# COVID-19 growth rates vs interventions

It’s been a couple of weeks since I posted regarding the growth rates of COVID-19 cases in various countries. Now, after these countries have implemented various control measures it is more clear how each measure effects growth rate. This post looks at the 10 countries with the highest number of confirmed cases.

# Death growth rates roughly match confirmed case growth rates

Firstly, to check whether the confirmed case growth rates roughly reflect the actual growth rates, they can be compared to death rates (suggested in a comment by Unnamed on my previous post). This doesn’t show what the ratio is between the number of cases and the number of detected cases, it only shows whether the fractional growth per day in detected cases roughly represents the actual growth rate of the virus.

Obviously the death growth rate doesn’t perfectly indicate the spread of the virus but it has different biases so if the two are similar then they probably at least somewhat reflect the actual growth rate.

A graph to compare growth rates of confirmed cases & deaths of the 10 worst affected countries is here. (Select country at the top)

(Note: For the y-axis I have used doubling time of infectious cases, which I’ve defined as 10 days fairly arbitrarily. The actual length of time doesn’t matter too much to the results but it’s important to have some limit otherwise the recent China and South Korea results particularly make less sense. This makes the definition of doubling time for deaths a bit odd but it’s good enough for the purposes of comparing the rates.)

Generally the rates match well. Iran and USA are the main ones where there is significant divergence (and both look more reasonable recently), the others all seem sensible.

In theory there should be a lag between cases and deaths which we seem to see in Italy and Spain but the data is too noisy to say for sure.

# Growth rates vs interventions

I’ve created 4 different displays of the cases data:

1. Cumulative confirmed cases

2. New cases per day

3. Doubling time for infectious cases

4. Fractional change in infectious cases per day

This last one is analagous to the effective reproduction number () and could be converted to that if I knew the mean infectious period. I could take a stab at this but I think it’s best to leave it as it is. It’s useful to know that 1 on this axis is the same as , even if there would need to be a scaling factor for the other points to convert them to .

I’ve annotated the graphs with what anti-COVID actions each country has taken and when. Apologies for anywhere I’ve got these wrong, if you see any massive errors for your country I’ll try to update.

Apologies also for the overlapping writing—mouse over the relevant points if it gets confusing and click the legend to toggle countries. Double-click to toggle between only that country or all countries.

There has been some form of lockdown for most countries but the exact extent differs between countries. I haven’t attempted to distinguish between them.

There is expected to be a ~5 day delay between actions taken and effects being seen in the confirmed cases statistics as people are usually tested when they are symptomatic.

## Uninhibited growth has a doubling time of 2-3 days

Refer to my previous post. I don’t really have much to add here, only that my initial calculations of doubling time had a small error so the doubling times are actually slightly lo (i.e. growth faster) than I initially reported.

## Growth with improved hygiene and social distancing has a doubling time of 3-5 days

I also mentioned in that post that it seemed as though the doubling time for each country was increasing over time. This seems to me to represent additional simple precautions starting to be taken—such as improved hygiene and social distancing (short of a lockdown).

4-5 days is probably the best that can be achieved by these methods. Many countries have put these in place but none have been able to slow the spread of the virus sufficiently without taking additional actions.

## Growth of virus with partial lockdown has doubling time >4 days

Different countries have enacted different strictness levels in their lockdowns. These haven’t been in place long enough to know exactly what’s happening but they have had an effect and in Italy’s case especially this has started to strongly increase doubling times.

## Growth with virus under control has halving time as low as 2-5 days

The indication of having (i.e. active cases decreasing) is that the doubling time becomes negative (and represents a halving time). This is probably seen better in the daily growth factor graph where a value <1 indicates shrinkage.

We have 2 examples of countries which have had significant outbreaks and brought them under control – China and South Korea. In both cases the doubling time starts climbing and keeps going until the active cases starts to decrease. Under full control the halving time of active cases was 2-5 days.

We don’t currently have any countries with a large number of cases where the doubling time is >6 days and holds steady for a prolonged period. The possible exception is Iran but I have less confidence in the data there due to the mismatch between growth rate of confirmed cases and deaths and in the last few days it looks like the growth rate is increasing again.

