Will Level 3 Get the Job Done?
Tags: Data Science, Covid 19, Lockdown, Ireland

Will our current Level 3 restrictions allow us to regain some control over this virus? Or will they only serve to delay a move to Level 5? This is a question that has dominated the past few days.

Obviously we don’t know what the future will hold, and it can be difficult to detect whether things are improving, especially when we focus on numbers of daily cases, which only seem to go up. However, if we look at changes in the relative growth of cases, then we may be able to see the early signs of progress, which might be just the encouragement that is needed.

This article looks at one such growth metric called the transmission ratio, T — which is designed to estimate how much virus transmission is occurring — and whether we can use it to gain early insights into what has been happening in Dublin and Donegal, where Level 3 restrictions have been in place for a little longer than elsewhere, which may help us to predict how the rest of the country will fare.

The conclusion is that there are early signs that Level 3 is working in Dublin in Donegal — cases are still growing by the rate of growth has slowed because transmission has reduced — but its going to be close, and we absolutely cannot afford to become complacent if Level 3 is to succeed.

Transmission Ratio (T=n/P)

The transmission ratio (T) is a simple measure, calculated by dividing the number of cases today (n) by the prevalence of cases 7 days ago (P). What is prevalence? And why 7 days ago?

Prevalence is the sum of the last 14 days’ worth of cases. It’s an estimate for the current number of active cases, given that people remain ill for about 14 days. Thus, prevalence 7 days ago is the sum of the cases during the period from 3 weeks to 1 week ago.

Why divide today’s cases by the prevalence from 7 days ago? Well, if someone is confirmed to be positive today, then it means they became infected some time ago — on average about 7 days ago, based on an incubation period of 4–5 days and a 2-day test confirmation cycle— and transmission occurred when there was a certain number of infectious individuals, which is approximated by the prevalence 7 days ago. Thus, T=n/P is the likelihood of transmission, from a week ago.

For example, as shown above, on October 5th the 7-day rolling average of confirmed cases in Cork was about 41, and the prevalence in Cork a week before (September 28th) was 291 cases. Therefore, the transmission ratio was 0.14; this is the transmission ratio for September 28th, not for October 5th, since transmission ratio for October 5th will only be known when we see how many cases are confirmed on October 12th.

A couple of points are worth making about T. First, it is based on confirmed cases and confirmed cases underestimate true infections, so doesn’t this make T inaccurate? No. Since the ratio of confirmed cases to true infections should be stable over a short period (2–3 weeks), this means that the ratio of confirmed cases to case prevalence should be a reasonable approximation for the true transmission ratio based on infections (the ratio of infections to infection prevalence).

Second, T is related to the now familiar R-number, the average number of new infections produced by an infectious individual, during their infectious period. The R-number is a more epidemiologically sophisticated metric, but it is complicated to calculate. We know that R=1 is an important threshold leading to infection equilibrium; when R=1 the number of infections remains stable. In a similar way, because prevalence is calculated as the 14-day sum of cases, then T=1/14 (~0.07) produces a similar steady-state; if T>0.07, then the number of cases will grow, whereas T<0.07 will lead to an eventual decay in case numbers.

Finally, like R, T is less useful when prevalence is very low. Therefore, in the analysis that follows we will only calculate T when case prevalence exceeds 20 cases per 100k of population.

Transmission Ratio in LOKdown

Before we look at the transmission ratio in Dublin and Donegal, let’s go back a little further in time to review what happened earlier in the summer when additional restrictions were placed on Laois, Offaly, and Kildare. The left-hand graphs below show the prevalence and daily cases (per 100k of population) in these counties from mid-July to mid-September; these graphs also show the minimum prevalence threshold of 20 per 100k as a horizontal line. The right-hand graphs show the corresponding transmission ratio (T), with the equilibrium threshold of 0.07 as a horizontal line. Each graph also marks the start and end of the restrictions (LOK) and the re-opening of schools (S) during the period under scrutiny.

The LOK restrictions were introduced on August 8th and cases and prevalence had been increasing in these counties for a week or two before that date. In fact, daily cases in Laois peaked before restrictions were imposed, and in all three counties prevalence peaked about 7–10 days later.

