One Finite Planet

One Finite Planet

Flatten the curve: Misleading?

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The phrase ‘flatten the curve’ has become popular, and perhaps, overused. The term has come to be associated with any strategy to manage Covid-19 that includes some measures to reduce the rate new infections. This page provides analysis of what different things can be done (at least in theory) to manage an outbreak, and the ‘curves’ that result, together with how to achieve the desired curve.

The end result is that textbook ‘flatten the curve’ diagrams do not reflect reality with Covid-19 in the 21st century, and mislead people into complacency.

This post describes four different ‘flattening the curve’ models. However, on review, every curve possible can be viewed as a variation of a ‘containment strategy’. The resulting curve being determined solely by the containment limits. Six scenarios which can be seen as covering the possible scenarios in almost every country are then mapped to the corresponding curves as that county meets the varying requirements and plans to ‘flatten the curve’.

Which countries how seen which scenarios? And what is the road forward in each case.

  • what is ‘flattening the curve’?
  • ‘flattened curves’
    1. the original textbook ‘flatten the curve as shown in popular diagrams: (not known to be applicable to Covid-19)
    2. a more practical flatten the curve result
    3. flattening to stop the curve
    4. stop and go: flattening for a ‘containment curve
  • aren’t all curves containment?
  • reading curve graphs
  • Scenarios, and their path to containment
    1. emergency! flatten the curve! people are dying! A very real Covid-19 Scenario in response to a catastrophe
    2. planned outbreak flattening: we will flatten the curve to have a controlled outbreak
    3. Avoidance lockdown: we will flatten the curve to peak to avoid being another country with a catastrophe
    4. eradication lockdown: stop the curve, flatten it until it is nothing!
    5. flatline: ‘flatten the curve’ with no curve, prevention of an outbreak
    6. flatline lockdown: containment has broken limits, add a lockdown to restore low case levels
  • Conclusion

What is ‘Flattening the Curve’

The original meaning ‘of flatten the curve’ is based on the curve produced when an outbreak infects a community and cannot be stopped, thus continues until there is ‘herd immunity’. The outbreak will result in bell curve like graph of cases as infection progresses through to herd immunity. Take that curve, and flatten it by slowing the spread of the virus, and you have matched the best outcomes experienced with the 1918 flu pandemic, and produced a ‘flattened ‘ curves that results in less deaths on the path to ‘herd immunity’. So despite pursuing herd immunity being a controversial topic, the term for optimising a herd immunity strategy has become popular, and now taken to mean ‘reduce infection rates’ from what infection rates would be if no action was taken.

While ‘flatten the curve’ has come to mean ‘anything to reduce cases’, there are different possible outcomes from such measures and successful management of an outbreak can only result from understanding those outcomes.

Confusingly, ‘flattening’ can mean any of:

  • case numbers per day still rising….but rising more slowly
  • case numbers per day no longer rising
  • case numbers per day falling

Usage can be ‘we are flattening the curve, but not yet seeing a peak’, or ‘the curve is flattening, but case numbers are not yet falling’, while others will describe ‘numbers falling’ as ‘flattening’. But generally, regardless of what the outcome is, measures to reduce case rates are still described as steps to ‘flatten the curve’.

Curve 1: Textbook Flattening As Per Diagrams.

Theory: Origin of Flattening

CDC graph – model not provided

In the 1918 flu pandemic, there was no vaccine or cure, so herd immunity as a result of infections was the only endgame. It was seen that places that reached herd immunity with a ‘flatter’ curve has less fatalities, than locations with a steeper curve.

However, from the diagram provided (this one by the Center for Disease Control in the USA) it is clear the conditions for the flatter curve had to be in place right at the outset, as the blue curve rises more slowly right from the very beginning. The only way to produce a curve like this, is to have a lower spread rate for the entire curve. Note that a country following the blue curve would have barely any cases at the time a country on the red curve has already reached the capacity of the medical system. The lower spread rates can be produced by having lockdowns, social distancing or other measures in place, and to produce a result such as the diagram, the measures must be in place at the time of the very first cases. Further, such measures must reduce the spread rate, but still have a spread rate greater than one, or the result will be ‘stop the curve’, rather than ‘flatten the curve’.

