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Sunday, October 12, 2014

Forecasting the 2014 Ebola Outbreak; Update #2

April 29, 2015
My latest update regarding the history of the Ebola epidemic can be found here:
December 17, 2014: 
The exponential growth rate of the Ebola epidemic continued for about two months after my original blog post in August, 2014.  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:
There is no clear end-point to the Ebola outbreak in West Africa.  The Ebola outbreak in West Africa, now involves about 9000 reported cases, with the likelihood of an equal number of unreported cases (10/17 update).   Like ripples from a stone thrown into the water, the epidemic has the potential to infect heavily populated areas of west Africa, to spread across porous borders by people fleeing the epidemic, and to jump from large cities to other large cities. The following figure shows a timeline extrapolated from the exponential growth of the disease, and marked by points representing populations at risk.  Africa is impoverished and lacks the resources to maintain a stringent defense against the epidemic.  There are no firebreaks, and the world will be able to do very little to stop a general pandemic if the case numbers continue to grow well into the coming year.

Reports from the World Health Organization (WHO) are increasingly bleak.  The WHO October 8th situation report states:

” …the situation in Guinea, Liberia, and Sierra Leone continues to deteriorate, with widespread and persistent transmission of EVD [Ebola Virus Disease]….the reported fall in the number of new cases in Liberia over the past three weeks is unlikely to be genuine.  Rather, it reflects a deterioration in the ability of overwhelmed responders to record accurate epidemiological data….There is no evidence that the EVD epidemic in West Africa is being brought under control….”

The international community is increasing support to the affected countries.  New teams and equipment are being sent from the United States, Great Britain, Cuba and Nigeria, among others.  However, the scale and speed of the intervention appears too little and too late.   At this point, we have to ask where this epidemic is going, and when will it get there.

This is the third post on the topic of Ebola on this blog.  The first two are here:
Course of the Epidemic
We can consider the future course of the epidemic by linking our charts of case numbers to the geographical distribution of actual populations.  Here is a map of Africa showing population density. 

The colored outlines show populations which might be at risk if the epidemic spreads across Africa.   For illustrative purposes, these populations are linked to the extrapolation of the cumulative number of cases, using a contagion model with Ro = 1.63.   In this model, each case of Ebola infects (on average) 1.63 new victims after a transmission and incubation period of 16 days.  This model fits the reported exponential growth of the disease to date.
A number of cases can be expected in developed countries as a result of infected individuals traveling to new countries.  The United States and Spain have already seen such cases and subsequent secondary infections.  However, rich countries with well-developed medical systems should be able to easily quell the epidemic, given only a few points of infection.  

The Ebola epidemic is much more likely to spread through the impoverished nations of Sub-Saharan Africa.   By the numbers, at the established rate of growth would spread the epidemic to every individual in Guinea, Sierra Leone, and Liberia, totaling 21 million people, by the mid-year 2015.  The epidemic could spread to every individual in West Africa, 412 million people, by autumn 2015.  By year-end 2015, the epidemic could spread to nearly a billion people in Sub-Saharan Africa by year-end 2015, showing the astonishing power of exponential growth.

Of course, the epidemic will not unfold as a neat mathematical formula.  There are factors which will impede the growth of the disease.  And there are factors which may accelerate the spread of the disease.  As the rate of infection rises in a population, several things may happen.  Some of these things will reduce the transmission of disease, and some things will increase the transmission of disease.  In affected populations, intense public health campaigns are being conducted, which should reduce risky behaviors and reduce the rate of infection.  Also, as the infected population approaches saturation, the rate of new infections should decline.  On the other hand, high infection rates will strain the fabric of society, overwhelm treatment facilities, and reduce the number of qualified professionals working to contain the epidemic.  And perhaps worst for the rest of the world, at some point large numbers of Ebola refugees will flee infected countries and carry the disease to new places.

Liberia Case Count
Recent data reported by the World Health Organization seem to show a slower rate of growth than in the earlier history of the epidemic.  However, there is serious doubt about the accuracy of recent reports, particularly from Liberia.  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 number of new cases appears to be declining; but this is unlikely to be true considering other evidence.

Since early September, reports from the Liberian Ministry of Health have been late, contradictory, and inconsistent.  The most recent reports from Monrovia and Freetown still indicate that all isolation units and hospitals are full, and that patients are turned away to transmit the disease at home or in the streets.  Although we can hope that public health measures are reducing the rate of transmission, we are in a state of uncertainty, without reliable numbers to support that conclusion.   

Adequate medical facilities are necessary to measuring the development of the epidemic.  If families of Ebola patients believe that they will receive treatment, patients will be brought to medical facilities, and data gathered about the progress of the disease.  If patients are consistently turned away from treatment facilities, there is no point to bringing stricken people to the clinics.   For the past two months, the number of beds in Ebola clinics in Liberia has been only about 20 % to 25 % of the number needed.   Some 75% to 80% of Ebola patients have been turned away, or left to suffer in the streets.  In this environment, it is understandable that families do not seek official care, but instead choose to care for loved ones at home.  This means that official figures no longer have any validity in measuring the epidemic, and virtually guarantees the spread of the disease. 

Ebola Clinic Beds
Specialized treatment centers are necessary for Ebola patients.  Without proper equipment and training, medical personnel are at tremendous risk of contracting the disease.  [As of October 8th, 416 healthcare workers have contracted the disease, and 233 have died.]  But throughout the epidemic, the number of beds in proper treatment centers has lagged far behind the need.   According to WHO, throughout the months of September and October, the number of beds available to patients has been about 25% of the number needed.

