COVID-19

Covid-19 discussion, bring your own statistics
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Bird on a Fire
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Re: COVID-19

Post by Bird on a Fire » Sun Feb 21, 2021 8:01 pm

KAJ wrote:
Sun Feb 21, 2021 7:45 pm
Bird on a Fire wrote:
Sun Feb 21, 2021 7:03 pm
Thought I'd have a play with some trajectories to look at the size of expected variations.

This is a super-simple individual-based model. Each infected individual infects n others, where n is drawn from a Poisson distribution with lambda (ie, mean and s.d.) equal to some reproduction number (so assuming a constant halving time). So with R<1, most individuals infect nobody, but some infect 1 or even 2 or 3 etc. This gives a mean that follows an exponential curve, but with stochastic variation around it.

Summing those new infections in each time step gives the total infected population, which then feeds forward to the next time step. A pretty bog standard Markov chain population model, in fact. I've done 1000 reps of each scenario just to get a look.
Just a couple of nit-picks. Please don't be offended.

"...Poisson distribution with lambda (ie, mean and s.d.)..." should be "mean and variance".
"...reps of each scenario..." In this context time is on an interval scale (Wiki) so the origin is arbitrary. That means the later scenarios (N0 = 10000, 1000, etc.) are just later evolutions of one outcome (the mean) of the first scenario (N0 = 30000). Effectively you've zoomed in on different regions of the tail.
Thanks! Pick away :)

I do see what you mean about each plot basically showing different regions of the same process, which is fair enough. Though I quite like the demonstration that you can get this fairly large spread of trajectories in 12 steps, starting from exactly the same point - if I ran these Markov chains for dozens of steps the error would compound going forward, as in the plot in my post above). (Other than that I was just recycling variable names in my code)

I suppose to simulate test data better, we'd want to consider each time step as a single day, and nudge Rt up accordingly.

(The lambda thing was a total brain fart.)
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Re: COVID-19

Post by shpalman » Sun Feb 21, 2021 8:16 pm

The errors compound as the square root of N, though, right?

What would nudging Rt up each day be modelling?
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Re: COVID-19

Post by KAJ » Sun Feb 21, 2021 8:32 pm

Bird on a Fire wrote:
Sun Feb 21, 2021 8:01 pm
I do see what you mean about each plot basically showing different regions of the same process, which is fair enough. Though I quite like the demonstration that you can get this fairly large spread of trajectories in 12 steps, starting from exactly the same point - if I ran these Markov chains for dozens of steps the error would compound going forward, as in the plot in my post above). (Other than that I was just recycling variable names in my code)
With R < 1 all the tracks will eventually* reach zero. Your latest N = 50 R = 0.9 graphs suggest that the number of time steps taken to reach zero varies between about 9 and substantially more than 25. IMHO, if I can mount this hobby horse one more time, this clearly demonstrates the inadequacy of a simple exponential model at low counts, and the need for a probabilistic model.

*Which might be a very long time!

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Re: COVID-19

Post by Bird on a Fire » Sun Feb 21, 2021 8:54 pm

shpalman wrote:
Sun Feb 21, 2021 8:16 pm
The errors compound as the square root of N, though, right?

What would nudging Rt up each day be modelling?
Ah sorry, I just meant that a daily (fixed) R value would have to be higher than one intended to cover a longer period, not daily increases.
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Re: COVID-19

Post by Bird on a Fire » Sun Feb 21, 2021 9:02 pm

KAJ wrote:
Sun Feb 21, 2021 8:32 pm
Bird on a Fire wrote:
Sun Feb 21, 2021 8:01 pm
I do see what you mean about each plot basically showing different regions of the same process, which is fair enough. Though I quite like the demonstration that you can get this fairly large spread of trajectories in 12 steps, starting from exactly the same point - if I ran these Markov chains for dozens of steps the error would compound going forward, as in the plot in my post above). (Other than that I was just recycling variable names in my code)
With R < 1 all the tracks will eventually* reach zero. Your latest N = 50 R = 0.9 graphs suggest that the number of time steps taken to reach zero varies between about 9 and substantially more than 25. IMHO, if I can mount this hobby horse one more time, this clearly demonstrates the inadequacy of a simple exponential model at low counts, and the need for a probabilistic model.

