The depressing math of swine flu, ICUs and ventilators

by revere, cross-posted from Effect Measure

One of the most feared outcomes of infection with influenza is Acute Respiratory Distress Syndrome (ARDS; in less severe form it mahy be called Acute Lung Injury, ALI). For reasons we still do not understand, cells deep in the lung that are involved in gas exchange (oxygen and carbon dioxide) become so damaged that the basic work of supplying the body with enough oxygen for life and getting rid of the carbon dioxide generated by metabolism is too much for the patient and either some intervention to relieve the lungs of some of the work is made or the patient dies. ARDS is so severe that often no intervention works, and fatality ratios of 50% are quite typical. The most common intervention is a mechanical device called a ventilator to do some of the work of breathing for the patient. Critical care respiratory therapy is much more than pumping air in and out of the lungs, however. It is a very complex and tricky art, and it is now believed by many that conventional mechanical ventilation can make ARDS worse and decrease the odds of survival. The literature on ventilation in ARDS is highly technical, and advanced methods using sophisticated computer-controlled devices are often needed.

Both the devices and the expertise to use them in ARDS are in short supply and will be a significant bottleneck if the flu pandemic is severe this fall. Even trying to figure out how serious the situation might be is fraught with difficulty, as a recent paper in PLoS Currents/Influenza demonstrates. It was an attempt to figure out how much stress will be put on Intensive Care Unit capacity given reasonable assumptions about how swine flu might evolve. The method used to make the estimate is easy to understand because it is is so simplistic, but even so, illustrates the difficulties.

The short paper, more of a back-of-the-envelope calculation, was done by researchers from a consulting group and hospitals in California and Washington DC (Swine origin influenza A (H1N1) virus and ICU capacity in the US: Are we prepared?, Zilberberg et al., PLoS Currents/Influenza). Here's what they did. They start with the population of the United States, which they give as 307,024,641 (NB: all their estimates are expressed with the same degree of meaningless excess of significant figures, so I am going to round their numbers for clarity; for example, in this case I'd just will say 300 million). They then assume that the attack rate (per cent of the population infected) will be 15%, but allow it might be as low as 6% or as high as 24%. Thus they are acknowledging uncertainty around the 15% number, which they will use in a way I'll describe shortly. These figures were taken from numbers given at a CDC press briefing on August 24, so they represent CDC's estimates at that time of the pandemic. Once flu season starts in earnest, attack rates of 30% may be more likely, but for this paper 15% was used. That attack rate represents about 46 million flu cases, but with the uncertainty included, they come up with numbers between 37 and 55 million. You can think of this as a confidence interval around the 46 million point estimate. How did they get these numbers?

They used something called a Monte Carlo simulation and it works like this. The 46 million number is easy to see. It's 15% of 300 some odd million. If you were using an Excel spreadsheet (which is what they used), you would enter the US population in one cell, the attack rate in another cell and then have a formula that multiplied the two together in a third cell. The problem with this is that while the population of the US is a fairly stable number, the 15% attack rate is a guess. There might be quite a lot of plausible attack rates, of which you judge that 15% is the most likely but many others possible. Think of a bell-shaped curve with 15% in the middle and many values higher and many lower. This bell curve can be very spread out or very sharp and peaked, depending on how sure you were about the 15%. What the authors did is construct a bell curve that used the two other figures expressing the uncertainty (the 6% and the 24%) in terms of its spread. Imagine, now, a gigantic pile of tokens with attack rates written on them (many will have 15% written on them, somewhat fewer that say 14% or 16%, and even fewer that a percent or two of numbers that are less than 6% or higher than 24%). Now instead of putting 15% into your spreadsheet cell, you reach into this gigantic pile of tokens and pull one out and put its value in the cell instead. Maybe it's 19% or 11% or 15% (since there are more 15% tokens than anything else, but depending on the spread of the bell, there still may not be that many). Now let the spreadsheet calculate the number of cases (the token-drevied attack rate times the population). That will give you a number of infected cases. Now put the token back in the pile and do it again. And again. And again. The authors did just that ten thousand times, although of course they used a computer and an add-on to Excel called Crystal Ball which allows you to stipulate various kinds of statistical distributions for your token pile. Theirs was bell shaped, as described. When you do this you get another bell shaped curve representing the range of infectives, complete with the peak value (the center of the distribution) and the 95% confidence interval (in this case the part of the bell with 95% of the stuff under it).

Now you can do the same thing for the rest of the calculation. Let's go back to the single calculation. You've got the number of infected cases in the cell with the formula (population times attack rate) and you multiply that by the proportion of cases you expect to be hospitalized. They used data from California for swine flu H1N1 which was estimated at that point to be 6% with a low of 2% (assumed) and a high of 10% (from the early Mexico data). Some of those hospitalized will be so sick that they wind up in the ICU on ventilators (12% from Mexican data, low and high guesses of 6% and 18%), and of those, again using Mexican data, 58% end fatally (low and high guesses of 40 and 60%). Now they do the random selection in each of these cells simultaneously 10,000 times, and that's how the joint confidence intervals are figured. Here are the results (rounded estimates and 95% confidence intervals):

  • US pop: 300 million
  • Cases: 46 million (37 - 55 million)
  • Hospitlizations: 2.8 million (2 - 3.6 million)
  • On ventilators: 132,000 (228,000 - 454,000)
  • Deaths: 192,000 (126,000 - 226,000)

This corresponds to an overall case fatality ratio (CFR) of out 0.5%, comparable to the early estimates from Mexico. Since some of the most important data is from those early weeks, this isn't surprising. But since there has been a lot of questioning of the applicability and accuracy of the Mexican CFR to the US setting. In a paper published in Eurosurveillance, a group from New Zealand used numbers from later in the pandemic and a variety of different CFR estimation methods and came up with dramatically lower numbers (0.06% to 0.0004%). Using a 30% infection rate (twice what the ICU paper used) still resulted in dramatically lower numbers for deaths. Instead of the roughly 400,000 deaths the US would expect under the ICU paper assumption (taking account of a 30% infection rate), the estimates from the Eurosurveillance paper would run from 3600 deaths to 54,000 deaths (there were no ventilator estimates).

Thus we have point estimates for the US that go from 3600 to 400,000 deaths, with some in between. The 3600 deaths estimate (a CFR of 0.0004%) seems wildly low, given that swine flu seems much like seasonal flu in virulence. Even the 0.06% figure (54,000 deaths) seems low, given the usual seasonal flu estimate is 0.1%. But in every case where assumptions have been made, they seem plausible, even if markedly divergent. Unfortunately the differences are of great significance. The ICU figures may be too high because of the use of Mexican estimates that were biased upward, but they only used a 15% attack rate which is likely too low. And the cases will not be coming into hospitals uniformly but in huge clumps or bunches. No hospital has a reserve of expensive mechanical ventilation equipment.

One can easily see that the scenario depicted by Helen Branswell in her piece, "War against H1N1 likely to be fought in intensive care units" is likely right on target. Given the numbers, it's hard to see how we can win that war.

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