In reality, in 1918, as with today, it is very difficult to predict what spread rate will result from measures to produce spread rate. Reducing spread rate in 1918 did save lives, but mostly from limiting overloading the medical system, which provided more people with medical treatment and as a result reduced fatalities.

Reality and Endgames.

For any country that did place measures in place at the time of the very first cases, and those measures resulted in a reduced spread rate that is still greater than 1.0 (i.e. the curve still grows), then the ‘blue’ curve would be the curve. It is possible that countries like Singapore and Japan actually achieved this ‘blue ‘ flattened ‘textbook’ curve by accident. Both countries put measures in place very early, and certainly with Singapore before ‘a curve’ had formed, and had relatively flat lines early with rise in cases delayed in comparison with other countries. Generally, countries that introduced measures at the very outset, were most likely trying to avoid an outbreak completely, rather than having a target curve in mind. No one introducing containment measures from the initial outbreak of Covid-19 would have been able to predict precisely the resulting curve in order to decide right at the start exactly what measures were needed for the ideal curve .

Herd Immunity: Delivering herd immunity using this approach without overloading the medical system would require being able to predict with great precision what the spread rate would be with ‘flattening’ measures in place. In theory, having a start matching to the blue curve could allow a lower level of cases through to the point of herd immunity, but flattening throughout the entire curve is really only going to happen if the best flattening available to society will still overload medical systems. Flattening applied at the outset will generally be too little, resulting in more cases than required, or will be sufficiently effecting as to make herd immunity take too ling. However if you want to achieve herd immunity, and you did know the exactly appropriate flattening measures, this approach will still take longer than alternatives. The reason for wanting to reach herd immunity, is to end the need for lockdowns, yet a flat curve from the outset means a very long lockdown. See ‘a more realistic flatten the curve below’. Flattening the entire curve, but yet trying to reach herd immunity, does not seem practical with Covid-19. The measures to contain the spread come at too great an economic cost to endure the longer time to herd immunity that this ‘flatten the entire curve’ delivers.

Stop and Wait (for Cure/Vaccine): Any country that had ‘curve flattening’ in place before cases started to rise, is very likely a country that hoped to have a spread rate of 1.0 or less, and therefore were planning avoid an outbreak and ‘stop the curve’. This curve has a spread rate that is reduced, but still above 1.0. From a slow curve such as this, stopping will normally still an option, just introduce further measures to reduce spread. Singapore has announced lockdowns following what has followed the pattern of a very flat curve. In practice, any location with one of these flat curves will likely choose the ‘stop, wait‘ endgame strategy.

Curve 2. ‘Practical Flattening’ the Curve

Spread reduced but still above 1.0. Graph produced from excel curve model, but flattening measures in place as indicated by the blue bar are never eased.

The ‘practical flattening’ curve to the right, is produced as a result of starting the flattening at the more practical time of when case numbers have started to rise. This means measures are lockdowns and any other measures as introduced as people are seeing cases already rising. Most countries only introduced measures to ‘flatten the curve’ after their curve had already started, so this curve was still feasible. Further, as the curve shows, the curve peak is still at the same level as with Curve 1 above, but the peak arrives earlier with substantially less of those economically costly shutdowns. Get the spread rate correct is a challenge and if the rate turns out at 1.0 or lower, you get a different graph (stop the curve). This ‘Curve 2’ graph results from measures that slow increase in cases, but do not stop the increase in cases. The graph here was produced by halving the spread rate. Note that peak cases is again reduced to 1/4 of the number that would occur without flattening the curve, but with Covid-19, even 1/4 of the raw curve level will still be a huge peak cases number.

Herd Immunity: The ‘single’ curve as above will deliver herd immunity, but only that level of herd immunity will only be effective while social distancing and any other measures introduced are to reduce spread rates remain in place. undefinedRemove the ‘social distancing’ measures, and cases will rise again until true herd immunity is reached. If ‘flattening’ measures (indicated by the blue bar) are removed after the curve peak, there will be a second peak, but also a reward of full herd immunity, and from the time measures are removed, borders can be opened and society returns to normal. This ‘two peaks’ curve has a short ‘flattening’ time, which keeps the economic cost low, provides full immunity and allows re-opening borders. At the outset, many countries may have been aiming for this curve. The problem with Covid-19, is that peak case levels can still be a burden on the medical system. Choosing a single set of measures to deliver this curve requires almost impossibly accurate modelling.

