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Tuesday, August 26, 2014

Forecasting the 2014 Ebola Outbreak

April 29, 2015
My latest update regarding the history of the Ebola epidemic can be found here:
August 26, 2014
The number of Ebola cases is growing exponentially. Using two methods, the cumulative number of cases is extrapolated based on the growth trend of the epidemic.  If the disease continues to grow unchecked, the extrapolation shows the cumulative number of Ebola cases might grow to one million cases within six months, and one billion cases within twelve to eighteen months.

Forecasting the 2014 Ebola Outbreak
The Ebola outbreak in West Africa is out of control, and the magnitude of the outbreak has been underestimated.  This is according to statements from an aid organization fighting the outbreak, and the World Health Organization (WHO).

Ebola virus disease is a severe hemorrhagic fever, contagious and generally fatal.  Mortality in previous outbreaks ranged between 60% and 90%.  The contagiousness of the disease has been downplayed in the media, because Ebola is not transmitted through the air like flu or colds, but rather through direct contact with body fluids.  Nevertheless, the experience of aid workers in West Africa shows that even experienced health professionals, using best practices of isolation and decontamination, often become ill and die from the disease. 

The World Health Organization reports the number of officially diagnosed cases and deaths, while cautioning that actual numbers may be significantly higher.  As of August 20, 2014, there have been 2615 diagnosed cases and 1421 recognized deaths.  Newly diagnosed cases are now appearing at a rate of about 100 per day. 

Rate of Growth
The number of new cases (and subsequent deaths) is rising sharply.  A chart of the cumulative cases and deaths shows the characteristic curve of exponential growth. 
Chart 1
By presenting the data on a logarithmic scale, we see the data appears as a straight line, indicative of exponential growth.  The history of the outbreak seems to divide into two phases, before and after mid-May 2014.   In the early phase of the outbreak, there was apparently very rapid growth, but this probably reflects late recognition of existing cases.   The curve then flattens on the logarithmic chart, showing a measure of success in containing the disease.  In the second phase, the data corresponds more directly to a straight line on a logarithmic scale.   In this phase, the disease has escaped the control of medical isolation, and is propagating at a constant rate.
Chart 2
The growth of Ebola cases is not linear, but obeys a power law.  Mathematically, this is the same as the power law that governs the size distribution of oil fields, the magnitude of earthquakes, the severity of stock market crashes, and the lives lost in terrorist attacks.   The power law can be determined by taking an exponential regression through the growth of Ebola cases to date, which is simply done in Excel.  The equation derived by the regression is as follows, counting days from May 23, 2014:
Number of Cases = 286*e(0.0239 * number of days) 
        [Counting Days from May 23, 2014.]
Chart 3
The first method of extrapolation in this post is based on the exponential regression.  The equation can be used to extrapolate the number of cases we might expect in the future, if the rate of growth continues as in the past.  Exponential growth is sometimes startling.  If the current rate of growth continues, cumulative cases of Ebola would total one million in about eight months, near the end of April, 2015.
Chart 4
Even more startling is to continue the extrapolation.  If the rate of growth continues unchecked, the number of Ebola cases would pass the one billion mark in less than a year and a half, in February, 2016.
Chart 5
Mortality due to Ebola virus is very high.   Death usually occurs 8 or 9 days after the onset of symptoms.  The data released by WHO necessarily contains a lag between the number of diagnosed cases and the number of deaths.  Assuming a nine-day lag, we can calculate mortality and survivorship for the cases represented in the WHO statistics.  Mortality fluctuates between 65 % and 75 %, with survivorship being the complement, ranging from 25 % to 35 %.  As noted by WHO, survivorship in the current outbreak is higher than in previous Ebola outbreaks. 
Chart 6
[Note:  The WHO website contains an erroneous calculation of survivorship, of 47%.   The figure presented by WHO appears to be the result of dividing the current number of deaths by the current number of cases (and subtracting one) without accounting for the lag between diagnosis and death.]

