THAT'S MATHS:A RECENT blockbuster, Contagion, painted a terrifying picture of the breakdown of society following a viral pandemic. The movie identified a key parameter, the basic reproduction number R-nought. This number measures how many new people catch the virus from each infected person, and is crucial in determining how fast an infection spreads.
In March 2003, an epidemic of severe acute respiratory syndrome (Sars) spread rapidly across the globe. The World Health Organisation issued a global alert after Sars had been detected in several countries. Since the spread of infections is greatly facilitated by international air travel, controls on movement can certainly be effective: with travel restrictions, the Sars epidemic was brought under control in months.
Epidemiological analysis and mathematical models are now essential tools in understanding and responding to infectious diseases such as Sars. Models range from simple systems of a few variables and equations to highly complex simulations with many millions of variables.
A broad range of mathematics, conventional techniques and methods emerging from current research, are involved. These include dynamical systems theory, statistics, network theory and computational science.
Public health authorities are faced with crucial questions: How many people will become infected? How many do we need to vaccinate to prevent an epidemic? How should we design programmes for prevention, control and treatment of outbreaks? The models allow us to quantify mortality rates, incubation periods, levels of threat and the time-scale of epidemics.
They can also predict the effectiveness of vaccination programmes and control policies such as travel restrictions.
Parameters such as transmission rates and basic reproduction numbers cannot be accurately estimated for a new infection until an outbreak actually occurs. But models can be used to study “what if” scenarios, to estimate the likely consequences of future epidemics or pandemics.
In a paper published in 1927, A Contribution to the Mathematical Theory of Epidemics, two scientists in Edinburgh, William Kermack and Anderson McKendrick, described a simple model with three variables, and three “ordinary differential equations”, which describe how infection levels change with time, that was successful in predicting the behaviour of some epidemics. Their model divided the population into three groups, susceptible, infected and recovered people, denoted S, I and R respectively. This Sir model simulates the growth and decline of an epidemic and can be used to predict the level of infection, time-scale and the total percentage of the population afflicted by it. However, many important factors are omitted from the simple Sir model.
The swine-flu epidemic in Britain reached a peak in July 2009 and then declined rapidly and unexpectedly. The key factor not included in the model was the effect on the transmission rate of the school holidays, with contacts between children greatly reduced. The growth of the outbreak was interrupted, but an even larger peak was reached in October, after school had resumed. When these social mixing patterns were included, the model produced two peaks, in agreement with the observed development.
The statistician George Box, a pioneer in time series analysis and the design of experiments, once remarked: “All models are wrong, but some are useful.” All models of epidemics have limitations, and those using them must bear these in mind. Given the vagaries of human behaviour, prediction of the exact development of an infectious outbreak is never possible. Nevertheless, models provide valuable insights not available through any other means.
Future influenza pandemics are a matter of “when” rather than “if”. In planning for these, mathematical models will play an indispensable role.
Peter Lynch is professor of meteorology at University College Dublin. He blogs at thatsmaths.com