*Which might be a very long time!
Yes, for sure. I think in the real world, it will be far easier to achieve further reductions in R with small case numbers (Australia is probably a good example), and probably in the UK the vaccine will start making a dent before then.

My impression, which I think is also what you're getting at, is that treating covid like a petri dish will work well with large numbers, but other sources of variation will have greater importance in the tails. With my ecologist hat on that's where granularity in things like behaviour and environment will matter more for the numbers (ie mechanisms, which would determine the probabilities in a probabilistic model).

In terms of the discussion at hand, even without any change in lockdown I think it's likely that at low numbers the case numbers could follow a path that doesn't much resemble an exponential, because of the variation inherent in what is basically an ensemble of Bernoulli trials. (In other words I think I agree with you, but might be expressing myself badly!)
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Re: COVID-19

Post by KAJ » Sun Feb 21, 2021 10:32 pm

Bird on a Fire wrote:
Sun Feb 21, 2021 9:02 pm
In terms of the discussion at hand, even without any change in lockdown I think it's likely that at low numbers the case numbers could follow a path that doesn't much resemble an exponential, because of the variation inherent in what is basically an ensemble of Bernoulli trials. (In other words I think I agree with you, but might be expressing myself badly!)
Yes. I think you've clearly demonstrated that as case numbers approach virus elimination, a simple exponential decline is not an adequate representation, even of an extremely simple [simulated] mechanism.

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Re: COVID-19

Post by shpalman » Mon Feb 22, 2021 10:24 pm

Biggest country has most cases and deaths

Its death rate is still lower than that of Italy or the UK.

(According to https://coronavirus.jhu.edu/data/mortality )

(The EU has a whole has had 515519 deaths.)
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Re: COVID-19

Post by wilsontown » Tue Feb 23, 2021 12:28 pm

Hi folks, I thought I would belatedly join in the party.

A simple plot which is just the percentage change in the 7-day case average (by date reported) from the previous day. Looks like the rate of decrease is falling and may even have reversed. Or maybe just a blip...
7day_case_rate_change.png
7day_case_rate_change.png (37.3 KiB) Viewed 551 times
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Re: COVID-19

Post by basementer » Tue Feb 23, 2021 4:00 pm

Hello! Welcome to the forum.
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Re: COVID-19

Post by Trinucleus » Tue Feb 23, 2021 4:39 pm

I would presume that number of cases is less important now than the hospitalisation and death rates, as the vaccine affect begins to come through?

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Re: COVID-19

Post by wilsontown » Tue Feb 23, 2021 4:56 pm

basementer wrote:
Tue Feb 23, 2021 4:00 pm
Hello! Welcome to the forum.
Thanks! I was previously on Bad Science (same username), but I didn't post all that much.
"All models are wrong but some are useful" - George Box

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Re: COVID-19

Post by Bird on a Fire » Tue Feb 23, 2021 5:11 pm

wilsontown wrote:
Tue Feb 23, 2021 4:56 pm
basementer wrote:
Tue Feb 23, 2021 4:00 pm
Hello! Welcome to the forum.
Thanks! I was previously on Bad Science (same username), but I didn't post all that much.
I remember you, wilsontown! Welcome back :)
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Re: COVID-19

Post by lpm » Tue Feb 23, 2021 5:36 pm

Everyone remembers you, because of the [REDACTED] incident. And you were a train. And you were one of the original 2006 blog-commenting crew that became the forum.

You are taking 7-day case average vs the previous day. The story is the same as the James Annan charts, which takes the day vs 7 days previously, and does a line for the 7 day average. Not sure what it means statistically - but the message is pretty concerning: there's been a distinct slow down in the rate of improvement over the past 10 or so days.
What ever happened to that Trump guy, you know, the one who was president for a bit?

