Day in and day out, the Director General of Health broadcasts the number of daily positive cases. Occasionally, the press release is sprinkled with additional details on deaths, ICU utilisation and the Rt (R at a particular point in time, t). R is the effective rate of reproduction, or effective reproductive number — it is the expected number of new infections caused by an infected individual in a population, where some individuals may no longer be susceptible.
Other commentators then jump in to provide more colour. Of late, the flavour of the day is the “daily test positivity rate” — the number of positive cases divided by the number of tests for the day (although we can safely assume the tests were not all taken on the same day). These commentators will go on to postulate how many more positive cases there will be if we were to increase the number of tests. We see viral news and videos, with nice charts and well-spoken doctors, saying how the number of positive cases will double if we double the number of tests. You sample a part of the population because you cannot test the entire population daily. The reason for sampling is to gauge its representation of the population, to facilitate making informed decisions, assuming the sample is unbiased (or corrected for intentional biasness). That is all.
But the point that more tests should and must be done is true. Elementary statistics tell you the larger the sample size, the smaller the margin of error — since the margin of error is zero if your sample is the entire population, but there are diminishing returns up to a point. And we will also argue in the rest of this article that more testing is critical to turn this pandemic into an endemic.
Many would quote the numbers and give their personal prognosis and prescriptions, like true epidemiologists. And with fear and anger, rooted in a deeply divided society, all of us fall into our own echo chambers, ignoring facts and science.
We are drawn to write this knowing we are not experts; we are not epidemiologists (although we have spoken to a few experts as well as to public healthcare officials). We know a little math and statistics and we are decently analytical and rational. We have some communication skills, which we hope can help us articulate and foster more understanding on this difficult subject. We aim to explain the mechanics behind the numbers and what they REALLY mean. We believe that knowing the limitations of what the numbers are (that positive cases and R are merely estimates of the reality) might help steer conversations meaningfully and remove “over-interpretation” of the results, thus giving a more appropriate level of confidence to those numbers and direct attention to others that matter also.
Another reason for writing this is to dispel the myth that there is no trade-off between lives and livelihoods. Those who argue that only lives matter, that we shut down the economy to drive this pandemic away quickly are likely those who do not have to work. It is understandable if you have inherited a fortune and are spending afternoons having tea, served by maids or waiters. The fact is that most peoples’ lives depend on having a livelihood. And it is this difficult balance that necessitates the need to understand the Covid-19 numbers and decide on the appropriate and balanced course of action.
Testing (sampling) will give us the key data necessary to manage the pandemic.
To understand how prevalent and widespread the virus is, how fast it is transmitting from one person to the next and where it is originating, we need massive data, which can be derived only from a massive number of tests since we cannot see the virus and, worse, it is likely that the majority of those who are carriers of the virus are asymptomatic (no signs of detectable sickness). And these tests need to be repeated on the same people over time intervals since the virus cycle appears to be about 14 days or less. Those tested negative today can come into contact with the virus tomorrow. And even those fully vaccinated could still be infected, as no vaccine offers complete protection.
This widespread testing is now more possible than before, with approvals given to use RTK Antigen test kits. The cost is now as low as RM12 ($3.86) for DIY test kits, if purchased in bulk, like for factories or offices. For an estimated RM50 to RM60 each, RTK Antigen tests can by performed by approved medical practitioners, and it is RM150 for the PCR tests. Perhaps it is time the government also implemented control pricing for these tests.
Understanding R and its importance, but also limitations
As mentioned above, R is the effective rate of reproduction, or effective reproductive number. It is the expected number of new infections caused by an infected individual in a population within the infectious period (for a more in-depth explanation, see “R value: How it is determined”).
Clearly, the R is very important, as the number of new infections will directly determine the number to be hospitalised. And knowing in advance (with a high degree of accuracy) how many hospital-ICU beds and ventilators would be needed means we can prepare for it — and therefore maximise the probability of recovery. The goal is better healthcare management — and to minimise the number of deaths.
The R will continuously change as the outbreak progresses, driven by public health intervention, people’s behavioural changes in response to the outbreak as well as virus mutations and vaccinations. For instance, lockdowns, mask mandates, improved personal hygiene and vaccinations can bring this figure down. This is what all governments are trying to do — to bring the Rt down to below 1. More on this later.
