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How Infectious is Ebola?

Ebola is the latest virus to warrant global fears over a pandemic which infects large numbers of people.  Throughout history we have seen pandemic diseases such as the Black Death in Middle Ages Europe and the Spanish Flu at the beginning of the 20th century. More recently we have seen HIV responsible for millions of deaths.  In the last few years there have been scares over bird flu and SARS – yet neither fully developed into a major global health problem.  So, how contagious is Ebola, and how can we use mathematics to predict its spread?

The basic model is based on the SIR model.  The SIR model looks at how much of the population is susceptible to infection, how many of these go on to become infectious, and how many of these go on to recover (and in what timeframe).  However given the nature of modelling diseases with very high mortality rates like Ebola, for our Ebola model the SIR stands for Susceptible, Infectious and Dead.

Another important parameter is R0, this is defined as how many people an infectious person will pass on their infection to in a totally susceptible population.  Some of the R0 values for different diseases are shown above.  Studies into Ebola estimate the R0 value at somewhere between 1.7 and 8.6.  Therefore whilst Ebola is contagious, it is nowhere near as contagious as a fully airbourne disease like measles.

The Guardian datablog have an excellent graphic to show the contagiousness relative to deadliness of different diseases. You can notice that we have nothing in the top right hand corner (very deadly and very contagious). This is just as well as that could be enough to seriously dent the human population. Most diseases we worry about fall into 2 categories – contagious and not very deadly or not very contagious and deadly. Ebola is in the latter category.

The equations above represent a SIR (susceptible, infectious, dead) model which can be used to model the spread of Ebola.

dS/dt represents the rate of change of those who are susceptible to the illness with respect to time.  dI/dt represents the rate of change of those who are infected with respect to time.  dR/dt represents the rate of change of those who have died with respect to time.

For example, if dI/dt is high then the number of people becoming infected is rapidly increasing.  When dI/dt is zero then there is no change in the numbers of people becoming infected (number of infections remain steady).  When dI/dt is negative then the numbers of people becoming infected is decreasing.

The constants β and μ are chosen depending on the type of disease being modelled.  β represents the contact rate – which is how likely someone will get the disease when in contact with someone who is ill.  ν is the recovery rate which is how quickly people recover (and become immune.

N is the total population

μ is the per capita death rate (Calculated by μ = 1/(duration of illness) )

N – let’s take as the population of Sierra Leone (6 million)

In the case of Ebola we have the following estimated values:

μ between 1/4 and 1/10 (because it takes an infected person between 4 and 10 days to die).  Let’s take it as 1/7 ≈ 0.14

β as approximately 0.6

N – let’s take as the population of Sierra Leone (6 million)

Therefore our 3 equations for rates of change become:

dS/dt = -0.6 I S/6,000,000

dI/dt = 0.6 I S/6,000,000 – 0.14 I

dR/dt = 0.14 I

Unfortunately these equations are very difficult to solve – but luckily we can use a computer program to plot what happens.   We need to assign starting values for S, I and R – the numbers of people susceptible, infectious and dead. We have 6 million people in Sierra Leone, and currently around 8000 reported cases of Ebola.  If we assume all 6 million are susceptible, then putting this all into the program gives the following outcome:

This graph is pretty incredible –  though it clearly shows some of our assumptions were wrong!  Given a starting point of 6 million people all susceptible to Ebola, and 8000 infectious individuals, then within 20 days the population would have crashed from 6 million to less than 1 million and within 60 days you would have nearly everyone dead.

Clearly therefore this graph is very sensitive to our initial assumed values.  Say for example Ebola was less contagious than we previous assumed – and so we had β = 0.15 with the other values the same.  Then we get the following:

This graph is very drastically different to the last one – you have infections remaining low – though this would still be enough to see a big population drop over the 3 years of the simulation.

Modelling disease outbreaks with real accuracy is therefore an incredibly important job for mathematicians.  Understanding how diseases spread and how fast they can spread through populations is essential to developing effective medical strategies to minimise deaths.  If you want to save lives maybe you should become a mathematician rather than a doctor!

If you enjoyed this post you might also like:

Differential Equations in Real Life – some other uses of differential equations in modelling predator-prey relationships between animal populations.

Modelling Infectious Diseases –  How we can use computer modelling to look at other infectious diseases like measles.

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

### New website for International teachers

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P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.