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A Small-World Network Approach Dr Michael Small and Prof. C.K. Michael Tse
Applied Nonlinear Circuits and Systems Research Group
In the Spring of 2003, SARS* affected thousands of people worldwide
and claimed over 900 lives. In Hong Kong, between
March and May, the SARS epidemic affected 1755 including 299 deaths.**
**For an official report on the SARS outbreak in Hong Kong, see http://www.info.gov.hk/info/sars/eindex.htm
About this project In this project we attempt to model the propagation behaviour of the SARS virus by an appropriate network topology and propagation dynamics. Specifically, we have developed a mathematical model for simulating the behaviour of the SARS epidemic that appeared in Hong Kong last Spring. Based on this model, a simulation software can be developed to help health professionals to predict the possibility and extent of community outbreaks. The model allows crucial parameters (e.g., probability of infecting a neighbour, probability of infecting a remote party, incubation period, probability of not infecting others due to treatment/recovery/quarantine, etc.) to be specified and hence provides a tunable platform for simulating the SARS epidemic.
Technical summary We considered a complex small-world network (SWN)*** model and showed that such a structure can be applied to reliably produce simulations quantitatively similar to the true data. The small-world network model not only captures the apparently random fluctuation in the reported data, but also reproduces mini-outbreaks such as those caused by so-called "super-spreaders", as appeared in the Hong Kong housing estate Amoy Gardens and the Prince of Wales Hospital. ***D. Watts and S. Strogatz, Collective dynamics of small-world networks, Nature, 393, pp. 440-442, 1998.
A glimpse at the model For simplicity, we use a simple grid network structure for illustration. Specifically, we assume that the network consists of N nodes and each node is connected to n1 immediate neighbours forming a rectangular grid. The behaviour of the epidemic is described in terms of four possible states of each node:
The crucial part of the model is the small-world network connectivity, in which each node is only connected to its n1 immediate neighbours and some randomly selected remote acquaintances. The remote acquaintances of any given node are selected according to an exponential distribution function. This network connectivity gives rise to the essential phenomenon of "super-spreading". The dynamics of the epidemic is modelled as follows:
 Simulation begins with a particular initial condition (number of infections in some randomly selected nodes), and an iterative process continues with a chosen set of parameters.
Illustrative animations The following Flash animations (requires browsers with Flash plugin) illustrate several possible outcomes corresponding to different choices of parameter values:
Results: fitting the Hong Kong data
The figure below shows several simulation runs, with the following sets of
parameters. The time-varying parameters emulate government
interventions such as introduction of quarantine measure, public
education, etc. 
 From a large number of simulation runs, we can also generate a probability density function for the number of reported cases, as shown in the following diagram (the black solid curve is the actual HK data):
Usefulness of the model The above model can lead to closed form expressions for the probability of an outbreak to occur and for the bounds of the probability of a containment, etc. These results can be found in a paper published in the International Journal of Bifurcation and Chaos. Furthermore, using the model, one can develop user-friendly simulation software to help reproduce the quantitative behaviour of the epidemic. Parameters can be adjusted to suit different epidemic behaviour. Using this software, the likelihood of a particular outcome, such as outbreak or eventual containment, can be predicted.
Applied Nonlinear Circuits and Systems Research Group |