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Cuchulainn
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Re: Models for Covid-19

March 25th, 2020, 7:26 pm

Paul,
How many houses in the system of ODEs? [$]H_i, i = 1,...N[$]

What's [$]N[$]?
http://www.datasimfinancial.com
http://www.datasim.nl

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Cuchulainn
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Re: Models for Covid-19

March 25th, 2020, 7:38 pm

I'll probably start a separate thread for modelling issues as opposed to numerical. Meanwhile this is the world we live in now.

2020-03-24_110017 - Model 1.jpg
Or this

Image
http://www.datasimfinancial.com
http://www.datasim.nl

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Paul
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Re: Models for Covid-19

March 25th, 2020, 7:58 pm

Paul,
How many houses in the system of ODEs? [$]H_i, i = 1,...N[$]

What's [$]N[$]?
I had ten. You can have as many as you like! You'll have to adjust the parameters though, specifically [$]\beta[$]. I should really non dimensionalize I guess!

I haven't yet figured out how to best aggregate/homogenise the many households.
 
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Re: Models for Covid-19

March 25th, 2020, 8:04 pm

kat, I think what is interesting about the simple model is how it shows that lockdown has to be early, before households have an infected member. Any ideas about how to estimate numbers? IC vs Oxford.
 
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Re: Models for Covid-19

March 25th, 2020, 8:07 pm

Paul,
How many houses in the system of ODEs? [$]H_i, i = 1,...N[$]

What's [$]N[$]?
I had ten. You can have as many as you like! You'll have to adjust the parameters though, specifically [$]\beta[$]. I should really non dimensionalize I guess!

I haven't yet figured out how to best aggregate/homogenise the many households.
For N = 100, this will be a system of 400 equations.
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
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Cuchulainn
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Re: Models for Covid-19

March 25th, 2020, 8:11 pm

kat, I think what is interesting about the simple model is how it shows that lockdown has to be early, before households have an infected member. Any ideas about how to estimate numbers? IC vs Oxford.
That would never catch on. That's why they should have done proper testing up front.
http://www.datasimfinancial.com
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Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
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Re: Models for Covid-19

March 25th, 2020, 8:17 pm

 IC vs Oxford.
Not really. It's which model is best. There are many institutes working on this.
Curiosly, neither IC nor Oxford have published the gory details. Until we see, it's all a bit of hot air at the moment.

What about Matt Keeling at Warwick?

https://homepages.warwick.ac.uk/~masfz/ModelingInfectiousDiseases/Chapter3/Program_3.4/index.html
http://www.datasimfinancial.com
http://www.datasim.nl

Every Time We Teach a Child Something, We Keep Him from Inventing It Himself
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katastrofa
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Re: Models for Covid-19

March 25th, 2020, 8:45 pm

kat, I think what is interesting about the simple model is how it shows that lockdown has to be early, before households have an infected member. Any ideas about how to estimate numbers? IC vs Oxford.
That would never catch on. That's why they should have done proper testing up front.
No such a thing as proper testing. The antibody test the gov buys has very low sensitivity and specificity (high number of false positives and negatives), afaik.
 
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Paul
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Re: Models for Covid-19

March 25th, 2020, 9:26 pm

Paul,
How many houses in the system of ODEs? [$]H_i, i = 1,...N[$]

What's [$]N[$]?
I had ten. You can have as many as you like! You'll have to adjust the parameters though, specifically [$]\beta[$]. I should really non dimensionalize I guess!

I haven't yet figured out how to best aggregate/homogenise the many households.
For N = 100, this will be a system of 400 equations.
That's misleading. You can ignore the [$]R_i[$]. And for deaths you only need [$]H[$] not [$]H_i[$]. So it's technically 201. 

But that's also misleading. 100 are identical, just different initial conditions. Ditto another 100. Not even different parameters! All that really matters is how these are averaged/aggregated, and it's not immediately obvious how to do it analytically. 

If it was really 400 or even 201 I'd be the first to complain! I even complained about seven!

Cuch, could you try it for ten, 100, etc. households and you'll see little change to results (as long as you scale parameters correctly).
 
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Paul
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Re: Models for Covid-19

March 25th, 2020, 9:31 pm

 IC vs Oxford.
Not really. It's which model is best. There are many institutes working on this.
Curiosly, neither IC nor Oxford have published the gory details. Until we see, it's all a bit of hot air at the moment.

What about Matt Keeling at Warwick?

https://homepages.warwick.ac.uk/~masfz/ModelingInfectiousDiseases/Chapter3/Program_3.4/index.html
I mean IC or Oxford initial conditions, not models. I still don't know what the IC model is, and I prefer my model to the Oxford one!

Oxford has many more already infected than IC (I think). 

The wordometers numbers underestimate infected, obvs. 

My best guess is 250,000 in UK infected currently.
 
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katastrofa
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Re: Models for Covid-19

March 25th, 2020, 10:03 pm

"Any ideas about how to estimate numbers?"

You could fit the model to the trends.
 
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Paul
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Re: Models for Covid-19

March 25th, 2020, 10:14 pm

Easy to solve for [$]I_i[$] in terms of [$]S_i[$]. Something like (not checked)

[$]\frac{d^2 (\ln S_i)}{dt^2}+\lambda \frac{d (\ln S_i)}{dt}=(\alpha-\beta)\frac{dS_i}{dt}-\beta \frac{d\sum S_j}{dt}[$]

Sum is over all households, inc. [$]i[$].

[$]\lambda=\gamma+\delta[$]

Prob not the best way to write it. 

And I'll have to figure out initial condition for [$]\frac{dS_i}{dt}[$].
 
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katastrofa
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Re: Models for Covid-19

March 25th, 2020, 10:57 pm

Why not run it through Mathematica? Alan? :-)
 
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Paul
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Re: Models for Covid-19

March 25th, 2020, 11:40 pm

Easy to solve for [$]I_i[$] in terms of [$]S_i[$]. Something like (not checked)

[$]\frac{d^2 (\ln S_i)}{dt^2}+\lambda \frac{d (\ln S_i)}{dt}=(\alpha-\beta)\frac{dS_i}{dt}-\beta \frac{d\sum S_j}{dt}[$]

Sum is over all households, inc. [$]i[$].

[$]\lambda=\gamma+\delta[$]

Prob not the best way to write it. 

And I'll have to figure out initial condition for [$]\frac{dS_i}{dt}[$].
Kat, could you check the above?! I'm going around in circles. (I'm getting different things for the simple [$]\alpha=\beta[$] case depending on whether I sum over [$]i[$] before or after solving for [$]I_i[$]!)

Easiest to write 

[$]\sum_{j\ne i}I_j =I-I_i[$]

to see what is going on.
 
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Paul
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Re: Models for Covid-19

March 26th, 2020, 1:18 am

Ok, integrated once, errors possibly corrected and with initial conditions included we are left with

[$]\frac{S_i'}{S_i}+\lambda \ln S_i -(\alpha -\beta) S_i-\beta S = c_i[$]

where

[$]c_i=\lambda \ln  S_i(0) -(\alpha - \beta) I_i(0) -\beta I(0)-(\alpha-\beta) S_i(0)-\beta S(0)[$],

[$]S=\sum_j S_j[$],

[$]I=\sum_j I_j[$],

and

[$]\lambda=\gamma+\delta[$]

Will need initial condition [$]S_i(0)[$] and [$]I_i(0)[$].
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