Some countries started with the goal of producing a very flat curve but gradually introducing more and more ‘flattening’ adjustment measures as case numbers increased. However, with the exponential growth early in an outbreak, and a time lag before measures take effect, this becomes a very risky strategy, resulting in countries changing course to a ‘stop’ strategy as per curve 3, below.

“The risk of a phased and gradual approach is continued epidemic growth, potential failure of the health system, and a far longer road to recovery,” Prof Macintyre said in a Medical Journal of Australia article.

daily telegraph

Stop and Wait (for Cure/Vaccine): If measures introduced to ‘flatten the curve actually produce this graph where cases continue to climb after measures to flatten the curve have started, then it is likely more measures will be required. With only having slowed infection rates, it is simply not very viable to consider eradication. Either new measures will change things to the ‘stop’ curves as below, or the result will have to be manage cases as best possible and wait for her immunity.

Curve 3: Flattening to Stop the Curve

Take Curve 2 above, and apply a stricter lockdown, and instead or new cases increasing at lower rate, case numbers decline once the lockdown takes effect. Given just how rapidly exponential growth gets out of control, it is safer to apply a stricter lockdown, so many countries see this type drop in cases once the lockdown takes effect. The above graph assumes sufficient lockdown remains to keep case numbers declining.

Curve Model with ”flattening’ removed too early as cases start falling.

Are we there yet? If lockdowns are released as soon as case number fall very low, then lockdowns may be needed again to prevent disaster. Note that the fall in case numbers seen in this scenario is a result of a strict lockdown, and not a result of herd immunity. If the lockdown is released, the outbreak would resume as show in this model with the spread rate return to the normal level for the virus when the lockdown ends. Any country mistaking the peak being reached under lockdown as being the same as the peak of the outbreak would see this result.

Herd Immunity: To reach herd immunity at least around half of the entire population must be infected. When the curve starts, a comparison between the number infected so far an the total population is required. It is feasible to keep going the current level of cases running until half the population has been infected? If yes, the curve must be transformed into the ‘stop and go curve containment curve, covered below as per ‘curve 4’ below.

Stop and Wait (for Cure/Vaccine/Eradication): If maintaining current case numbers until herd immunity can be achieved is not appealing, the other choice is to follow the initial graph in this section and set a low limit or even zero limit for containment. With a sufficiently low level of cases, track and trace may substitute for lockdowns.

Curve 4: Stop and Go, a ‘containment’ curve

Curve 4 brings everything from earlier curves together. Curve 1 showed ‘flattened’ curve for a virus, together with a ‘raw’ or natural curve for the virus. Curve 2, introduced making changes as the curve progresses, introducing ‘flattening’ and then later removing flattening. Curve 3, reflects how a curve progresses when ‘flattening’ measures create a ‘false peak’, and can reduce cases far before the point a curve would normally peak, and revealed the risks of removing ‘flattening’ measures in such a situation.

Curve from model with coarse containment for illustration purposes

This curve, ‘Curve 4’, is to make use of ‘flattening’ to halt and then resume a curve to deliver a case load that remains within a target band of infection rates. The curve shown here was created by completely removing flattening as case levels drop. The graph here was crated by allowing cases to rise at ‘full speed’ between use of ‘flattening’ to apply the breaks. In practice, relaxed flattening measures would produce a smoother curve, and require less changes, but the exaggerated ‘waves’ of this graph work for illustration purposes. Every ‘containment curve’ will have an upper caseload limit, with flattening to ‘stop the curve’ needing to be introduced sufficiently in advance to prevent exceeding the upper limit, and a lower limit with a plan to relax lockdown measures as the lower limit approaches.

Herd Immunity: Containment to achieve herd immunity requires an ‘upper limit’ caseload within the capacity of the health system to cope if the upper limit is reached. In practice, that upper limit would need to be reduce to provide a ‘buffer’ due to the challenge getting flattening measures to ‘stop the curve’ to deliver the stop at a desired point. The lower containment limit when targeting herd immunity must be sufficiently high that the case load at this lower limit will result in the required percentage of people having been infected within a target timeframe.