I suspect that having a sustained high doubling time is possible if is just above 1 but so far either a country is not doing enough (doubling time of 2-5 days) or they are doing enough and the cases are about to start decreasing. If is large to start with it’s hard to find that perfect amount of intervention which takes to 1 so that the number of new cases stay manageable. Possibly as China and South Korea loosen their restrictions they are starting to find that point.

# Country summaries

The above is based on looking at the performances of various countries as described below.

## China

China successfully applied a quarantine in Wuhan which reduced a rapidly growing epidemic to a handful of new cases per day. This quarantine was very strict compared to other countries on this list and the halving rate was 2-5 days. Other, less strict quarantines are likely to shrink more slowly.

More recently (11th March), the restrictions in Wuhan were eased to allow citizens to go back to work. Since 18th March the virus has started growing again, so far averaging a doubling rate of 7 days or so. So far they are performing the dance successfully.

This seems to me the most likely next stage for Western countries. Exactly what rate the virus is managed to before it needs to be suppressed again is unclear. China at the moment should have plenty of tests and protective equipment so whatever they achieve is likely to be fairly close to the best possible scenario. Successful contact tracing could allow it to pause indefinitely without a full lockdown.

## Italy

Doubling times have been increasing as the government has implemented additional control measures.

Lombardy (main outbreak in Italy) was locked down fairly early on. This increased the doubling time to 3-4 days. Later lockdowns which eventually covered the entire country increased this further such that the number of new cases per day appears to be levelling off. Most recently the Lombardy lockdown was tightened to decrease spread rate further. I don’t think it will be long before the number of live cases starts to decrease.

## USA

The growth rate in the USA shows the least evidence of slowing down. The growth rate in deaths is less so there may be something confounding the data on confirmed cases, such as increasing coverage of testing.

Many states, including the main centres appear to have implemented lockdowns in recent days so these should start having an effect shortly. Some counties in the Bay area implemented a lockdown earlier but John Hopkins have started aggregating by state and any effects haven’t shown up in the California figures yet.

## Spain, Germany, France, Switzerland, UK

These have all followed fairly similar paths. Schools have closed between 2k and 5k cases (Switzerland ~1k). Lockdowns have happened between 5k and 10k cases.

Some countries seem very keen to say there is no lockdown (e.g. Germany, Switzerland) and their actions are correspondingly less strict. However they do entail a large curtailment of freedoms even if they are less strictly enforced.

France seems to have been most strict with their measures in enacting fines for violators although I don’t know how effective these are.

If I borrow VipulNaik’s taxonomy, all of them are somewhere between level 2 and level 3 lockdown.

The UK and Switzerland are a bit behind the other countries in terms of cases but their actions have similarly lagged so haven’t taken advantage of their initial advantage.

## Iran

Iran is a strange one in that their total number of new cases per day has been fairly flat for a couple of weeks. I’m not sure whether they have achieved a perfect or whether their data is a bit funny – their deaths data don’t really reflect their confirmed cases although more recently they start to match more closely.

## South Korea

South Korea are my favourite COVID-19 dealing country.

They essentially had it under control until the infamous patient 31 infected a large number in his church who then went on to infect more until there were >5k cases associated with the church (more than half of the total number of cases in the country).

Despite the government never imposing any particularly strict orders, the entire city of Daegu was deserted within a couple of days of the patient 31 outbreak being confirmed. Between that and the intensive contact tracing and testing program the outbreak was quickly brought under control so that the hospitals weren’t overrun and the fatality rate was kept down to 1.3%.

In 2015 South Korea experienced the second worst outbreak of MERS. There were 168 confirmed infections and 38 people died. This article has an interesting summary of how the lessons from that outbreak fed into the COVID-19 response.

The halving time during the reduction phase was 3-5 days.

Arguably they have now entered their dance phase as .

## Japan

I haven’t included Japan on the graphs as nothing much has happened there which is pretty amazing. Japan is probably what South Korea would look like if there had been no patient 31.

They have taken precautions similar to Western countries before the latter implemented stricter lockdown. However they have managed to contain every cluster of cases before any have got out of control.

There has been a lot of talk about Japan not doing enough testing and that their numbers are artificially suppressed. My prior for this is pretty low – this seems like an unlikely thing for a government to do, especially as it wouldn’t take long before the truth came out as the death toll rose.

As for evidence against that hypothesis, Japan have done a lot of testing compared to the number of cases − 19 out of 20 tests come back negative (even if the absolute numbers are low). If they are deliberately suppressing their numbers then they’re doing a really good job at testing the wrong people.

