Ebola NOW! The Exponential Growth of a Deadly Outbreak

A photograph showing one of the containment-suited Ebola doctors caring for a yound victim of the illness during the Liberia epidemic in 2014.The Ebola Crisis is Growing Exponentially

“The West Africans are scared” said Ban Ki-moon at a meeting of the United Nations in Washington U.S., discussing the growing threat of Ebola.  And you could feel the sense of urgency as World leaders discussed the Ebola crisis.  Not enough money has been put forward to tackle the disease.  We are late in our response.  And the clock is ticking…  

The “good” thing about Ebola is that the virus is unlikely to mutate into a version that can spread through the air, as other viruses have done.  And infected people cannot pass it on to others during the incubation period, which we know is between 2 and 21 days.  Only when they have developed detectable symptoms, such as a fever, can the virus carriers become infectious to others, and only then by the transfer of bodily fluids.

Now, there are three bad things about Ebola:

  1. The bodily fluids that can transmit the infection include the tiniest droplets of sweat, meaning that the slightest touch can pass the virus on.
  2. The death rate of those who become infected is up to 70%.
  3. The U.S. government’s Centres for Disease Control (CDC) warned recently that we could see 1.4 million cases of Ebola by January 2014.

A graph showing Ebola's Worst Case Scenario by January 2015. Source: CDCAs the number of known cases is only around 9,000 so far, this suggests that the number of new cases currently roughly doubles every 3-4 weeks.

This is called exponential growth.

And before discussing the Ebola crisis any further, here comes the science bit!


What is Exponential Growth?

An exponential growth (or geometric growth) is the growth of a system in which the amount of units being added to the system is proportional to the amount already present.  The bigger the system becomes, the greater and the more rapid the increase. 

For example, the number of microorganisms in a culture increases exponentially.  Typically, the first organism cell will split into two daughter organisms, who then each split to form four, who split to form eight, and so on and so forth, until an essential nutrient is exhausted.  Such a growth development applies to many type of systems, living systems or otherwise.


Of Rice and Men

Human and animal populations grow exponentially, provided the number of births and deaths per individual per year remain constant.  Evolution and the genetic complexity of life on Earth has consistently doubled every 376 million years.
The Legend of the Chessboard: If you put one grain of rice on the first square of a chessboard, two on the second, and keep doubling the grains every square, there are not enough grains of rice in the World to get you to the 64th square.
The Legend of the Chessboard: If you put one grain of rice on the first square of a chessboard, two on the second, and keep doubling the grains every square, there are not enough grains of rice in the World to get you to the 64th square.

Each uranium nucleus, in an uncontrolled fission reaction, produces multiple neutrons.  Each one of those neutrons can in turn be absorbed by adjacent uranium atoms, causing them to undergo fission.

If the probability of neutron absorption exceeds the probability of neutron escape, the production rate of neutrons and induced uranium fission increase exponentially.

Even, internet traffic increases exponentially…

Put simply, this means that in the case of a disease epidemic, like the one caused by the Ebola virus, the number of cases does not increase linearly, that is 1, 2, 3, 4, 5, 6…

It increases exponentially, that is 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,1024, 2048…

Basically, exponential growth implies a runaway expansion.

Calculating Exponential Growth:

Populations, Bacteria and Ebola Cases…

A graph comparing Growth Functions to Exponential Function. Source: MathWarehouse
This graph compares three types of mathematical growth functions.  It illustrates how exponential growth (green) catches up and surpasses both linear (red) and cubic (blue) growths.  Source: Mathwarehouse.com

An exponential function has a constant as the base of a variable exponent.  When the exponent assumes a positive integer value, the function merely equals the product of that many bases:

2x  is equal to  2 x 2 x 2 = 8  when x = 3

3x  is equal to  3 x 3 = 9  when x = 2

10x  is equal to  10 x 10 x 10 x 10 x 10 = 100,000  when  x = 5.


When you take away certain assumptions, such as x being an integer or positive, the calculations get more complex.


Euler’s Number

Of particular interest to scientists and engineers, the number

e = 2.718281...

This irrational number, called “Euler’s number”, or the base of the natural logarithm, has the interesting property that ex has a slope at x that is equal to the value of ex itself.


Growth Rate

In order to calculate a future population size, you have to calculate the growth rate by considering known sizes at two time periods.  Divide the later population size by the initial size, then take the natural log of the result.

The result is the growth rate k between those two times.

Divide the growth rate by the number of time units, to calculate the periodic growth rate.