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Re: COVID-19

Post by KAJ » Tue Feb 23, 2021 7:24 pm

lpm wrote:
Tue Feb 23, 2021 5:36 pm
Everyone remembers you, because of the [REDACTED] incident. And you were a train. And you were one of the original 2006 blog-commenting crew that became the forum.

You are taking 7-day case average vs the previous day. The story is the same as the James Annan charts, which takes the day vs 7 days previously, and does a line for the 7 day average. Not sure what it means statistically - but the message is pretty concerning: there's been a distinct slow down in the rate of improvement over the past 10 or so days.
As discussed before, I prefer to allow for the day-of-week effect by fitting it rather than averaging over the week. Then looking at cases by publish date does show a small positive second difference in the final few fits, although the first difference is still very negative.
PubCases.png
PubCases.png (15.5 KiB) Viewed 411 times

Code: Select all

At 23/02 fit = 8242.0 with halving time = 26.3 days. That time doubling in 36.7 days 

Analysis of Variance Table

Response: log(PubCases)
              Df  Sum Sq Mean Sq F value    Pr(>F)    
poly(date, 2)  2 10.8175  5.4087  887.34 < 2.2e-16 ***
day            6  0.4195  0.0699   11.47 6.594e-07 ***
Residuals     33  0.2011  0.0061                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Call:
lm(formula = as.formula(aform), data = aDF, weights = weights)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.12345 -0.05140  0.01057  0.05514  0.16957 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     9.87404    0.03189 309.624  < 2e-16 ***
poly(date, 2)1 -3.19435    0.07916 -40.354  < 2e-16 ***
poly(date, 2)2  0.30515    0.07810   3.907 0.000438 ***
dayMon         -0.06185    0.04509  -1.372 0.179402    
dayTue         -0.12143    0.04513  -2.691 0.011096 *  
dayWed          0.04617    0.04526   1.020 0.315114    
dayThu          0.11823    0.04518   2.617 0.013284 *  
dayFri          0.20106    0.04512   4.456 9.07e-05 ***
daySat          0.07148    0.04509   1.585 0.122400    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.07807 on 33 degrees of freedom
Multiple R-squared:  0.9824

	PubCases	date	fitted		dfit.dx 	d2fit.dx2
1	8489	2021-02-23	9306.145	-0.02632030	0.0007172448
2	10641	2021-02-22	9554.338	-0.02703754	NA
3	9834	2021-02-21	9816.187	NA		NA
Again as I've remarked previously, I prefer cases by specimen date to by publish date. For those I zero weight the latest 5 dates as being "incomplete" and for that data the second difference is practically zero, that is the fit is practically straight. That said, the zero weighted data is consistently above the fitted line, can only increase as it becomes complete, and I expect to see this reflected in the regression as the points are included in the next few days. In summary, I see some weak evidence that the rate of reduction in cases has decreased over recent days,
SpecCases.png
SpecCases.png (17.25 KiB) Viewed 411 times

Code: Select all

At 21/02 fit = 6265.9 with halving time = 16.4 days. That time halving in 633.0 days 

Analysis of Variance Table

Response: log(SpecCases)
              Df  Sum Sq Mean Sq  F value    Pr(>F)    
poly(date, 2)  2 10.4842  5.2421 1486.028 < 2.2e-16 ***
day            6  1.1611  0.1935   54.856 9.231e-16 ***
Residuals     33  0.1164  0.0035                       
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Call:
lm(formula = as.formula(aform), data = aDF, weights = weights)

Weighted Residuals:
      Min        1Q    Median        3Q       Max 
-0.160331 -0.021775  0.008375  0.026510  0.092967 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     9.70275    0.02449 396.206  < 2e-16 ***
poly(date, 2)1 -3.85773    0.07834 -49.246  < 2e-16 ***
poly(date, 2)2 -0.01881    0.07873  -0.239    0.813    
dayMon          0.48038    0.03430  14.005 1.93e-15 ***
dayTue          0.39394    0.03433  11.475 4.63e-13 ***
dayWed          0.35132    0.03443  10.203 9.76e-12 ***
dayThu          0.31460    0.03437   9.154 1.41e-10 ***
dayFri          0.27420    0.03433   7.988 3.24e-09 ***
daySat          0.04057    0.03430   1.183    0.245    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05939 on 33 degrees of freedom
Multiple R-squared:  0.9901