I interpolated data from WHO in order to obtain a daily record of the number of cases.  The chart is shown below.  The data are noisy, but also show two phases in the progress of the outbreak.  In the initial phase, medical intervention contained the disease almost to the point of elimination.  In the subsequent phase, the number of new cases rose sharply, and continues to rise.
Chart 7
Epidemiologists use a variable, Ro, to represent the infectiousness of a disease.  This factor is the basic reproduction number, or the number of uninfected people who will catch the disease from a single infected person.   Ro for many childhood diseases is quite high; for instance, Ro for measles is 15.  Smallpox is 6, and the deadly 1918 Spanish flu was 3.  The typical seasonal flu has an Ro of only 1.2 or 1.3, but nevertheless, many, many people catch the flu.  

The other factor necessary to estimate the rate of contagion is the time between subsequent infections.  The Ro value for HIV/AIDS is 3.5, but years may pass between the initial infection and subsequent infection.  Therefore the spread of the disease is relatively slow.   For 2014 Ebola, I calculated the value of Ro, while varying the time lag until the subsequent infection.  I found the best match with a lag of 8 days, representing the time between diagnosis of the first infection and diagnosis of the second infection.  

Chart 8
The following chart represents Ro for the 2014 Ebola outbreak, assuming an 8 day lag between infections.  I have to assume that the reservoir of unreported cases is neither adding nor subtracting from the development of new cases.  For the second phase of the outbreak, represented by a straight line on the logarithmic chart, Ro has ranged from about 1.0 to 2.0.

Using a lag of 8 days between subsequent infections, I varied Ro to match the curve for the growth of new cases.   I found the best match using an Ro value of 1.31. 
Chart 9

Nate Silver, in his book “The Signal and the Noise”, cautions that early estimates of Ro are subject to large uncertainty, because of noise inherent in the data.  On the other hand, later (but more accurate) estimates of Ro are likely to be useless in preventing epidemics.  I believe my estimate of Ro = 1.31 is conservative; the 1995 outbreak of Ebola reportedly had an Ro of 1.8.

The second method of extrapolation in this post is based on a contagion model, derived from the fit to existing data.  We can use the contagion model developed with Ro to extrapolate the progress of the epidemic, to one million and one billion cases.   This model gives a more rapid growth rate to the disease.  Extrapolation of this model yields one million cases in less than six months, and one billion cases in slightly over a year from today.
Chart 10
Chart 11

Pathogens evolve rapidly.  A small number of viruses rapidly reproduce to become billions within a single body; and the exponential increase in victims provides more orders of magnitude in the number of viruses reproducing.  That is why flu vaccines must constantly be re-formulated, to match the current strains of the disease in circulation. 

The evolution of viruses favors those which are most likely to infect additional victims.  Thus, infectious diseases tend to evolve to new forms that are more catching, and allow their victims to survive longer, infecting additional victims.  A virus which immediately kills its host is less likely to propagate than a virus which allows the host to linger.  Diseases evolve to new forms which are more infectious, but less deadly.  We can expect the same progression in Ebola, but to what degree and in what time frame are impossible to say.

From Extrapolation to Prediction
Throughout this post, I have been careful to use the word “extrapolation” rather than “forecast” or “prediction”.   The mathematical projection of a trend is a long way from the real world.  Still, I believe the extrapolation of the early trend of the Ebola epidemic shows what might happen, if it becomes a global pandemic.  I dislike alarmist or extremely dire warnings about social or environmental hazards, but in this case, the dire warning seems to be a direct result of objective analysis.

We tend to believe that such a horrific epidemic as Ebola can only happen in backwards and undeveloped places (except for movies about the zombie apocalypse).   We take comfort in our modern hospitals, our sanitation systems, our education and wealth.   There is a sense that “it can’t happen here.”  But I think this is misplaced over-confidence.  There have been very stringent efforts to control the disease in West Africa, which have been unsuccessful.  I do not think it would necessarily be easier to control an Ebola outbreak in New York City than in subsistence villages in West Africa.  I think the developed world should consider the risks and consequences of an Ebola outbreak, and make contingency plans accordingly. 

An uncontrolled Ebola outbreak in the developed world would have severe consequences.  Some breakdown of social order should be expected, and this breakdown might make medical response and containment more difficult, or impossible.  It might be difficult to keep medical teams in place, if there is substantial loss of life among those treating the disease.  Economic consequences would be considerable, and there would be collateral damage to quality of life. 