If a colony of bacteria grew from 2,000 individuals to 6,000 in two hours, then

\frac {6,000}{2,000} = 3

and k = \ln({3}) = 1.099 for the two hours.

The hourly k-value is then

\frac {1.099}{2} = 0.5495.


The k Factor

A graph showing the number of cases of the illness by country, as reported by 4 August during the West African Ebola 2014 Outbreak.

Using the k-value, you can estimate a future population size under the assumption that the growth rate remains constant

Multiply the growth rate by the number of time periods, apply your calculator’s exponential function ex (e to the power of x), and multiply this by the initial population size.

To calculate the size of the bacterial colony after 10 hours, raise 2.71828 to the power of 10 times 0.5495 – the hourly k value- and multiply by the initial population size of 2,000:

e^{10 \times 0.5495} \times 2000 = 486943.

Population size after 10 hours is roughly 486,943 bacteria.


A virus (like SARS, influenza, smallpox or ebola) will typically spread exponentially at first, if no artificial immunisation is available, as each infected person can infect multiple new people.


The History of a Killer Organism So Far…

The first known outbreak of Ebola virus disease (EVD) was only identified after the event occurred.  The Sudan outbreak infected 284 people and killed 151 in Nzara, South Sudan between June and November 1976.

A second outbreak occurred when village school headmaster Mabalo Lokela began displaying symptoms on August 26th, 1976, having returned from a trip the Ebola River in Northern Zaire, near the Central African Republic border, between 12-22 August 1976.

An infographic about the Ebola Outbreak and the History of the virus. Source: WHO



Early signs and symptoms typically start between two days and three weeks after contracting the virus, with a fever, sore throat, muscle pain and headaches.  Then, vomiting, diarrhea and rash usually follow, along with decreased function of the liver and kidneys.  At this time, generally, patients begin to bleed both internally and externally.

Eventually, death occurs typically six to sixteen days after first symptoms appear, often due to low blood pressure from fluid loss.


“This is an unforgiving disease.

We must take action.”


No specific treatment for the disease is yet available. Efforts to help those who are infected are supportive and include giving either oral re-hydration therapy or intravenous fluids.  This supportive care can improve a patient’s outcome.

It might be months or longer before the disease outbreak peaks.  But when it does, the disease will begin dying down.  Nothing can experience constant exponential growth forever.  Exponential growth always slows down eventually, but in the case of Ebola, the question is when?


Fighting Ebola

An image of the VP40 protein of the Ebola virus. Source: Stanford Education
VP40, a protein of the Ebola virus, can arrange itself into three very different shapes, shown in blue, each with a distinct function.  Image: Nikola Stojanovic/SLAC and Zachary Bornholdt/The Scripps Research Institute

A vaccine would slow it down, and the British pharmaceutical giant GlaxoSmithKline already has one under development, but it is still in an early stage of testing. 

ZMapp is a monoclonal antibody vaccine: an experimental biopharmaceutical drug comprising three humanised monoclonal antibodies under development as a treatment.  The limited supply of the drug has been used to treat a small number of individuals infected with the Ebola virus.  Although some individuals have recovered, the outcome is not considered statistically significant.
ZMapp has proved effective in a trial involving Rhesus macaque monkeys, but it has not been subjected to a randomised clinical trial to prove its safety or its efficacy in patients.

Human volunteers are now being given the vaccine to check for unforeseen side effects.  If no serious side-effects are found, the vaccine will then be given to health workers in West Africa.

A process that normally takes years is being compressed into mere months, and 10,000 doses of the vaccine are already being produced (for the health workers).  But it will be the end of the year before we know if it actually gives a useful degree of protection from the virus.

If it does, then millions of doses would have to be produced and injected into the people of Liberia, Sierra Leone and Guinea, where Ebola is already an epidemic – or tens of millions of doses if the disease spreads to more populous countries, like Ivory Coast, Ghana, Senegal or, worst of all, Nigeria, which has 175 million people.


Preparing for the Worse, Hoping for the Best…

An infographic about the Ebola virus spread prevention measures and symptoms diagnosis.

The incubation period of up to 21 days makes the early detection of Ebola difficult.

Outbreak control requires a coordinated series of medical services, along with a certain level of community engagement.

The necessary medical services include rapid detection and contact tracing, quick access to appropriate laboratory services, proper management of those infected, and proper disposal of the dead victims through cremation or burial.