	SpecCases	date	fitted		dfit.dx		d2fit.dx2
3	6813	2021-02-21	6265.915	-0.04223398	-3.336159e-05
4	7617	2021-02-20	6536.217	-0.04220062	NA
5	10273	2021-02-19	6817.952	NA		NA


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Re: COVID-19

Post by OffTheRock » Tue Feb 23, 2021 7:31 pm

Down to 45 known active cases (142 this time last week) in Guernsey with 0 new cases for the 1st time in a month. Unlockdown started yesterday so food places are now open for takeaway, some non public facing businesses can reopen with restrictions on the number of staff at work at once (5). All households can now form a bubble with one other household. Still being very cautious with schools as far as I can tell.

Doubt they'll be loosening border restrictions any time soon given this brush with the Kent variant.


Would half term explain the slow down? There seemed to be a lot more people out and about last week than there has been since the start of January. Particularly in groups.

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Re: COVID-19

Post by lpm » Tue Feb 23, 2021 7:45 pm

But KAJ, you've got to focus on the exam question, not the model. The only reason for analysing daily data is to get early warning of a problem fast.

Here's the very simple James Annan chart from last week, data up to Thurs 18 Feb. If the red line and blue line will upwards from the purple, that's an alarm bell.

Image

Here's the latest chart, data up to Monday 22 Feb. Sure enough, the red line is moving upwards. And it'll move another millimetre up when he adds today's data. A problem is visible just from eyeballing alone. Adding in complications and proper modelling can obscure a signal, even if it makes the signal more reliable.

Image
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Re: COVID-19

Post by wilsontown » Tue Feb 23, 2021 8:21 pm

lpm wrote:
Tue Feb 23, 2021 5:36 pm
Everyone remembers you, because of the [REDACTED] incident. And you were a train. And you were one of the original 2006 blog-commenting crew that became the forum.

You are taking 7-day case average vs the previous day. The story is the same as the James Annan charts, which takes the day vs 7 days previously, and does a line for the 7 day average. Not sure what it means statistically - but the message is pretty concerning: there's been a distinct slow down in the rate of improvement over the past 10 or so days.
Not sure if I was in right at the ground floor quite as far back as 2006, but that was me. Nice to be remembered, even if it is for the [REDACTED] incident...

Maybe in a few days case numbers will have continued to fall rapidly and this will look like a blip, but everything seems to be hinting in the same direction just now.
"All models are wrong but some are useful" - George Box

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Re: COVID-19

Post by Woodchopper » Tue Feb 23, 2021 8:39 pm

Recommend long article on why countries like India and Nigeria appear to have such a low mortality rate from Covid-19.

https://www.newyorker.com/magazine/2021 ... han-others

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Re: COVID-19

Post by Woodchopper » Tue Feb 23, 2021 9:08 pm

Countries with fewer cumulative deaths have generally suffered less severe economic consequences as measured by real GDP growth, but the relationship is loose and there are many exceptions.
https://www.thinkglobalhealth.org/artic ... g-pandemic

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Re: COVID-19

Post by KAJ » Tue Feb 23, 2021 9:49 pm

lpm wrote:
Tue Feb 23, 2021 7:45 pm
But KAJ, you've got to focus on the exam question, not the model. The only reason for analysing daily data is to get early warning of a problem fast.

Here's the very simple James Annan chart from last week,
Hmm.

I have the luxury of setting my own exam questions :D . "Early warning of a problem" would be of very little benefit to me, and only of substantial interest if I was confident of a low false-positive rate. My reasons for analysing daily data are principally curiosity, as a pastime, and because I can.

The specific objective I set myself was defensible, quantitative, simple, description of trends in the data. Where simple means simple to me, and my background includes statistical data analysis so `log(y) ~ poly(date, 2) + day` counts as simple. When that gives R-sq of greater than 95% I also regard it as a practically complete description.

