# 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.