R value: How it is determined

The effective reproductive number, R, shows how many new infections can be caused by an existing infectious individual. As such, it is an important indicator in helping us understand how fast a virus is spreading.

For instance, assuming there are 100 people infected with the Covid-19 virus in the population today, an R of 2.5 means these people are going to infect 250 others by the next generation cycle, D (the infectiousness period in days). These newly infected 250 people will then go on to infect another 625 people (250 x 2.5) by the next D and so on. So, when R is more than 1, the number of new cases rises exponentially. Conversely, the number of new cases would drop over time when R is less than 1 (see Chart 1).

To put this another way, think of

R as compounding interest paid by the bank on money deposited. The main difference is that interest is paid on all money in a real bank account, whereas R is paid only on new money in our hypothetical account. This is because money in the bank generates interest income as long as it remains in the account, but an infected person “generates” new infections only for a limited number of days ( D).

Measures implemented by governments during the pandemic — such as the Movement Control Order, mandatory mask wearing, social distancing and vaccinations — serve the same purpose, which is to reduce R to less than 1 and thus reduce the expected number of new infections.

Knowing the actual R is crucial, as it will help governments determine the extent of a lockdown, whether an extension to the existing lockdown is required or whether movement restrictions can be safely relaxed without triggering a fresh outbreak.

During an outbreak, when R > 1, we can reduce R by lowering the average number of contacts by a fraction, p, and the level of risk of each contact by another fraction, q. Based on mathematical derivation, we know that R can be lowered when (1 - p) x (1 - q) ≤ 1/R. Thus, a higher

R will require more stringent movement controls (higher p) and standard operating procedures such as face masking (higher q) to contain the spread.

Similarly, when the number of cases is declining ( R < 1), we can plan to reopen the economy. This will mean an increase in the average number of contacts but, based on the equation, we know the number of new cases can continue to decline ( R remains below 1) as long as

(1 + p) x (1 + q) ≤ 1/R. Thus, a lower R can still accommodate a more rapid reopening (higher p) without risking another outbreak (for further reading, here is the full report: https://www.hbs.edu/ris/ Publication%20Files/20-112_4278525dccf2-4f8a-b564-2e95d0e7ca5b.pdf).

Clearly, R is an important parameter, but it is almost impossible to determine the actual R, owing to limitations in data gathering. We can only estimate R, the accuracy of which is largely dependent on the correctness of the mathematical model used and the quality of data available.

An important element in the calculation is that the probability of a person infecting others changes over time, starting from the day he contracted the virus. For instance, a person normally becomes infectious only three days after contracting the virus, thus the probability of him infecting others during the first few days is low. Assuming an incubation period of five days, the likelihood of him infecting others is the highest on the fourth and fifth days, when he is infectious but asymptomatic. When the person becomes symptomatic on the sixth day, he is likely to be isolated and, thus, the probability of him infecting others falls from that time onwards.

The probability distribution can be represented using the curve in Chart 2. Note that the infectiousness of a person increases from the time he gets infected, peaks around the time when he becomes symptomatic and decreases to zero eventually as he recovers. The whole cycle is the aforementioned “generation cycle”. When a virus is transmitted from one person to another, a new generation cycle is created, and the number of infections caused during the new generation cycle is R.

In short, to calculate a meaningful

Rt (R at time t), we need to be able to a) identify as many infected individuals as possible; and b) determine the number of days since they got infected. Currently, there is a significant time gap between the moment a person contracts the virus and when he is tested positive. Pinpointing the moment when one first contracts the virus is difficult, especially when more than 70% of reported cases are unlinked to any known source.

The only way to have a more accurate estimation is to detect the infections as early as possible — through intensified and regular testing. By ramping up testing, the number of reported cases will also be closer to the number of actual infections, thus improving the accuracy of calculated Rt. Only by having better-quality data can we make better decisions for controlling the outbreak.

Many input assumptions are needed to estimate the probability distribution — for instance, the distribution function (Bayesian or gamma distribution), generation cycle time and appropriateness of the test sample. A more comprehensive review of such estimates and historical public health responses is found here: “A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics” (https://academic.oup.com/aje/ article/178/9/1505/89262).

It is why epidemiologists and other experts are required to make this assessment. Because the Ministry of Health does not share its data with the public, however, it is near impossible for these epidemiologists to offer their conclusions, which may or may not be the same as that determined by the MoH. Even in the best of scenarios, complicated modelling is riddled with uncertainties and, where decisions are predicated on data sampling and statistical assumptions, perhaps greater transparency will lead to more independent analyses and generate higher confidence with intellectual discussions.