Network Theory and SARS
The paper cited below contains an analysis of the effectiveness of various network structures in modeling the spread of Severe Acute Respiratory Syndrome (SARS) through Canada in 2003. A simple model for disease spread, the compartmental model, divides the total population into various compartments, and relates the populations of each compartment mathematically. For example, for the chicken pox, it would make sense to divide the population into those who are susceptible, those who are infectious, and those who are immune. Properties of the system, such as the duration of the disease, the number of people one comes into contact with during the day, etc., are used to relate the populations of the compartments. This very simple model, however, falsely assumes total homogeneity, that all individuals are the same. In fact, some individuals have many more interactions with the sick, such as nurses and doctors, and some are more susceptible.
Different Network Structure Models: (A) urban, (B) power law, and (C) Poisson networks
We can think of individuals as nodes in our graph, and points of contact as edges. We can call these undirected graphs “contact networks”. In an effort find a model which more accurately captures the network, the group tested three graph structures against the data collected about the spread of SARS. They modeled urban areas, specifically Vancouver, as regions consisting of homes, schools, hospitals, work places, and shopping centers, each with its own subdivisions (classrooms, medical wings, stores), and nodes– different types of individuals (teachers, family members, children, doctors, managers), and connected the nodes based on intuition and social structure. They then selected a set of edges by probabilistically connecting the nodes (members of the same home are guaranteed connection, hospital staff have a 0.3 chance of contact, etc.)
Node Degree for Network Structures
As points of comparison, they also tested against probabilistically generated networks, one following a Poisson distribution, and the other following a power law. Of relevance to this course is the motivation behind the choice of a power law distribution. Rather than assuming that all individuals have the same degree, as does the compartmental model, the power law model generates a graph in which, while the majority of edges are held by the majority of people (a big head), there exists a minority of individuals who have many more edges than might be expected. This minority, when infected, have the potential to spread the disease very, very quickly, and can bring an outbreak into an epidemic. The graph below displays the probability of an epidemic occurring, given the degree of patient 0 (the first person to catch the disease).
Probability of Outbreak given the degree of Patient Zero
-JB
Sources:
Network theory and SARS: predicting outbreak diversity
Compartmental models in epidemiology