http://www-personal.umich.edu/~mejn/papers/mpnsb.pdf 

In November 2002, a respiratory disease occurred in
China, and within eight months spread to over 25 countries, taking over 700 lives.  The cause of the disease, which came to be known as the Severe Acute Respiratory Syndrome (SARS), was a never before seen coronavirus, and its rapid spread became the subject of study in the field of network epidemiology.  This paper reveals that traditional models of disease spread were inadequate in reproducing the actual spread of the virus, and presents possible explanations.  It also presents a more accurate approach to modeling the spread of infectious illnesses.

In traditional models of disease spread, a quantity called the basic reproductive number, R₀, is defined as the average number of people to whom an infected person spreads the illness.  One can easily see that according to this theory of spread, if R₀>1, an epidemic results.  Estimates of R₀ in the case of SARS were between 2.2 and 3.6; this should have resulted in anywhere between 30,000 and 10million cases of SARS within the first 120 days.  However, observed incidence of the illness was much lower, and the authors present possible reasons for this.  Briefly, though, all members of a population are not equally likely to be infected, or to infect others.  For instance, many transmissions took place in crowded places, and particularly in hospitals.  A small portion of the population, then, runs a higher likelihood of being infected (or infecting others), while the majority are at considerably lower risk.

The authors also highlight the importance of the ‘initial conditions’.  The first infected person in a population is called ‘patient zero’; treating ‘patient zero’ as a node in the network, the initial condition is the degree of ‘patient zero’, which we recognize from class as the number of edges leaving this node.  Figure 4A in the paper illustrates how the risk of an epidemic increases sharply with the degree of ‘patient zero’.  The authors compare the spread of SARS in Toronto and
Vancouver.  In Toronto, ‘patient zero’ was the matriarch of a large family who died at home, whereas ‘patient zero’ in
Vancouver returned to an empty home and was hospitalized soon after his return.  More than 200 cases appeared in Toronto, but only 3 in
Vancouver, and these seemed to have been imported. 

This pattern is not surprising; the behavior of networks is highly nonlinear, and it is easy to see that the spread of a disease like SARS would vary exponentially with the degree of nodes in the network.  The point, though, is that the topology of the network is of critical importance in determining the spread of illnesses, and needs to be modeled accurately.  Quantities like R₀ are useful only to an extent, and traditional models do not adequately take network topology into account.  I have left out the details of the analysis presented in the paper, but briefly, the authors examine three (topologically) different networks with the same values of R₀ and demonstrate that the networks exhibit very different probabilities of an epidemic occurring.  In particular, the occurrence of individuals who are labelled ‘superspreaders’ and ‘supershedders’ has a tremendous impact on what happens.

(If you’ve got this far, you might want to take a look at the paper.  It’s not that long, surprisingly readable, and actually quite interesting.)