Stop and Wait (for Cure/Vaccine): Containment for eradication is similar to containment for herd immunity with lower limits. The lower upper limit is the trigger level to reapply ‘stop the curve’ as cases are deemed to have returned if this level is reached, and the lower limit is point at which it is felt safe to relax ‘flattening’ measures.

Aren’t All Curves Containment?

Curve 4 is described as a containment curve, but in reality there are also upper and lower limits for the first three curves, which means every curve is about containing cases within limits.

Curve 1, the ‘textbook flattening’, is an imprecise attempt to contain cases below an upper limit, and if this curve was put into practice it would require a ‘lower limit’ to trigger removing social restrictions.

Curve 2, more practical flattening, is a more realistic and efficient version of curve 1 so the same comments apply.

Curve 3 if used for ‘eradication’ still requires an upper limit where ‘stop measures’ would again apply if cases rise, and a lower limit to know when to relax measures. The only way to use curve 3 for ‘herd immunity’ is to follow curve 4.

In summary, any strategy introducing ‘flattening’ measures is a containment strategy. Unfortunately, governments are often not clear themselves on what their upper and lower limit levels should be.

Reading Curve Graphs: Cumulative Curves.

To track an actual outbreak and see if case numbers are rising or falling, an actual graph of case numbers, as provided in typical ‘flatten the curve’ diagrams (and so far in this page) is the most informative option. Sadly, most graphs on outbreaks of Covid-19 plot ‘cumulative cases’ against time, as in sensationalist appeal, the largest number, ‘total cases to date’ is the headline number. So in graphs like this one …

See source for interactive version. Flat lines are ‘killed curves’, gentle slopes are ‘flattened curves’

the lines only ever go upwards, as what is graphed is a cumulative total cases from all days so far, not cases per day as with the ‘curve’ graphs .

Unfortunately, graphs like the above, ‘our world in data’ graph above also uses a logarithmic vertical scale, so it is still difficult to see actual trends.

Using spreadsheet models as described in the curve, will produce cumulative ‘total infected’ data, so it is easy to produce cumulative graphs like this:

However, they are not as illustrative of the curve of curve as graphing ‘cases per day’. Hopefully as the outbreak continues and countries get curves under control, more ‘cases per day’ graphs will start to appear for easier comparison.

Scenario 1. Emergency Lockdown Flattening

Plan: Emergency! flatten the curve! People are dying!

Examples: Wuhan, Italy, Spain, United Kingdom.

In each case, virus infections are already out of control prior to any measures to ‘flatten the curve’. People are dying, the fear (or reality) of medical systems being overloaded demanded the strongest response possible as soon as authorities recognized the severity of the problem.

In some cases, like the UK, there was a plan to have a phased response as per ‘planned flattening’ below, but it quickly became clear things were not running to plan.

Lock downs to ‘flatten the curve’ in this emergency situation generally consisted of governments putting in place every step they considered step possible all at once. The goal was ‘stop the curve’, or at least slow or flatten the curve as much as possible. In all of these cases, it does seem ‘stop the curve‘ has been achieved. Having ‘stopped’ the curve, but after applying the maximum possible lockdown measures at great economic cost, what is next? Each country must now decide where the limits for a containment plan lie.

Exiting Lockdown – Containment goals

Containment plans can be designed to achieve any of the following:

  • continue lockdown measures to head to eradication, containing the outbreak at the lowest level possible
  • remove all measures sufficiently and resume the curve
  • introduce a policy to make adjustments as are needed to contain the outbreak within upper and lower bounds at the highest levels tolerable until herd immunity

Wuhan went for eradication. Eradication means continuing the imposition of economic and social constraints on the population, but means the lowest death toll. Eradication is one of only two ‘endgames’ which are compared in a separate post.

Resuming the curve is really only an option if there have been sufficient cases already that the point of herd immunity is already very close, so the second peak will not overshadow the first as in the model show above. At this time, nationally, no country appears to have had even 1/20th of their population infected, so simply removing all measures would appear extremely dangerous.

Even relaxing measures as part of ‘containment strategy‘ that is targeting herd immunity, on the bases of cases identified so far, would seem ‘bold’. Despite this, in some countries, civil unrest and statements on lifting ‘flattening measures’ suggest relatively high settings for ‘upper limit’. Perhaps regional percentages (e.g. New York city specifically) are higher than national cases percentages?

