Posted on 04/17/2005 12:10:40 PM PDT by Lessismore
The Angolan Ministry of Health has announced that the death toll from the epidemic of Marburg hemorrhagic fever has jumped to 233 from a total of 255 cases recorded in Angola until Saturday.
According to a press release issued here by the ministry on Sunday, all the reported cases of the Ebola-like disease had originated in the northwestern province of Uige, though patients have died elsewhere, including Luanda, Zaire, Malange, Kuangza Norte, Kuanza South and Cabinda provinces.
The rising death toll in the absence of patients in the isolation wards in Uige and Luanda supports the observation that patients are dying in neighborhoods and avoiding hospitals. These numbers would also suggest that infected individuals are moving out of the area. However, it is difficult to see why such movement would not result in transmission in the new areas, because the families would not be trained in infection control. The large number of fatal Marburg cases in health care workers indicates that transmission associated with close contact with dead or dying patients is quite efficient.
The increase in patients follows a period of reclassification of Marburg cases. Health reports from last week suggested the requirements for inclusion in the case tally were being tightened. All of the new reported cases in Uige for April 13 were laboratory confirmed, suggesting lab confirmation may be a new requirement for inclusion as a reported case. That requirement may be more easily met in Uige, since there is now a new lab set up with rapid turnaround of testing data. However, a lab confirmation requirement may produce a significant undercount, especially in more distant provinces. Sample collection may be absent in some cases, and in others transport of collected samples to testing facilities may produce sample degradation and false negatives.
The reclassification did eliminate cases from Kwanza Sul, where there were cases clustered in 3 adjacent municipalities. The reported cases in Zaire were also eliminated administratively by reclassifying them.
Efforts to manage an outbreak by simply administratively eliminating cases because of unrealistic requirements can lead to gross undercounts and undetected spread of virus. The analogies with the bird flu management in southeast Asia have some striking parallels. The bird flu undercounts were driven by several factors, including unreported data, untested patients, and using tests with poor sensitivity. False negatives due to poor testing procedure or sample degradation also contributed to the gross undercount.
The data management practices have been particularly disastrous in Vietnam, which is likely to be the epicenter of the next flu pandemic, which may have already begun. In spite of almost daily press releases citing provinces that were bird flu free because of no detectable virus, a recent report indicated that 71% of the ducks in all 11 provinces in the Mekong Delta were positive for H5N1, as were about 25% of the chickens. These numbers were so high, that management via culling was considered to be a poor option.
Similarly, the size of human clusters continues to grow. Confirmatory data from the largest clusters is being withheld while 1000 samples are sent to the CDC for analysis. Since these samples were collected over 3 weeks ago, it would seem that most of the samples are positive, and CDC is analyzing sequences to identify which molecular changes are correlating with which clinical diseases. However, it seems quite clear that while Vietnam was administratively eliminating the virus, the H5N1 virus itself became so entrenched that the number of options has diminished.
Unfortunately, the administrative management of Marburg via press release and reclassification appears to be on the same misdirected path. The approach eliminates reports of virus that don't match models or wishful thinking, while the virus continues to spread beyond control.
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Thanks.
You are both added to the ping list. b'ville, if that's not okay, please let me know.
As with many of you I have been troubled by the lack of consistency in the data, so I decided to make a more careful analysis. My original analysis was a simple regression: I fit the data to a wide variety of functions, and the exponential was the closest fit with a time constant (e-folding time) of 9 days. That lead to extraordinary predictions (e.g. a million people infected by July 1).
However, there are several problems with my original analysis:
1. It indicates a basic model for 1st order DE growth, namely that the growth rate depends on the size of the infected population. This model makes sense for the propagation of most diseases because people don't die. HOWEVER, WITH MARBURG, EVERYBODY DIES AND THIS CHANGES THE DYNAMICS.
2. We had an important piece of data which we did not use, namely the ratio of cumulative deaths to cumulative infections, which was running about 92%. (Everybody dies, but they don't die immediately).
3. The WHO plotted the data and got a linear growth rate.
4. The growth rate seemed to slow dramatically recently.
So.... I reworked the analysis and the data. The differential equation now includes the fact that people die, and when they die they no longer infect others because they are buried (there is a slight modification to this which I will include later). The new equation gives some interesting results. Define:
alpha = infection rate = rate one person with Marburg infects others
tau_d = average time to die after symptoms and infectiousness appears
I = Instantaneous number of infected persons (not dead)
I_c = cumulative number of infected persons
I_o = growth equation constant
D_c = cumulative dead people
The solution to the equation is:
I = I_o exp((t/tau_d)((alpha*tau_d -1) / tau_d))
This equation is also exponential but it has some interesting features. Consider the product term: alpha*tau_d. This product term is easily interpreted as the number of people a person infects during the time he has symptoms and is still alive. So, if a person exhibits Marburg symptoms and infects 3 people before he dies, then alpha*tau_d = 3. If:
alpha*tau_d = 1 ==> steady state, no growth
alpha*tau_d < 1 ==> containment, the epidemic dies out
alpha*tau_d > 1 ==> exponential growth
alpha*tau_d = 2 (for example) ==> exponential growth with tau_d e-folding time.
Therefore, the growth / containment of this epidemic is strongly dependent on this parameter. A modest change of alpha*tau_d changhes the nature of this thing entirely. The result is common sense. Assume that tau_d (average time to live after symtoms present) is 9 days (approximately consistent with multiple reports). Then if each infected person infects just 2 others in that 9 days, this thing grows with the 9 day efolding time. However, if they infect only 1 person in that 9 days, we achieve steady state (defined as I remains constant and I_c increases linearly). Modest changes in behavior could effect this, perhaps.
