December 6, 2021

The science magazine of Imperial College

Mathematics And Ebola

Andrew McMahon takes a look at how we can use maths to help fight disease

The Ebola outbreak that began in West Africa in March 2014 tapped into the public imagination across the world as representing all that can be frightening about a virus. Many commentators have spoken about its horrifying symptoms and high mortality rate, with a World Health Organization status report at the start of November declaring a total of 4,818 reported deaths from Ebola since the outbreak began.

Ebola, however, has not been the only disease to cause widespread fear this century. After the SARS outbreak of 2002-2004 and the swine flu pandemic of 2009, modern history has taught us that viruses can spread very quickly throughout local populations and even across the world. These recent examples left many ordinary people asking: how do we understand when an outbreak will become a pandemic? When is it best to start investing in mass vaccination or producing large amounts of combative drugs? How can we predict how severe an outbreak will be?

The answers to these questions can be found, in part, using mathematics.

By applying the tools of mathematics to epidemics, scientists and public health officials can make more informed decisions about the best course of action to take. For example, if a mathematical model can accurately describe what effect mass vaccination will have on an outbreak, a decision can be made about whether this is the most effective way to spend (ultimately finite) resources. Minimising the risk associated with these types of choices is one of the main goals of the mathematical study of epidemics.

It is clear that having a predictive model of a disease would be useful, but how does this work in practice? To write a mathematical model of an infection, you first need to identify what the important variables are: how contagious is the disease? How long does it take before an infected person becomes infectious to others? Will the rate of infection change with time?

In fact, almost all basic models of epidemics will contain variables describing the answers to the above three questions. Respectively these are called the reproductive number, the generation time and the epidemic growth rate. These numbers give you the basic information about any epidemic, but in order to capture the correct behaviour, more detailed models take into account other factors such as demographics, migration, transport networks, seasonality, or even sexuality and drug use (as in the case of AIDS).

Using the above numbers you can write down an equation that should tell you how many individuals in a population are infected at any one time. If your model is good enough, you can then use it to predict the effect of introducing new variables into the system, for example the effect of having a mass vaccination program. This sort of information can prove invaluable to public health officials as it allows for the analysis of several possible scenarios without wasting the precious resources of time and money testing them in the real world, ultimately saving lives.

Once a model is developed, the predictions can be somewhat surprising. For example, some models of the H1N1 virus (swine flu) produced during the 2009 pandemic predicted that if developed world countries focussed on vaccinating and treating only their own population, a serious global pandemic could still not be avoided. The interconnected world of the 21st century meant that resources would have to be shared between developed and developing countries in order to beat the virus. In the end however, it turned out that H1N1 was a mild strain with a relatively small mortality rate, so the pandemic was not as serious as predicted. The results from this model could, however, still prove useful if and when a serious pandemic hits, as many experts believe could occur with a future strain of avian flu.

There are limitations in terms of what mathematics can offer, however. Many patterns of infection require very sophisticated models in order to understand them, with many not being successfully modelled yet at all. The fact that mathematical epidemiology is such a vibrant area of research is a testament to the fact that there is always more to do. That said, in the age of ‘Big Data’ and vast computing resources, epidemiological models are getting better all the time. This will hopefully mean that when the next potential pandemic hits, we can be confident of containing it.

Andrew McMahon is a first year PhD student in Physics

Images: Ebola 2014 outbreak map of Guinea, Liberia, and Sierra Leone, Wikimedia; Ebola virus, NIAID