Japan’s cases started to get serious in mid-Feb. I think it’s clear that they managed to avoid any out of control outbreaks until at least the beginning of March, otherwise there would be so many cases by now that it would be obvious. If they can keep the virus in control for 3 weeks then they can probably keep it in control for a couple more up until now. If a cluster does get out of control in Japan then I expect it to go the same way as South Korea.

Of course the Japanese government could be lying about everything but again if they are I would expect better evidence from citizens/​journalists by now.

# Summary

Doubling times

Improved hygiene and basic social distancing: 3-5 days

Lockdown with work allowed: 5+ days (possibly cases decreasing)

Halving times (single sample each)

Full lockdown: 2-5 days

Flexible lockdown + Epic contact tracing: 3-5 days

• There’s a huge confounder here which is testing ramp up: it’s hard to say how much of growth in confirmed cases is growth in actual infections vs growth in testing. For example if you look at the graph of cases vs graph of tests in the US they track closely, and the % of positive tests hasn’t changed much.

However there’s another dataset which doesn’t have this problem—the kinsa smart thermometer dataset, and it indicates that social distancing has been highly effective at curbing all flu-like infections in the US.

• Comparing confirmed cases to deaths should identify that confounder if it’s there. Interestingly the US is one of the countries which showed up as possibly confounded at the beginning. More recently I suspect this is less of an issue.

My analysis suggests about a 40% decrease in R due to hygiene and social distancing. R0 is ~3 for COVID-19 so this bring R down to ~1.8 which means the virus is still growing fairly fast. For flu R0 is ~1.3 so after these measures R is ~0.8 and therefore is shrinking.

• I am puzzled at how mild interventions don’t show a much bigger decrease.

In Ottawa, where I live, we have social distancing and have shutdown non-essential places of business. If you work in a closed business, you have probably reduced your person-to-person contacts from (guessing) 300 per week to maybe 50 -- those 50 being people at grocery stores, etc. Moreover, the intensity of those instances of contact has dropped. You may have played poker with 10 people, mutually touching cards and chips and sitting together for hours. Now, you stand within 2m of a store cashier for a couple of minutes.

Just this 83% drop—which I think is conservative—should push R down from 3 (without intervention) to 0.5.

Add in hygiene improvements and more aggressive quarantining of those with symptoms, and R should drop even farther below 0.5.

If my numbers and logic are reasonable, the reason we haven’t seen a lot of dropoff yet must be because of legacy cases (from before intervention) still coming in and obscuring the current trajectory. (We’ve only had serious social distancing for 12 days or so.)

Unless there’s there something wrong with my calculations or logic. Are my estimates of contact frequency (300, 50) badly off?

What am I missing?

• A couple of additional points to leggi:

Elizabeth calculates roughly 25% of people are in essential roles. These people are less able to reduce numbers of contacts.

At least initially many people don’t take social distancing seriously so the effects are likely to ramp up over time.

In that case it makes sense that initially doubling times increase over 5 and over time they keep increasing.

In China the distance was enforced and Koreans took it seriously right away so it didn’t take long for their doubling times to increase.

• Some first thoughts:

• It only takes one person to infect you.

• How many people is the person serving in the grocery store coming into contact with?

• How strict are they all with their precautions?

• What about other members of the same household and all their contacts?

• The recommended distance between people may not be sufficient to prevent transmission.

• It’s easy to break the distance rule (might just be a second or two even if being v. careful).

• Fomite transmission (inanimate carrier of infectious diseases)

• Pre-(noticed) symptomatic transmissions. What if someone has a fever during the night, how many people would notice it/​associate it with COVID? (It always amazes me the denial some people can have about their symptoms.)

QUESTION - has anyone come across data about duration of a COVID-fever? (although there’s a massive potential for variability between individuals so not sure the data would actually be useful/​representative/​meaningful but it’d be good to have whatever information is out there...)

• All these points make sense. But aren’t they also (with the exception of the one about members of the same household) subject to the logic that they reduce roughly proportionally to reduced contacts? For instance, even in the unlikely case my contacts’ contacts are not reducing, I am still reducing contacts with my contacts’ contacts by reducing contacts with my contacts.

