Mathematical modelling of infectious diseases M. J. Keeling*, and L. Danon Biological Sciences, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Introduction: Mathematical models allow us to extrapolate from current information about the state and progress of an outbreak, to predict the future and, most importantly, to quantify the uncertainty in these predictions. Here, we illustrate these principles in relation to the current H1N1 epidemic. Sources of data: Many sources of data are used in mathematical modelling, with some forms of model requiring vastly more data than others. However, a good estimation of the number of cases is vitally important. Areas of agreement: Mathematical models, and the statistical tools that underpin them, are now a fundamental element in planning control and mitigation measures against any future epidemic of an infectious disease. Well- parameterized mathematical models allow us to test a variety of possible control strategies in computer simulations before applying them in reality. Areas of controversy: The interaction between modellers and public-health practitioners and the level of detail needed for models to be of use. Growing points: The need for stronger statistical links between models and data. Areas timely for developing research: Greater appreciation by the medical community of the uses and limitations of models and a greater appreciation by modellers of the constraints on public-health resources. Keywords: modelling/swine flu/prediction/uncertainty Introduction The progress of an epidemic through the population is highly amenable to mathematical modelling. In particular, the first attempt to model and hence predict or explain patterns dates back over 100 years,1 although it was the work of Kermack and McKendrick2 that estab- lished the basic foundations of the subject. These early models, and many subsequent revisions and improvements,3,4 operated on the prin- ciple that individuals can be classified by their epidemiological status��� most simply susceptible to the infection, infected and therefore infec- tious, and recovered and hence no longer infectious. (We stress that British Medical Bulletin 2009 1���10 DOI:10.1093/bmb/ldp038 & The Author 2009. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org Accepted: September 24, 2009 *Correspondence to: Prof. M. J. Keeling, Biological Sciences, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK. E-mail: m.j.keeling@ warwick.ac.uk British Medical Bulletin Advance Access published October 24, 2009

this classification is based upon an individual���s ability to host and transmit a pathogen, and may be relatively unconnected to their medical status.) In this review, we focus on how such models can be used to predict the future outcome of an epidemic process (or the impact of control measures) however, models may also have a more theoretical use as explanatory tools elucidating fundamental principles of transmission and the factors driving epidemic behaviour. The so-called SIR model is one of the simplest and most fundamental of all epidemiological models. It is based upon calculating the pro- portion of the population in each of the three classes (susceptible, infected and recovered) and determining the rates of transition between these classes (Fig. 1). In the simplest model of a single epidemic, births and deaths can often be ignored, and so, only two transitions are poss- ible: infection (moving individuals from the susceptible to the infected class) and recovery (moving individuals from the infected to the recov- ered class). It is generally assumed (and supported by epidemic data) that the per capita rate that a given susceptible individual becomes infected is proportional to the prevalence of infection in the popu- lation 5 while for simplicity it is often assumed that infected individuals recover at a constant rate.2 To make progress even with this simple model requires modellers to estimate two parameters: the proportional- ity constant for infection and the recovery rate. This illustrates the fun- damental relationship between models and statistics without a good Fig. 1 From left to right: a pictorial representation of the flow of individuals between classes in the SIR model. The basic differential equations for the SIR model which give the rate of change of the proportion in each class (negative values reflect flows out of a class, whereas positive values reflect flows into the class). The result of numerically solving the SIR model, showing how the proportion of susceptible, infected and recovered individuals in the population is predicted to change over time. M. J. Keeling and L. Danon Page 2 of 10 British Medical Bulletin 2009