Whichever path is chosen, the introduction of measures that directly stop the growth in cases number per day, ends any resemblance an ‘outbreak curve’ as seen in flatten the curve diagrams.

Scenario 2: Planned Flattening

Plan: “we will flatten the curve and have a controlled outbreak”
Examples: Sweden, Australia, UK (Initially), USA (state dependent)

Everyone had seen the diagrams. A flattened curve is desired. An outbreak is inevitable, but the outbreak can be managed to deliver herd immunity without health systems being overrun. Or so it was thought.

Countries with this strategy typically started with goal of containment using a ‘curve 2’ style approach. It soon becomes obvious that the goal of preventing overloading the medical system, yet minimizing economic costs, is best served by introducing ‘flattening’ measures in stages as cases rise.

Places such as Australia and the US were delaying and staging measures to flatten to curve until the curve had reached threat levels. Much of the US then went directly to emergency shutdown, and Australia went to safety shutdown.

The key indicator a government is seeking to follow textbook ‘flatten’ the curve, and therefore still achieve herd immunity, is a plan to introduce measures in stage, as case number rise. Stricter and stricter measures introduced in stages with a goal of containing case numbers just below the limit of the medical system.

To maintain the characteristics of a outbreak curve, the peak of the curve arrives as a result of herd immunity reducing spread rates.

In summary, no country has managed to remain with ever rising case numbers of an outbreak curve as seen in diagrams, without either entering into a cautionary lockdown and thus progressing to scenario 3, below, or finding some other way to ‘end the curve’ and thus find themselves with an artificial curve peak, as the same choices on what to do next as discussed in emergency lockdowns above.

Scenario 3: Cautionary Lockdown.

Plan: Outside an emergency, use a lockdown to significantly ‘flatten the curve’ to allow greater preparedness for next steps. Significantly delay any risk of reaching case levels such as those seen in Italy.

Example: Australia, Denmark

Italy started lockdown on March 9, with signs of a crisis already clear.

Most countries planning to follow scenario 2 (an improved version of a textbook flattened curve), had plans to introduce measures with a phased approach, but soon sensed that there may not be time for their phased approach and introduced a lockdown apply the brakes more heavily.

While some countries had already committed to an eradication plan, remaining countries not already in a state of emergency were often left in an ‘in between’ land where eradication seemed to difficult and ‘herd immunity’ to inhumane. The focus for now could be lockdown to buy time without yet having committed to an endgame.

Lockdowns for eradication and to address an emergency are typically as strict as possible with a goal or reducing case numbers as quickly as possible. A precautionary lockdown does not have the same urgency and need not be as strict. A curve is slowed significantly may be all that is required.

While having a lower percentage of citizens already infected, countries in a ‘precautionary lockdown’ now face the same three choices as countries any other country exiting a lockdown:

  • continue lockdowns to progress towards eradication
  • abandon lockdowns and allow the curve to resume
  • move to more relaxed lockdowns that will see cases again rise

Typically, the lower the case number so far, the more appealing eradication may be, and the less appealing abandoning lockdowns would be.

The problem with allowing case numbers to rise again is that the higher the caseload, the greater the expense of dealing with that caseload, so the path makes little sense without an endgame of herd immunity.

Scenario 4: Eradication Shutdown

Plan: The curve is all about herd immunity. The curve shape is the result of the continual increase in herd immunity as more people are infected. But the path to herd immunity appears to mean too many deaths. If herd immunity is not the goal, then end the curve immediately. No more curve!

Examples: Wuhan (China), New Zealand.

For anyone feeling infecting most of the population to reach herd immunity is unacceptable, then there is an alternative.

Textbook flattening the curve has both social distancing and herd immunity in play before the curve trends downwards. China did ‘stop the curve’ in Wuhan if reports are to be believed. But China did so reportedly with methods difficult to reproduce in a freer society. So the idea of bringing cases towards eradication without requiring herd immunity could be seen as a challenge. But given that herd immunity comes at such a great cost in terms of suffering and death, perhaps that challenge is worthy of being accepted.

