Dynamics of modern epidemics

The application of mathematical modelling to the spread of epidemics has a long history and was initiated by Daniel Bernoulli’s work on the effect of cow-pox inoculation on the spread of smallpox in 1760 (Bernoulli 1760). While most studies concentrate on the temporal development of diseases and epidemics, their geographical spread is less well understood. The key question and difficulty is how to include spatial heterogeneities and to quantify the dispersal of individuals (Keeling et al. 2001; Smith et al. 2002; Keeling et al. 2003; Lipsitch et al. 2003). In a well established class of models spatial dispersal is accounted for by ordinary diffusion (Murray 1993). This approach admits a description in terms of reaction-diffusion equations which generically exhibit epidemic wavefronts propagating at constant speeds. These wave fronts were observed for instance in the geotemporal spread of the Black Death in Europe from 1347–50 (Langer 1964; Noble 1974; Mollison 1991; Grenfell et al. 2001). However, today’s volume, speed, and non-locality of human travel (Fig. 11.1) and the rapid worldwide spread of severe acute respiratory syndrome (SARS) (Fig. 11.2) demonstrate that modern epidemics cannot be accounted for by local diffusion models, which are only applicable as long as the mean distance travelled by individuals is small compared to the geographical scope of the model. In this chapter, we focus on mechanisms of the worldwide spread of infectious diseases in a modern world in which humans travel on all scales. We introduce a probabilistic model which accounts for the worldwide spread of infectious diseases on the global aviation network. The analysis indicates that a forecast of the geographical spread of an epidemic is indeed possible, provided that local dynamical parameters of the disease such as the reproduction number are known. The model consists of local stochastic infection dynamics and stochastic transport of individuals on the worldwide aviation network which takes into account the national and international civil aviation traffic. In broad terms, our simulations of the SARS outbreak are in good agreement with published case reports. We propose that our model can be employed to predict the worldwide spread of future infectious diseases and to identify endangered regions in advance. Based on the connectivity of the aviation network we evaluate the performance of different control strategies and show that a quick and focused reaction is essential to inhibit the global spread of epidemics.