As a challenge to myself I also included code to generate a text description of the last part of the fitted curves, you may not have noticed them in the code blocks where they are the first lines:
PubCases: At 23/02 fit = 8242.0 with halving time = 26.3 days. That time doubling in 36.7 days
SpecCases: At 21/02 fit = 6265.9 with halving time = 16.4 days. That time halving in 633.0 days

And any description of a data set, short of the complete data set, entails some kind of model. My experience makes me prefer my models explicit, but the model is always there, whether acknowledged or not. Even use of a simple arithmetic mean makes assumptions about the distribution of the data.

If the data is used to draw conclusions (e.g. "there's been a distinct slow down in the rate of improvement over the past 10 or so days") then some kind of probabilistic model (whether explicit or not) is also being used to decide that the evidence is not just random variation. I don't trust any of the common probabilistic models in this context, so try to to avoid inferences - hence the weasel wording of the conclusion in my last post "I see some weak evidence that the rate of reduction in cases has decreased over recent days", which was intended (like the whole post) as a direct response to the last sentence of your post "Not sure what it means statistically - but the message is pretty concerning: there's been a distinct slow down in the rate of improvement over the past 10 or so days".
(BTW the statistics I present in code blocks often include inferential statistics, but that is just because R's summary.lm includes them, when I read my results I ignore such statistics.)

With respect to the James Annan chart, you say "A problem is visible just from eyeballing alone. Adding in complications and proper modelling can obscure a signal, even if it makes the signal more reliable".
That first sentence expresses a certainty which I find unjustified - how common would such a pattern be in the absence of a "problem"? 4 out of 6 higher than the recent trend?
Complication is in the eye of the beholder, I find numerical statistics add precision and clarity to graphs. Avoiding "proper" models means using inappropriate models. Although I may not approve, I don't object to others informal data analysis - chacun à son goût.

[dismounts hobby horse]

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Re: COVID-19

Post by jimbob » Tue Feb 23, 2021 10:56 pm

Bird on a Fire wrote:
Tue Feb 23, 2021 5:11 pm
wilsontown wrote:
Tue Feb 23, 2021 4:56 pm
basementer wrote:
Tue Feb 23, 2021 4:00 pm
Hello! Welcome to the forum.
Thanks! I was previously on Bad Science (same username), but I didn't post all that much.
I remember you, wilsontown! Welcome back :)
Same here, a train
Have you considered stupidity as an explanation

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Re: COVID-19

Post by jimbob » Tue Feb 23, 2021 10:58 pm

lpm wrote:
Tue Feb 23, 2021 5:36 pm
Everyone remembers you, because of the [REDACTED] incident. And you were a train. And you were one of the original 2006 blog-commenting crew that became the forum.

You are taking 7-day case average vs the previous day. The story is the same as the James Annan charts, which takes the day vs 7 days previously, and does a line for the 7 day average. Not sure what it means statistically - but the message is pretty concerning: there's been a distinct slow down in the rate of improvement over the past 10 or so days.
The Covid.joinzoe.com/data shows it too
Have you considered stupidity as an explanation

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Re: COVID-19

Post by jimbob » Wed Feb 24, 2021 9:09 am

Bird on a Fire wrote:
Sun Feb 21, 2021 9:02 pm
KAJ wrote:
Sun Feb 21, 2021 8:32 pm
Bird on a Fire wrote:
Sun Feb 21, 2021 8:01 pm
I do see what you mean about each plot basically showing different regions of the same process, which is fair enough. Though I quite like the demonstration that you can get this fairly large spread of trajectories in 12 steps, starting from exactly the same point - if I ran these Markov chains for dozens of steps the error would compound going forward, as in the plot in my post above). (Other than that I was just recycling variable names in my code)
With R < 1 all the tracks will eventually* reach zero. Your latest N = 50 R = 0.9 graphs suggest that the number of time steps taken to reach zero varies between about 9 and substantially more than 25. IMHO, if I can mount this hobby horse one more time, this clearly demonstrates the inadequacy of a simple exponential model at low counts, and the need for a probabilistic model.