In the 8th October report, WHO indicates that there are about 1100 beds available to Ebola patients in qualified treatment centers.  The current need is for over 4300 patients, a shortfall of 3200 beds.   There is adequate care for only 25% of the patients.

The United States and Britain have mobilized military resources to fight the epidemic.   The United States will build and staff treatment centers containing 1700 beds in Liberia, while the British army will build and staff treatment centers containing 700 beds in Sierra Leone.  Both programs will require about 60 days for implementation. This effort will still leave a shortfall of 800 beds, according to current needs.  It is likely that other organizations will also contribute new beds, and perhaps bring the total number of beds to 4300 by the end of November.  

The problem is the human tendency for linear thinking, whereas problems in nature are generally exponential.  By my estimates, the Ebola epidemic is doubling about every twenty-nine days.  (The CDC calculates the epidemic is doubling every 20 days, by assuming a large number of unreported cases).   In 60 days, the epidemic will have grown four-fold.   In mid-September, when the US Army intervention was announced, about 1800 new beds were needed, for a total of 2425 beds.  If the epidemic continues to grow exponentially, by mid-November about 10,000 beds will be needed. 

Death Rate
In my first post, I noted that diseases evolve quickly, and that evolution causes the disease to become more contagious, and less deadly.  Living patients propagate the virus better than dead patients.  As expected, the mortality rate for Ebola patients appears to be falling, from about 70 percent to about 55 percent.
The chart shows cumulative reported deaths as a fraction of cumulative reported cases, with a lag of eight days.  I would like to calculate a daily death rate (an instantaneous function), but the official figures are reported too sporadically for a meaningful chart. 

Future Course of the Epidemic
The equation used by epidemiologists to forecast an epidemic looks like this:
Cumulative Cases = ((Ro/1+d)t)t
The term “d” is a decay parameter, which reduces the rate of transmission over time.  This could represent effective medical intervention, or a public awareness campaign, or disease saturation within the population at risk.  We might hope that the declining number of reported new cases in Liberia reflects a real decline in new infections, but all other evidence indicates that the people of Liberia increasingly distrust the official treatment centers, and no longer bring patients to be counted.

We might also consider a parameter which increases the rate of transmission over time.  Such a parameter might represent breakdown of social order, growing distrust of medical facilities, or dispersion of the population fleeing the epidemic.  In any event, the data quantifying the epidemic are now uncertain, and there is little basis for assigning values to parameters which may reduce or increase the rate of transmission. 

While we must make every effort to stop Ebola within the current area of active transmission, we should also realistically consider what will happen if those efforts fail.  It is likely that neighboring nations will receive refugees through porous borders, and some of those refugees will carry the virus.   Ebola can be expected to spread along the West African coast through coastal communities, and jump from infected large cities to other large cities.  Like ripples from a stone thrown in the water, Ebola has the potential to expand through heavily populated West Africa, and ultimately affect the entire continent.  At some point, perhaps at 20,000 active cases, or 100,000 active cases, the disease will not be stoppable with the procedures that ended previous epidemics.   According to the rate of growth and the distribution of African population, the 2nd half of 2015 may be a very bad time for Africa.

Some people do survive Ebola, and they are surviving in increasing numbers.  This shows that the human organism can mount an effective immune response to Ebola.  Medical science has developed vaccines for other deadly viruses.  It is inevitable that an effective vaccine will ultimately be developed and produced on a large scale.  The question is how soon the vaccine can be produced and brought to protect the people at risk.

In considering the impact of the Ebola epidemic, I am strangely reminded of something a colleague said to me on the morning of 9/11/2001.  The second plane had just struck the World Trade Center.   Being quicker and brighter than me, she turned and said, “You realize, nothing will ever be the same again”.  Of course, I had not realized the impact of the event, and didn’t fully understand until several years later.  In the same way, something with global impact has happened in West Africa.  The world does not yet realize that everything has changed, and nothing will ever be the same again. 
*  Cumulative Cases = ((Ro/1+d)t)t
 I used a simpler exponential function for my first model, and a simple rolling formula in Excel to create my contagion model with Ro.  In the equation shown, one “t” must represent the time between initial and subsequent infection, and the second “t” must indicate the life of the epidemic.  I will update this post after I have some time to play with the equation.
Obsolete as of November 7, 2014, see latest post

Chart Updates to my original post, using the original extrapolations.
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.
New Cases (yellow circles) as of WHO Roadmap update, 10/25/2014.   Reporting dates by country are variable.
WHO Situation Reports, Ebola Response Roadmap

Ebola disease outbreak news

Population, Landscape, and Climate Estimates, v3: Population Density 2010, Africa
National Aggregates of Geospatial Data Collection

It = ((Ro/1+d)t)t   where is scaled in generation time, with Rthe basic reproductive number, and d a “control parameter” that causes incidence to decay. 

October 12, 2014
Personal report by Drew Hinshaw, WSJ, on awful situation in Monrovia.   Patients now told to stay at home; families told to leave patients alone.  Increasing numbers of patients on street, as compared to several weeks ago.   Mobs of people moving on streets; people trying to leave.

US military intervention to fight Ebola; commitment of 3000 soldiers and 1700 beds.

UK commits to building five treatment facilities with 700 beds in Sierra Leone.  Estimated to allow treatment of 8800 patients over six months.

Lots of facts about Ebola.

Lots of numbers and graphics showing that the global response will never catch up with the epidemic.

Infographic showing the rate of increase in cases over two months, and the number of beds available for treatment.   Also shows the US commitment to build additional beds over the next two months, which will provide about 1.5% of the needed beds in November.

Public education programs to convince population that Ebola is real; overflowing clinics; people touching and praying over Ebola patient.

Ebola is transmitted even under stringent conditions of containment.