = 136 and rate 122.8 + 1/5 = 123.0.

$$\bar{R} = \frac{136}{123.0} = 1.106, \qquad \text{SD} = \frac{\sqrt{136}}{123.0} = 0.095$$

where $\bar{R}$ is the posterior mean of Rt for days 8 to 10 and SD its posterior standard deviation. The 95% credible interval from the gamma quantiles is 0.93 to 1.30, and the posterior probability that Rt is below 1 is 0.13.

4. Sensitivity to the serial interval. With probabilities 0.1, 0.2, 0.3 and 0.4 (mean 3.0 days), total infectiousness in the window becomes 35.8 + 39.2 + 41.9 = 116.9, the gamma rate is 116.9 + 0.2 = 117.1, and the posterior mean rises to 136/117.1 = 1.161, further above 1, as Gostic and colleagues describe.

5. Effect of a vaccination campaign. Suppose 55% of the population is susceptible and a campaign protects a further 10% of the population, all previously susceptible, leaving 45%. With homogeneous mixing and an unchanged transmission rate, Rt scales with the susceptible share and falls to 1.106 × 0.45/0.55 = 0.905. The new value is below 1 when the current Rt is below 0.55/0.45 = 1.222, which the posterior puts at probability 0.89.

6. Cases over the next four generations. Starting from 100 cases and holding Rt fixed, the next four generations contain 100 × (1.106 + 1.223 + 1.353 + 1.496) = 518 cases without the campaign and 100 × (0.905 + 0.819 + 0.741 + 0.671) = 314 with it, about 204 fewer.

7. Interpretation. Transmission is probably still slightly above the threshold, and the campaign is likely to take it below 1. The 204 figure is a short-run sketch that ignores further depletion in both arms and assumes perfect mixing, so a cost-effectiveness estimate would need the dynamic model itself.

Rt in dynamic transmission models and economic evaluation

In economic work Rt is more often a model output or a check on a model than a direct input. Pitman and colleagues list changes in the effective reproduction number among outcomes appropriate for reporting from a dynamic model, alongside the long-run equilibrium of infection and the likelihood of elimination. An Rt series estimated from surveillance data can serve as a target in calibration of a transmission model, since a model that cannot reproduce it is a weak basis for valuing an intervention.

Timing matters for value as well as size. Wallinga and Teunis estimated that delaying SARS control measures by 1 week would have nearly tripled the epidemic size and lengthened the expected epidemic by 4 weeks, so how quickly an Rt estimate detects a change has economic weight. Uncertainty in an Rt estimate and its serial interval belongs in probabilistic sensitivity analysis, while the choice between instantaneous and case measures, or between delay adjustments, is closer to structural uncertainty.

Rt beside neighbouring quantities

Nearby measures describe transmission in other units. The force of infection is the rate at which susceptible people are infected per unit of time, while Rt counts cases per case and has no time unit. Converting an exponential growth rate into a reproduction number requires the generation interval. For an exponentially distributed interval, Wallinga and Lipsitch give $R = 1 + rT_c$, with $r$ the growth rate and $T_c$ the mean generation interval. The herd immunity threshold is the immunity level at which Rt reaches 1, and the resulting indirect protection is covered under vaccination externalities. The SIR model and SEIR model pages give the compartment structures behind it.

Misreadings of Rt

Many errors with Rt come from treating an estimate as more precise or more causal than its data allow. Most concern what the number leaves out.

1. Reading Rt as a measure of size. A value of 0.9 in a large epidemic still produces many cases and costs.
2. Taking a fall as proof that a measure worked. Depletion of susceptible people, seasonality and changes in contact all lower Rt, and separating them needs a model.
3. Ignoring the lag. Estimates from admissions or deaths describe transmission days or weeks earlier.
4. Trusting the credible interval as complete. Standard intervals leave out uncertainty from reporting delays and changing ascertainment, and from the generation interval unless it is sampled.
5. Treating 1 as a decision rule. The value of a measure depends on the cases and costs it averts over the whole epidemic, which an Rt estimate on one date does not show, so the comparison belongs in a full cost-effectiveness analysis.

Sources

• Cori A, Ferguson NM, Fraser C, Cauchemez S. A new framework and software to estimate time-varying reproduction numbers during epidemics. American Journal of Epidemiology 2013;178(9):1505-1512. https://doi.org/10.1093/aje/kwt133
• Fraser C. Estimating individual and household reproduction numbers in an emerging epidemic. PLoS ONE 2007;2(8):e758. https://doi.org/10.1371/journal.pone.0000758
• Gostic KM, McGough L, Baskerville EB, Abbott S, Joshi K, Tedijanto C, et al. Practical considerations for measuring the effective reproductive number, Rt. PLoS Computational Biology 2020;16(12):e1008409. https://doi.org/10.1371/journal.pcbi.1008409
• Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. Dynamic transmission modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health 2012;15(6):828-834. https://doi.org/10.1016/j.jval.2012.06.011
• Thompson RN, Stockwin JE, van Gaalen RD, Polonsky JA, Kamvar ZN, Demarsh PA, et al. Improved inference of time-varying reproduction numbers during infectious disease outbreaks. Epidemics 2019;29:100356. https://doi.org/10.1016/j.epidem.2019.100356
• Wallinga J, Lipsitch M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proceedings of the Royal Society B 2007;274(1609):599-604. https://doi.org/10.1098/rspb.2006.3754
• Wallinga J, Teunis P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. American Journal of Epidemiology 2004;160(6):509-516. https://doi.org/10.1093/aje/kwh255
• World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes, 2nd edition (WHO/IVB/19.10). Geneva: WHO; 2019. Section 6.1.1. https://iris.who.int/handle/10665/329389