*Which might be a very long time!
Yes, for sure. I think in the real world, it will be far easier to achieve further reductions in R with small case numbers (Australia is probably a good example), and probably in the UK the vaccine will start making a dent before then.

My impression, which I think is also what you're getting at, is that treating covid like a petri dish will work well with large numbers, but other sources of variation will have greater importance in the tails. With my ecologist hat on that's where granularity in things like behaviour and environment will matter more for the numbers (ie mechanisms, which would determine the probabilities in a probabilistic model).

In terms of the discussion at hand, even without any change in lockdown I think it's likely that at low numbers the case numbers could follow a path that doesn't much resemble an exponential, because of the variation inherent in what is basically an ensemble of Bernoulli trials. (In other words I think I agree with you, but might be expressing myself badly!)
We also know from early in the pandemic that Covid seems particularly "clumpy" compared to other infections in the way it spreads - remember the focus on superspreader events when prevalence was low?

That would imply that simple exponential models will start to fall over at higher prevalences than for other infections
Have you considered stupidity as an explanation

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Re: COVID-19

Post by jimbob » Wed Feb 24, 2021 7:52 pm

jimbob wrote:
Wed Feb 24, 2021 9:09 am
Bird on a Fire wrote:
Sun Feb 21, 2021 9:02 pm
KAJ wrote:
Sun Feb 21, 2021 8:32 pm

With R < 1 all the tracks will eventually* reach zero. Your latest N = 50 R = 0.9 graphs suggest that the number of time steps taken to reach zero varies between about 9 and substantially more than 25. IMHO, if I can mount this hobby horse one more time, this clearly demonstrates the inadequacy of a simple exponential model at low counts, and the need for a probabilistic model.

*Which might be a very long time!
Yes, for sure. I think in the real world, it will be far easier to achieve further reductions in R with small case numbers (Australia is probably a good example), and probably in the UK the vaccine will start making a dent before then.

My impression, which I think is also what you're getting at, is that treating covid like a petri dish will work well with large numbers, but other sources of variation will have greater importance in the tails. With my ecologist hat on that's where granularity in things like behaviour and environment will matter more for the numbers (ie mechanisms, which would determine the probabilities in a probabilistic model).

In terms of the discussion at hand, even without any change in lockdown I think it's likely that at low numbers the case numbers could follow a path that doesn't much resemble an exponential, because of the variation inherent in what is basically an ensemble of Bernoulli trials. (In other words I think I agree with you, but might be expressing myself badly!)
We also know from early in the pandemic that Covid seems particularly "clumpy" compared to other infections in the way it spreads - remember the focus on superspreader events when prevalence was low?

That would imply that simple exponential models will start to fall over at higher prevalences than for other infections
And coincidentally, this article appeared in my email inbox just now:

https://www.nature.com/articles/d41586- ... b-44602497
Experts say that we already know enough about the main factors of superspreading to use this phenomenon to our advantage. They are calling on policymakers to harness this knowledge to target control measures that will slow — or even stamp out — the pandemic. One of the most basic steps is closing crowded, indoor hotspots to prevent superspreading events. Researchers also recommend following Japan’s lead, by using backwards contact tracing to uncover superspreading events.

Explosive transmission
On average, each person who contracts SARS-CoV-2 will pass it on to between two and three others. But that tidy population-level estimate — known as the basic reproduction number (R0) — hides immense variation at the individual level. In reality, most infections arise from just a handful of people (see ‘Infection connections’). Endo’s early analysis estimated that around 10% of cases in countries outside China accounted for 80% of secondary infections up to the end of February2.
Have you considered stupidity as an explanation

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shpalman
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Re: COVID-19

Post by shpalman » Thu Feb 25, 2021 6:14 pm

Nearly 20,000 new cases in Italy today, and there's a quite obvious upward trend developing. The province of Como has trended up from ~140/week/100,000 ten days ago, to >250/week/100,000. Lombardy is just going over 200/week/100,000, the nation as a whole is 170/week/100,000.

See what the local news says, whether Lombardy risks becoming an Orange zone and/or whether Como will be a Red or "dark Orange" province following the next round of evaluation.
molto tricky

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