Prevention includes decreasing the disease transmission rate from infected animals to humans.  Potentially, this may be done by only handling potentially infected bush meat while wearing protective clothing, and thoroughly cooking it before consumption.  It also includes wearing proper protective clothing and washing hands when around a person with the disease.  Handling samples of body fluids and tissues from infected patients should be handled with special caution.

Until and unless a vaccine becomes available in very large quantities, the only way to stop the exponential spread of Ebola in the affected countries is to isolate the victims, a task that is very difficult in mostly rural countries with very minimal medical facilities: Liberia with 4.2 million people, had only 51 doctors and 978 nurses and midwives at the start of the crisis, and some of those have already died or fled the outbreak frontline.

It is worth noting that you do not need to find and isolate everybody who gets the disease to break the exponential pattern.  Just isolating 75% of the cases as soon as they become infectious would drastically slow down the spread of the epidemic.

At the moment, in the three most affected countries, only an estimated 18% of the victims are being taken to treatment centres where most of them will die.

A photograph showing the quarantine precautions undertaken due to the Ebola outbreak.The most important intervention has been the dispatch of 3,000 U.S. troops to Liberia, with the primary goal of creating 17 large tent hospitals and training 500 nurses to work in them.  Britain is providing 200 new hospital beds in its former colony of Sierra Leone, with 500 more in the next few months.  Cuba has sent 165 health workers, China has sent 60, and France has sent various teams to help its former colony, Guinea.  With the exception of the American aid to Liberia, it is all woefully inadequate.

Ten months after the first case of Ebola was confirmed in Guinea, we are still playing catch-up, and playing it rather badly.  Are the developed countries not at risk if the virus continues to spread?

Even without vaccine, the governments of the developed countries are confident that their health services can find and isolate any infected people quickly enough to prevent Ebola from becoming an epidemic in their own countries.  And they may well be right.  In most cases, they see the limited help they are sending to West Africa as a charity endeavour, rather than a vital self-interest.  They may be wrong.


What Now?

The World Health Organization (WHO) said it did not anticipate how big the Ebola virus disease crisis in West Africa and beyond, was going to become…

At least, it is now clear that the Ebola crisis requires an immediate global response coalition.  We must look forward and plan how to get the epidemic under control.  The speed and effectiveness of resources mobilisation is what is really needed in order to beat this disease.

At the same meeting of the United Nations in Washington, Mr Ban’s announcement followed that of Christine Lagarde, the head of the IMF (International Monetary Fund) who announced her full financial support to halt the epidemic:

“It is good to increase the fiscal deficit to cure the disease”.


A photograph showing the dreadful human toll of the Ebola outbreak. Source: ABCAs Professor Peter Piot, who first identified the Ebola virus in 1976, said in a recent interview with Der Spiegel:

“I am more worried about the many people from India who work in trade or industry in West Africa. It would only take one of them to become infected, travel to India during the virus’s incubation period to visit relatives, and then, once he becomes sick, go to a public hospital.

“Doctors and nurses in India often don’t wear protective gloves. They would immediately become infected and spread the virus.”

By that point, Ebola would be on the loose in a country of more than a billion people, millions of whom travel abroad each year.  All hope of confining the disease to Africa and driving it back down to almost nothing, as was managed in past outbreaks, would all be gone.

The World must intervene immediately, and rally around the three worst-affected West African countries, where the spread of Ebola is entrenched and accelerating in the capitals of the affected nations of Liberia, Sierra Leone, and Guinea.

Over 200 health workers are already among the victims.


The Ebola crisis may have been vastly under-estimated.


The news report of dozens of new cases everyday.  In some places, the number of cases is doubling everyday.  Experts warn of potential economic decline for the region.

As the World is playing catch up, Ebola quickly moves on.
A photograph showing the RFA Argus Hospital Ship.
Royal Navy Medical Ship RFA Argus deployed from Falmouth on 17 October 2014. It is en route to Sierra Leone to step up the effort against Ebola.

The WHO is currently reporting figures of nearly 10,000 cases, and over 4,500 deaths from the disease, mainly in West Africa.  And remember, these are the cases we know about, but it is likely that the Ebola virus crisis has been vastly under-estimated.

On 17th October 2014, the Royal Navy medical ship RFA Argus left Britain for Sierra Leone, carrying three Merlin helicopters and a crew of around 380.  The supply ship will act as an offshore base as the troops set up Ebola treatment centres in Sierra Leone.  Those on board include doctors, nurses, surgeons and Royal Marines.  RFA Argus will reach West Africa at the end of the month.

The time to stop Ebola is now.


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