Equally, as important as the R is, we must also remember that its accuracy depends on — and therefore is also limited by — the integrity of the data that we have today. Therein lies the problem.
A crucial dataset required to calculate R is the actual number of infected cases. But we have only an estimate for this data, which is based on the number of tests done as well as whom they are conducted on (in other words, the sample population, since we cannot test the entire population at one go). This is why as we explained previously that testing is sampling. And the test results are only as good as the sampling. We demonstrate this point in the accompanying diagram (“Test results are only as good as the sampling”).
• If repeat testing is performed on the same sample population, say, twice a week, the number of confirmed cases (the test positivity rate) will be very low, maybe even zero;
• If random testing is performed on a sampling of the asymptomatic population, positive cases are also expected to be low. But if the number of tests is expanded, then we will likely get a few more positive results; and
• If testing is targeted at known clusters, the number of positive cases will likely be high — and higher if we expand the number of tests done.
Clearly, the results (number of positive confirmed cases) — depending on whether our sample population is 1, 2 or 3 — are very different. The interpretation will be different and the appropriate policy response will most certainly be very different. In other words, the absolute number of cases must be read in the context of the sample taken. Higher or lower case numbers on their own have no real-world meaning.
Why have confirmed cases remained so high after weeks of lockdown — and does it really matter?
From the diagram, we know that the number of positive cases depends on:
• The sampling (which sample population); and
• How large the sampling size is, that is the number of tests conducted. R can be understated if:
• A fall in the number of tests (smaller sampling size) results in fewer sampled positive cases; and/or
• There is a drastic change in the population sampled — from a high-risk concentration segment of the population to a lower-risk or random sample of population.
Conversely, R would be overstated if the opposite happens. Both an understated and overstated R leads to bad policy decisions.
Why have confirmed cases remained so high after weeks of stringent lockdown? The most probable answer should be fairly clear by this point. The number of tests conducted in the country has been way too low. Case in point: Even after increasing testing in mid-May, Malaysia’s number of tests daily averaged just above 2,800 per million population. Prior to this, for the first 5½ months of this year, the number of tests daily averaged only 1,600 per million population. The UK, by comparison, performed nearly 12,000 tests daily per million population over the same period, which has been further raised to more than 13,000, on average, since mid-May.
A consistently low number of tests very likely translated into the relatively low number of confirmed positive cases from February to May. This also means that the number of undetected and unreported infections — the majority probably asymptomatic — has been high for months and the daily reported case numbers were severely understated (and could still be).
This would explain why case numbers shot up in recent weeks even though test numbers have not (rising test positivity rate) — the virus is already pervasive in the community (see Chart 1). Poor detection means the infected are not isolated and therefore continued to spread the virus to others. This corroborates the huge increase in the number of cases classified as “sporadic” — that is, origin unknown (see Chart 2).
This also means the earlier drop in Rt — which is calculated using the existing dataset of only the reported cases — was an illusion (see Chart 3). And that led to the wrong decision to relax movement restrictions, which resulted in the current, very serious wave of new cases.
In short, Malaysia could have done better in early case finding time (with extensive use of digital tools), testing and following up with rapid tracing, isolation of and support for the infected, or the often-quoted FTTIS. What is done is done. Hindsight is always 20/20. We cannot go back in time to rectify the error — though this should be a lesson learnt to prepare us for the next pandemic — but we can certainly do so going forward, by rapidly expanding testing.
There are some who believe that increasing testing now is futile, especially in the Klang Valley, where the virus is already so pervasive. And perhaps they are correct. But we think that making the decision to abandon testing (and, critically, isolation) at this point, without the support of empirical evidence, could turn out to be yet another huge mistake — with deadly consequences. If the virus is indeed as pervasive as they think, would the Klang Valley not have achieved herd immunity by now? That would mean high positive cases but hospitals would not be overrun and deaths would not be rising. At the very least, substantially increased testing today would give us useful data.
Testing, testing, testing
Clearly, data integrity is of utmost importance. And this can be achieved by ratcheting up the intensity and consistency of testing, beyond targeted testing at known clusters (as is currently being done).