The transmission ratio was actually falling in all of these counties before restrictions were imposed even though cases and prevelence were still growing; presumably as locals adjusted their behaviour in response to growing case numbers and the possibility of new restrictions. In other words, the transmission ratio provided an early signal about the state of outbreaks that was otherwise masked by case numbers and prevalence. By the time restrictions were imposed the transmission ratio was already near or below the equilibrium threshold in Laois and Offaly, and it continued to fall in these counties in the days that followed, allowing restrictions to be lifted on August 22nd.

Kildare had been exposed to much higher transmission ratios in the days before August 8th, and transmission rates were still high (>0.07) when restrictions were imposed. But T continued to fall, and by the end of August restrictions were also lifted from Kildare.

It is worth noting, in the days that followed the lifting of restrictions, the transmission ratio rose again in all three counties. Although restrictions had the desired effect, their benefits were short-lived, which is a cautionary tale for us all today.

Dublin & Donegal

Here are the corresponding graphs for Dublin and Donegal during September and October. A version of Level 3 (L3) was introduced in Dublin on September 18th, and in Donegal on September 25th; so-called wet pubs (W) were permitted to open in Donegal on September 21st. On a per 100k of poulation basis, cases and prevalence were growing more rapidly in Donegal than in Dublin, and the transmission ratio peaked in Donegal at a much higher level (~0.4) compared to Dublin (<0.2). However the transmission ratio was high (>0.07) in Dublin for longer than in Donegal and it was operating on a higher baseline prevalence (>50 cases per 100k in Dublin vs. 20 cases oper 100k in Donegal, during much of September) .

As we saw in Laois, Offaly, and Kildare, the transmission ratio actually started to fall before restrictions were imposed, presumably as people began to moderate their behaviours in accordance with the media warnings of the time. This provided a certain positive momentum going into these Level 3 restrictions.

The good news for Donegal is that the transmission ratio has continued to fall, albeit somewhat more slowly, since restrictions were imposed. However, transmission remains high in Donegal and it has not fallen far enough for the situation in Donegal to improve significantly.

The good news for Dublin is that the transmission ratio is now at the equilibrium level of 0.07. If this trend continues then cases will begin to reduce in the coming days. Given the size of Dublin, and its high prevalence, it will be important to to push the transmission ratio even lower to collapse case numbers in the coming weeks.

What does it all mean?

So where does this leave us? Will Level 3 do the job? The evidence above suggests that Level 3 can work, but only if we are all concerned enough to change our behaviours, and not just for a week or two but for a sustained period of time. The graph below shows a summary of the current transmission status of counties. The location of each county is based on its current transmission ratio (the x-axis; actually it is the mean transmission ratio over the last 3 days) and the weekly rate of change of the transmission ratio (the y-axis; the mean transmission ratio over the last 7 days divided by the mean transmission ratio for the 7 days before that); remember that the transmisison ratio is calculated up to a week ago. The size of each marker, and its colour, corresponds to its case prevalence (per 100k of poulation).

We want all counties to be in the lower left quadrant of this graph — low transmission ratio and decreasing, or at least not increasing — once there cases and prevalence will soon start to fall. Unfortunately only Dublin qualifies at the moment (just). However, Donegal and a few others are at least falling transmission rates relative (weekly change<1), which means they will soon move into this quadrant if that continues.

There is a cluster of counties with moderate transmission ratios (e.g. Wicklow, Kildare, Louth, Carlow). Hopefully they will begin to turn the corner in the days ahead as Level 3 starts to have an impact. However, there are others with much higher transmission ratios (Limerick, Kerry, Sligo, Clare, Laois), and they are still growing, which may require a more sustained effort to rectify.

One thing is for sure, it is still too early to say whether last week’s country-wide Level 3 restrictions will have the desired effect. The first signs should start to appear around October 14th. If we can produce and sustain falling transmission rates then daily cases and prevalence will follow, and if transmission levels fall far enough (<0.07), and for long enough, then we will benefit from the type of exponential decay that is the positive flip-side of the exponential growth that we usually hear about. This is what we saw in the spring lockdown and it brought us to near zero-COVID for much of the summer. The question is whether we can achieve this in Level 3 today, for the winter, based on what we learned back then. I hope we can.