Under these plans herd immunity (unless by vaccination) has already been abandoned. Exiting lockdown takes on a slightly altered perspective. At the end of each lockdown period, only two choices are left:

  • continue lockdowns to progress towards eradication
  • abandon lockdowns and allow the curve to resume
  • move to more relaxed lockdowns that will may see cases again rise

Once cases numbers are sufficiently low, a ‘flatlining’ plan can then be followed.

Scenario 5: Flat lining

Plan: When case numbers sufficiently low, significant recourses can be applied to new case detection and ‘track and trace’. If such measures are applied from the outset, then case numbers need never rise. There will never be a curve.

Examples: Taiwan, Hong Kong, Singapore.

Can you stop the curve before there is even a curve to stop? If you could eliminate cases as soon as they appear right from the outset, then you would not need all shutdowns and social distancing to stop the curve. The curve would never even start. You would have the same challenge as after the curve is stopped: how do your prevent a new outbreak from building into a curve?

The first thing to realise, is that if you can achieve flatlining, then you never have very many cases, and that would avoid the need for social distancing, and restaurants, businesses, schools to remain open.

Taiwan is possibly the best success so far of applying a ‘flatline’ strategy, with Singapore seeming initially on track, only to find foreign worker camps not sufficiently contained initially.

A flatline strategy requires the backup of a plan to lockdown should cases exceed what is regarded as a limit to the ability to fully track and trace. If the limit is exceeded, then the flatline lockdown scenario then applies.

How to flatline?

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Basically, every time there is hint of a spike of life from the virus, you must kill it off immediately.

The first step is to be extremely vigilant, testing temperatures, looking for symptoms and ensuring there are is a an early test everytime any symptoms are detected.

Secondly, when symptoms are detected, the person must be strict quarantined until test results are in, which in practice is going to be impossible unless test have a fast turn-around time. Then, if tests are positive, the case must kept in strict quarantine until there is certainty the they are no longer contagious.

Thirdly, all contacts must be tracked, traced and tested, even if asymptomatic. There will be asymptomatic people would will only be discovered by there links to cases that do have symptoms.

If all this sounds expensive, they yes, per case it is very expensive. The key is to a maintain a very low number of cases, and be very aware if the real costs both economically and in terms of lives that will be required if this fails.

Flatlining, what is in a name?

You may be wondering where I get such knowledge of

This image has an empty alt attribute; its file name is main-qimg-b92fdee4f3ae26a2f63876ce676c2072.webp

all these terms: flatlining, ‘stopping the curve’. If have heard actual experts speak of the difference between ‘stopping the curve’ and ‘flattening the curve’. However, ‘stopping the curve’ is descriptive rather than a specific term. Flatlining is actually my own name, as I have not found a better term for treating each individual case as a ‘spike’ to be killed off.

In summary, ‘flatlining’ is part of a containment strategy that works with very low case numbers as limits. The resources per case allow ‘flatlining’ without a lockdown, provided low cases numbers allow allocating the resources per case. If cases rise above an the ‘upper limit’, then move to ‘flatline lockdown’ becomes necessary. If cases remain below a ‘lower limit’, then it would be possible to consider easing any restrictions in place at the time.

Scenario 6: flatline lockdown

Plan: Flatlining has not provided sufficient containment, use a lockdown to restore low case numbers in order to return to an improved flatline.

Example: Singapore

Singapore started as one of the scenario ‘flatlining’ countries, as described in this video. However, after this video Singapore found there were clusters not being controlled, particularly among migrant workers and has recently needed to introduce a lockdown.

The goal of a flatline lockdown is to reduce cases below the level at which ‘flatline’ containment measures can again be used to control the outbreak.

Conclusion: What Curve?

No scenario has been found where a continuous curve will continue throughout an outbreak of Covid-19. Every scenario become some version of ‘cuver-4’ where there is more than one peak, and thus no longer a continuous curve.

A big mistake is to still think of there being a curve with a single peak, and one that peak once reached, ‘the worst is over’.

Every situation plan should expect multiple peaks Covid-19. With an eradication plan, the first peak may dwarf any subsequent peaks if thing go perfectly. But Singapore has shown even eradication plans should plan for future peaks. The main lesson is that for Covi-19, a peak does not signal the end of the outbreak. No scenario will run to the end with a curve that actually looks like textbook ‘flatten the curve’ diagrams. ‘Flatten the curve’ never means what the diagrams show.

Table of Contents


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