Originally Posted by
invisible
Understood. At the same time, we can do back on the envelope calculation with current growth factors how many hospital and ICU/CCU beds would be required if hospitalization time is 7, 10, 14 or 21 days and if serious/critical percentage of hospitalized are 5, 7, 10, 15%. Right?
I posted an analysis of this problem (hospital bed, or equivalently ICU bed, demand) earlier in the thread (seems an age ago now the way things have developed since then ... ). It is good to see though that now more focus is being placed on this aspect of the epidemic. This is a little more sophisticated than the one referenced in the twitter thread linked some way above by
karenkay , but still relies on estimates of some average numbers for the key parameters (i.e. doesn't consider a distribution of values).
For anyone that it interested you can play around with the numbers using this spreadsheet here
(link to Excel file):
The inputs required (and factors that influence the demand for hospital beds are:
(1) the initial number of cases;
(2) the doubling time;
(3) the fraction infections that require hospitalization;
(4) the average number of days after infection that these patients need hospitalization;
(5) the average duration of hospital care needed.
The same model can be used for the demand on ICU beds (changing the inputs in (3-5) appropriately).
In turns out that the fraction of total cases needing a bed settles to a fixed value based on the values of (2-5) above. Interestingly the duration of care needed doesn't have a large impact, as for most sensible values this is already much larger than the doubling time (so the number of patients admitted in the last few days greatly outweighs the much smaller number admitted several doubling-times before).
For anyone interested in the mathematics, I give a walk-through of the derivation of this steady state in the spoiler-box below.
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If we assume exponential growth, the number infected on day t is N(t)=t0exp(Lt), where t0 is the number on day 0 and t is time, taken here to be units of days. This is the standard definition of exponential growth.
In this case L is the growth constant, and can be most easily understood in terms of the doubling time d, which we also need to define in days to keep the units correct, by L = ln(2)/d = 0.693147/d.
The new infections on day t can then be written as N(t) - N(t-1).
This just says "the new infections today are the difference between the total number infected today, N(t), and the total number that were infected yesterday, N(t-1)".
Similarly the number of new infections between any two days t1 and and t2 = N(t2) - N(t1), as this is just the difference in total numbers on these two days.
We want the know the number of people in hospital on day t . Let's call this H(t).
If the average patient needs a bed for b days, then on day t the number of required beds is the sum of all new hospital admissions over the previous b days.
To relate this though to the number of infections we need to remember also that hospital admission is not required immediately after infection, but after some days. Call this "treatment incubation period" an average value of a days.
Also the number requiring hospital beds is just a fraction f of the total number infected. So of the people infected on a given day, only a fraction of these, f, will need a hospital bed a days later.
So the number we actually need is the fraction f of the total number of new infections from day t-b-a (read that as "t minus b minus a") to day t-b (t minus b). This corresponds to a period of b days starting from day t-b-a, and can be written as N(t-b) - N(t-b-a). Note that new infections on day t-1 (or any day counting back b days) don't matter (yet) as these patients are still in the phase of developing symptoms.
If that still isn't clear note that the patients admitted on day t will be a fraction f of the the newly infected population on day t-a. Similarly the patients admitted the previous day t-1 will be the fraction f of new infections on day t-a-1. And so on back to day t-b, where the new admissions are from those infected on day t-b-a.
Now we just need to combine all these relationships:
H(t) = f[N(t-b) - N[(t-b-a)] = f[t0exp(L(t-b)) - t0exp(L(t-b-a))]
We collect terms outside the bracket, and separate the exponential terms
H(t) = ft0[exp(Lt)exp(-Lb) - exp(Lt)exp(-L(a+b))]
We can now express this as a fraction of the total cases by day t, by dividing H(t) by N(t) = t0exp(Lt)
In doing this the t0 and exp(Lt) terms both cancel out (hence the rearrangement above) giving:
H(t)/N(t) = f [exp(-Lb) - exp(-L(a+b))]
i.e. the ratio of the number of patients in hospital compared to the total number of infections is a fixed number, that depends on the values of a, b, L (given by the doubling time, d), and f.
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In other news ... some updates from here in Beijing regarding implementation of social distancing:
(1) Large shops/supermarkets have to stick to a maximum customer density, so in theory you many need to queue to get into a shop. So far this has not happened to me (though it is clear they are keeping track of the number in the shop at any time)
(2) Banks and other places with a service counter follow the same system, but here queues outside are more common, and most have used tape-markings on the sidewalk/pavement to indicate 1m or 2m distance between people in a queue. Remarkable thing, despite much former evidence to the contrary, it turns out that people in China do know "how to queue" - they just needed some tape markings to help with the organizational aspect
I had to go to a bank a couple of days ago - what would normally have taken maybe 10 minutes took 40 (with only 2 customers allowed in the bank at any time - as just two bank staff were serving customers).
(3) Restaurants are still mostly either closed or doing (good) business on a take-out/delivery basis. Where they are open the rule mentioned above about "1 person per table" does seem to be implemented as far as I can tell, though apparently this is not a Beijing govenrment rule (only "advisory").
(4) One of the main hurdles to getting the city fully functioning is the large number of workers that would need to use public transport to get from the suburbs to work. The company my wife works for (a state-owned legal office) is ready to start working as normal, but most likely won't as most of the work can be done from home, and companies in this position are encouraged to maintain the "home-office" model, so as to reduce the demand on the bus/metro system for employees who cannot work from home.
(5) In that regard I read that there was a trial of a "pre-booking" system for travel on the metro during rush hour. This works by each day people having to reserve a time-slot for travel on the metro, with only a certain number of people allowed into the metro system during rush hour (in practice done by getting a QR code on a smart-phone that allows access to the metro), This trial was carried out I think at two or three stations - but I haven't heard any feedback yet regarding how it worked, or if it will be extended.