The Ebola Virus. Found at: http://commons.wikimedia.org/wiki/File:Ebola_virus_em.png

When diseases start to spread unusually or appear more threatening than usual, we have to figure out what the probability is that the disease will spread and become an epidemic, and this is most often done using mathematical models. If we can model the spread of the disease, using certain parameters specific to the pathogen, we can get a sense of how many people could be infected and how we should respond to the threat.

This is becoming most pertinent now because currently there is an outbreak of Ebola in Africa, spreading from Liberia to Nigeria, and there has been some concern on the news about an epidemic spreading to the United States. This would be especially problematic because there is no known vaccine or treatment for Ebola. But, will it actually spread to affect many other countries in the world?

The most common model for diseases is known as the SIR model (which stands for Susceptible Infected Resistant). In this model, people in the population are either susceptible to the disease (as in they have not contracted it yet), infected with the disease, or they have been infected and recovered and they are now resistant to the disease. We can plot the spread of the disease as a function of the number of people in the population, the transition rate, and the number of people with the disease at a certain time t, and we get a curve that looks like this:

Found at: http://commons.wikimedia.org/wiki/File:Logistic.png.

Ignoring what the axes mean in this graph, this curve is known as a logistic growth curve. The disease starts out slow, infecting only a few people, then as more people get it, they pass on the disease to more people, and the disease spreads faster. Then, it hits a time where so many people are infected and now resistant that the spread slows down and eventually stops.

But, as far as we know, Ebola is not one of the diseases where people who get it once are necessarily resistant. So then, how does the model change?

Then we can use the SIS model (susceptible infected susceptible), where people can recover from the disease but once they have recovered, they are then susceptible to the disease again.

So, say that people are recovering at some rate a, and the transmission rate is t. If a > t (or if people are recovering faster than the disease transmits) then the disease won’t spread. Say the contact rate (the rate of infected people contacting not infected people) is c, then with some math we can figure out that the disease will spread if ct-a is positive, and if ct-a is negative, the disease won’t spread. We call that number ct-a R, or the basic reproduction number, which tells us the number of susceptible people infected from a single infected person. If R<1 the disease does not spread, and if R>1, the disease spreads, which makes logical sense because if a single person can infect more than one person, the disease will spread. If a single person cannot infect one person, than the disease will slowly decline.

We have calculated the R’s (basic reproduction numbers) of certain common diseases; for example, the R of measles is 15 and the R of the flu is 3 (meaning both diseases spread easily, measles way more than the flu). On a side note, this is why the recent resurgence of measles cases, potentially due to a fear of vaccinations, is so worrisome.

The R for Ebola is hard to say for sure, because R is calculated using data from past cases. Unfortunately, as of 2004, there have only been a few big Ebola outbreaks, including one in Congo in 1995 and one in Uganda in 2000. From these two cases, a group of researchers determined that the R is about 1.83 for the Congo outbreak and 1.34 for the Uganda outbreak. Another group of researchers found the R value to be around 2.7 instead. This does not necessarily mean that this will be the R value for the current outbreak because there isn’t enough data to say for sure, but it does suggest that the Ebola virus does spread, but not as well as diseases we interact with a lot like the flu.

Why might Ebola have a (comparatively) low R value? The transmission rate could be low; ebola requires direct contact with bodily fluids with an animal or another human, which occurs far less than contact with viruses that spread in the air or just by touch. Ebola patients are also contagious for a relatively short amount of time before showing symptoms, so if Ebola is correctly identified quickly (and that is a big if because Ebola is notorious for being misdiagnosed), patients may not be able to spread the disease to many people. Finally, Ebola tends to kill its host (or show bad enough symptoms that an infected person is hospitalized and under quarantine) fairly quickly, making it harder for an infected person to transit the disease to many people.

Now, this is not to say that the possibility of an Ebola epidemic is zero, because this is currently the biggest Ebola outbreak and it is spreading farther than is usual for Ebola.

However, there is some good news to come of mathematical models – some researchers (their paper can be found here http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2870608/) looked into modeling an Ebola outbreak after medical interventions have occurred (including medical treatments, hospitalizations, and quarantines) and found that the R value dropped significantly (to 0.4 and 0.3, meaning the disease would not spread). They also found that the time it took for these interventions to happen was one of the determining factors in how big the epidemic would be (as well as decreasing the rate of transmission after death; there have been cases where people have contracted the virus while burying victims of the disease).

Models are never perfect; for example, this particular one did not take into account animal to human transmissions. However, even simplifications can tell us a lot.

If anyone is interested in modeling, there’s a really cool (and free!) online class on Coursera (coursera.org) called Model Thinking and it’s all about going through different types of models and it actually goes through epidemiological models too. You should check it out if you’re interested! https://www.coursera.org/course/modelthinking.

**Works Cited**

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2870608/

http://mtbi.asu.edu/files/Mathematical_Models_to_Study_the_Outbreaks_of_Ebola.pdf

http://math.lanl.gov/~mac/papers/bio/CHCFH05.pdf

http://www.who.int/mediacentre/factsheets/fs103/en/

http://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf

http://www.ncbi.nlm.nih.gov/pubmed/15178190

http://ebola.emedtv.com/ebola/ebola-incubation-period.html

Coursera Course, Model Thinking