Widespread and repeat testing on the same sample populations — for example, twice a week testing at factories, construction sites and schools as well as regular testing in offices — will enable quick isolation of infected cases and more effective tracing to cut the transmission chain. This will reduce the time the virus has to spread from one to others.
And, of course, more tests mean more accurate estimates of the granular data, including better-defined geographic localities. This allows for more targeted intervention measures instead of broad-based lockdowns.
When there is sufficient and consistent testing on an unbiased sample population (or corrected for necessary bias), the resulting data will give us an R value that is much more useful — on which to form the basis for better decision-making and improved public healthcare management. Ultimately, the number of deaths is the most accurate metric to measure the severity of the outbreak (even if the numbers could still be undercounted). Unfortunately, this is a lagging number — by at least two to three weeks. It is all the more reason that testing is so important, to detect and isolate those infected as soon as possible.
While the capacity for PCR testing is limited, the number of RTK Antigen testing is unlimited, given that no lab work is required. There are several available RTK Antigen test kits in the market currently, all approved by the Medical Device Authority (MDA). The cost is very affordable, especially when compared to the alternative — the cost of extremely disruptive lockdown measures.
One likely consequence of widespread testing (larger sampling) is a sharp spike in positive case numbers in the short term. It will probably trigger more unhappiness and anger among the people. But we must view this positively rather than with fear. Turning a blind eye to a problem does not mean the problem does not exist, nor will it simply go away!
Vaccination will reduce transmission — this is a mathematical certainty
What we are saying is that it is imperative that testing be increased significantly and the sample population widened. Larger sampling will yield a better, more robust dataset, which, in turn, will give us a better representation of the outbreak — a more accurate R — and that must lead to better decisions in managing the pandemic.
That said, it is also a mathematical fact that, as vaccination is ramped up and as the percentage of the population inoculated increases, the severity of the outbreak must eventually peter out. Earlier in the article, we explained that Rt would change as the outbreak progresses — depending on the response from public health authorities and people’s behavioural changes.
For instance, reducing contact among the people through movement restrictions and changing behaviours such as social distancing and masking will lower R, the effective rate of reproduction. Let us call this percentage reduction “p”.
We can also slow the spread of the outbreak by reducing the riskiness of infection in each contact between two persons. And it is clear that vaccination of the population achieves this. We will call this percentage reduction “q”.
Mathematically, to reduce R and stop exponential growth of new infections, the combination of the above two fractions, p and q, must be less than or equal to 1 /R.
That is, (1 - p) x (1 - q) ≤ 1 /R.
In short, q will rise over time as more and more of the population is inoculated. Thus, even when the lockdown is lifted, the left side of the equation must eventually grow smaller and smaller as q approaches 1. It is for this reason that we believe this pandemic will end soon — and why we are bullish and fully invested for our portfolios.
The UK is providing the world with critical knowledge by being ahead in the outbreak in terms of testing and vaccination. And the data is highly optimistic. The country has been reopening gradually and positive cases are rising anew. This is to be expected as it maintains a high level of testing and especially with the more contagious Delta variant.
Critically, the number of deaths has stayed low. This is the clearest evidence yet that the vaccine works — and that the absolute number of positive cases needs careful interpretations! It may not stop people from getting infected — no vaccine offers 100% protection — but it does very effectively lower the risks of severe illness and death (see Chart 4). This is what Malaysia must aim to achieve so that lives and livelihoods can return as close to normal, as soon as possible.
Looking even further ahead, Singapore’s road map in a post-vaccinated world in which the disease is endemic provides additional insights. The country will do away with the daily reporting of Covid-19 cases altogether, as the numbers are no longer relevant, just like the number of people down with the flu is not broadcast on a daily basis.
Widespread rapid testing will be carried out in airports, seaports, office buildings, malls, hospitals and educational institutions, for screening purposes. The sick will be sent home. The quality of data gathered will translate into better public healthcare management where variants of concern can be identified quickly, hospitalisation and deaths monitored closely and counteractive measures taken where necessary.
Box Article: 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_4278525d-ccf2-4f8a-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.
- Box Article Ends-
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