• Western-style lockdowns may be slightly effective for a week or two, then very effective thereafter, due to the disease running through all members of an infected household. Do we yet have data on this possibility?

• A few considerations that may change the historical observation that death rates match confirmed case growth rates.

Consider the lancet article below, where the mean time to death for persons who die from infection is 18.8 days, with a coefficient of variation of 0.45, which calculates out to a standard deviation of about 8 days. Sustained growth for many more doublings than is seen in prior countries, will result in an accumulation of latent mortality (That is, 15% of cases contracted today, that result in death, won’t do so for >27 days, and 50% wont die for 18.8 days). Therefore the more rapid the growth, and the more sustained over successive doublings, the greater the latent mortality within the dataset, because mean time to death is fixed whereas the time required to double the aggregate confirmed cases is variable. This doesn’t mean the death rate wont match the confirmed case growth rate, as much as recognizing that there is an inverse relationship between deaths per case early on, and the death rate may surge when the diseased cohort “matures”.

https://​​www.thelancet.com/​​journals/​​laninf/​​article/​​PIIS1473-3099(20)30243-7/​​fulltext

More to the point, the current death to case ratio in the US is about 3%, where other countries with a more mature epidemic are around 9-12 percent. This suggests that if were were to fully arrest the epidemic 100% today, and limit it to 330,000 cases, then a 10% death rate would be a total of 33,000 deaths, of which at least 85% would predictably occur within 27 days (one standard deviation). Since we have around 10,000 deaths, then 23,000 will die in the next 27 days no matter what we do.

However, the public health response may further break the curve in this case, where we ee explosive growth in deaths soon, and an overwhelm of the health care systems capacity around the country. We may very well see growth in death rates exceed case growth rates for the cohort.

• Yes, we definitely expect to see a lag between growth rates of cases and deaths, it is odd that even when this seems to be present it is only a couple of days to a week. I think this may be partly due to delays in diagnosis. 17.8 days is between onset of symptoms to death. However there is normally a lag between onset of symptoms and diagnosis (onset to hospitalisation I think is generally a bit less than a week) but even this still leaves a theoretical 10+ day lag.

That is all based on relative numbers within a country. Comparing CFR (case fatality rate) values between countries is notoriously unreliable due to testing capability. Looking at naive CFR I think the UK are about to overtake Italy as having the worst CFR in this set of 10 despite being earlier in their epidemic. This is either due to being worse at testing or better at diagnosing deaths as being COVID related (some countries aren’t counting deaths which don’t occur in hospital—source). CFR in the US is low compared to where other countries were at similar points in their epidemic so I guess it won’t reach 10% but it is likely to reach 5%.

• Thank you—this is very helpful to me.

We don’t currently have any countries with a large number of cases where the doubling time is >6 days and holds steady for a prolonged period.

I spent a few days looking at this, doing simulations etc. The zone where you slow the growth rate to a low rate, not exploding and not collapsing, is very narrow. It is roughly R0 = 0.98 to 1.10.

So, given

1. The effect on R0 of a given set of measures is quite uncertain and hard to predict, and

2. To flatten the curve without causing a collapse in cases you need to hit a very narrow zone,

the implication is that to be confident that you will not have an explosion, you need to target an implosion in cases. If you think you can finesse “how much can we get away with” you are probably kidding yourself.

• Yes, holding at a high number is tricky and not particularly desirable. If you get doubling time above 6 days then it’s likely that you’ll start decreasing cases.

I think that the most important thing if trying to hold at a low level whilst relaxing restrictions is ensuring that the doubling time is longer than the incubation time (which is the main lag in your control loop). That way if you have made an error the virus isn’t too far gone before you start to notice and contact tracing for containment remains viable.

• This seems to suggest we will experience a cyclical pattern to these events. When we transition to a R0 < that .98 we seem to have things under control and will likely start seeing either relaxing any imposed controls or just relaxing our self imposed constraints on interactions. Then we’ll have a few new infections, and R0 then returns to a value above the range so we’re back in the epidemic spread phase again.

Does that seem right from this?

• I also notice a lot of countries have had two peaks. Possibly this is a combination of stopping overseas visitors, combined with testing focused on overseas arrivals (and neglecting local transmission which is harder to find), combined with a ramp-up of testing which produces a spike in apparent cases.

Do the initial lull in confirmed cases is likely to be a false dawn.