With a little more algebra, we also can calculate the parameter alpha*tau_d in terms of the ratio of the cumulative infected cases and the cumulative deaths. The result is:
alpha*tau_d = I_c/D_c
Well: I_c/D_c = 1/.92 (it has been running 92% for most of the data)
Therefore: alpha*tau_d = 1.086
This is very close to 1, which means that the growth (although it is actually exponential) appears linear, just like the WHO data showed. (To make this consistent recall that the first two terms of the Taylor expansion of the exponential are a constant and a linear term). Assuming that tau_d (the time from symptoms to death) is nominally 9 days, then the actual exponential growth rate is 9* (1/(1/.92 - 1)) = 103 days!!! That is a lot slower.
Well, this is pretty consistent. The death to cumulative infection rate is consistent with both the growth of new cases and also the data plotted by WHO. Moreover, the change in behavior is consistent.
Consider a modification to this model. Consider the case that people infect others after they die because of the burial customs. How do we account for that?? The solution is a simple extension of the model. Define tau_db = time after death until buried. Assume that alpha, the infection rate remains the same until burial (we don't have any other information otherwise). Then, make the simple substitution:
tau_d ==> tau_d + tau_db
Then the relevant parameter is alpha*(tau_d + taud_db), which must be larger than simply alpha*tau_d.
In this case the growth rate could have been much larger because of the extreme sensitivity of the results to this parameter and becuase alpha*tau_d is so close to 1. We might have had real exponential growth. With the behavioral changes and, frankly the fear involved, tau_db may be effectively zero (i.e. no one gets infected anymore by dead bodies). Moreover, there is better isolation, so that alpha*tau_d has dropped.
We used to have exponential growth, but it has slowed dramatically. If we can drop alpha*tau_d to 1, the epidemic stops.
This also gives us another way to analyze the data and track the epidemic. We now should watch the ratio of cumulative deaths to cumutive infections. As this approaches one, the epidemic stops. If, for some reason, this number drops below one, we have a problem.
Final note for those of you who are mathematically inclined. You may note that in order to have a decaying case, alpha*tau_d < 1, which means that I_c < D_c, which is clearly impossible. The reason for this discrepancy is that the model is linear but the process is non-linear because alpha*tau_d actually varies. Not to worry. If the total deaths equals the total infected cases, this thing stops.
uhhhhhh? Yeah, ok, whatever you say....thanks for being the smart one in the barrel :-)
Sorry, it's the scientists' disease, we feel we have to explain everything.
Just believe the conclusion: if the total # deaths = # cumulative infections, this thing stops.
If we have a lot more infections so that that total # deaths is much less than cumulative infections, it grows.
If you think about it, I have managed to spend a lot of effort on mathematics to prove what you could probably infer from common sense.
And the graph looks like????
Sure, that would work fine.
Originally, I didn't do it that way because alpha and tau_d were defined differently (and intuitively) when I did the derivation. But you change works nicely.
The other reason I separated the effects was that the behaviors when someone is alive is different than when they are dead, so keeping separate constants allowed for this. However, since we don't have any behavioral data, your approach makes sense too.
You've gone to a whole lot of fuss for us, and I can't thank you enough.
I could follow your rationale to about halfway through. I'm going to try again later after about a gallon of espresso. ;-D
You are amazing, FRiend.
Good point! May as well design in as much flexibility to the model as you can in the beginning. It's sure easier than adding terms later!
Hmmm...
I hate to say it, but I do not think we have accurate enough data to be doing any meaningful statistical analysis. About the best I think we can say with any confidence is that the number of cases is still increasing.
Putting on my old, dusty statistics professor hat from 25 years ago, I seem to recall a basic tenet of statistics: Garbage in, garbage out. I don't teach statistics anymore, but I occassionally testify as an expert witness involving heavy use of statistics. Right now we're at the "fun to play with the numbers stage", but without confidence in the source data, it is not possible to have any confidence in the results.
My pleasure.
Thanks for the kind words.
I have sent a new plot to Covenantor, so we can see the graph of this.
I certainly don't disagree with that. Indeed, the actual data we have been working from and regressing to seems all over the place.
However, there are some observations that seem stronger than others, and I argue it makes sense to develop a model on the consistent points of the data rather than fitting to data that are all over the place.
The two essential observations that seem robust are:
1. Everyone dies.
2. The ratio of cumulative deaths to cumulative infections has remained very constant at .92, regardless of the data set.
So, I have corrected my earlier results.
Gotta agree. Sigh.
2. The ratio of cumulative deaths to cumulative infections has remained very constant at .92, regardless of the data set.
I can see how that might be a meaningful ratio. But we don't know how many people have died, or how many are sick but not yet dead, so even if the ratio is valid we don't have meaningful data.
I would add a 3rd point to your list: The number of cases is still increasing.
Beyond that, it looks to me that we are in "wait and see" mode. I doubt the government of Anglola will be the first to tell us, but if it breaks out in a big way in Luanda we will still find out. If it wipes out most of Uige we will find out. If it shows up in the U.S., we will know. Etc.
Certainly
However, I would argue that as with all data, there is a level of uncertainty. If it shows up in the US, we have a low level of uncertainty, as you point out. The data so far have a very high level of uncertainty, i.e. very serious undercounts. But, by using what we do know (everyone dies and the instantaneous death rate is 92%), we have a better level of uncertainty than we do without it. We can also explore the underlying dynamics by modeling it independent of the raw data.
I would not, of course, argue that the model is ultimate truth. It is simply better than what we had before.
Wow. Probably need to take a night class to wade thru that :)
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