What I have tried to do here is show the mathematical possibility for a global Ebola pandemic within the next twelve to eighteen months.  It isn’t a lot of time.   There will not be time to develop vaccines or experimental drugs, or even to manufacture new drugs if a miracle treatment did exist.  The Ebola epidemic is moving very quickly. 

I do not believe that we will see one billion cases of Ebola within eighteen months.   But I think it is a possibility, unless the threat is met with serious counter-measures at the earliest possible time.   I do believe that the epidemic will pass the mark of one million cases in Africa, and will cause disruption to international travel for some time. 

As Nate Silver has written in “The Signal and the Noise”, the most difficult predictions to make are those which involve choices and public policy.  There is feedback between the prediction and public policy, which can change the conditions controlling the prediction.   In the case of predictions regarding public welfare, the very best predictions are those which are self-defeating.  When a prediction regarding a public hazard is met by a change in public policy, mitigating the hazard, it has been successful, even while the prediction itself is wrong.  

December 17, 2014: 
The exponential growth rate of the Ebola epidemic continued for about two months after my original blog post.  The exponential trend was broken in mid-October, 2014, thanks to global relief efforts and effective public health programs in the affected countries.  Updates to my original charts can be found here:

Obsolete as of November 7, 2014, see latest post.

Update, October 31, 2014
Here are key charts from this post, with data from WHO updated through October 25.

In the October 29 report, WHO presents revised figures that add about 3700 cases to the previous total.  These cases were recognized through study of patient databases, and occurred throughout the epidemic period, and not only since October 22.  The additional cases return the cumulative case number to my original exponential extrapolation, first presented on August 26th.

It is uncertain whether the apparent flattening of the cumulative cases, observed through the month of October, is real or the result of under-reporting.  Case reporting is increasingly late, and WHO cites data missing for a number of dates.

I have now seen two anecdotal reports that give a more optimistic appraisal of the situation in Monrovia, indicating fewer patients are reporting to Ebola clinics, and fewer bodies are being collected from the city outside the clinics.  Authorities disagree on whether the drop in patients shows a real decline in the epidemic, or avoidance of the clinics.

Extrapolation to one million cases:
Extrapolation to one billion cases:
Obsolete Updates
Update, October 10, 2014
Reported data from Liberia show a falling number of new cases; however, the WHO and CDC believe that the situation continues to deteriorate.   Since early September, official data from Liberia have been late, contradictory, and inconsistent.  WHO and CDC believe there is substantial under-reporting of new cases.  At best, there are only about 25% of the number of beds required in treatment centers.  After weeks of seeing patients turned away from treatment centers, it seems likely that families are now caring for victims at home, causing under-reporting of official cases -- and transmitting the disease to new victims.
The October 8th report from WHO bluntly states “Evidence obtained from responders and laboratory staff in the country indicates beyond doubt that there is widespread under-reporting of new cases, and that the situation in Liberia, and in Monrovia in particular, continues to deteriorate from week to week.”  The CDC estimates that there may be as many as 1.5 times as many unreported cases as reported cases. Needless to say, forecasting the growth of the epidemic will become very difficult if there are no reliable reports on the number of cases.
Update, October 27, 2014
Here are key charts from this post, with data from WHO updated through October 18.  The latest point is the first point to fall below my original extrapolation.

The trend of officially reported cases has flattened, showing a reduced rate of transmission.  But the trend must be taken with a grain of salt, considering anecdotal evidence for many unreported cases.
New case numbers from Liberia continue to be delivered much later than data from Guinea and Sierra Leone.  I am interpolating numbers from Guinea and Sierra Leone to obtain a consistent single reporting data for the epidemic.

I found one anecdotal report from Monrovia which is cautiously optimistic.   The report appeared on the website

WHO continues to say that the situation is deteriorating in Liberia and Sierra Leone.
Updates with additional text can be found below.
Update #1:
I posted an update with text on September 15.  I tweaked the models and looked at some of the containment efforts and procedures.

Update #2:
My second update is located here:
This post compares the geography of populations at risk to the extrapolated cumulative cases.
The post also discusses the accuracy of the case count from Liberia, the number of beds needed in West Africa, and the trend of the mortality rate.
World Health Organization

Various News Sources


Nate Silver, 2012,  The Signal and the Noise, Why so Many Predictions Fail, But Some Don't, Penguin Press, New York, 534 p.
Copyright 2014, Doug Robbins
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