Spatial contact networks
From Santa Fe Institute Events Wiki
According to Grenfell et. al. , "Understanding the spatial contact network for parasite transmission is a holy grail because it will allow prediction about the spread of emerging pathogens and ultimately guide public health and veterinary intervention programs."
Our goal is formalize a method for determining spatial contact networks from time series data of disease transmission. We have weekly measles outbreak reports from 60 urban cities in England and Wales dating from 1944 to 1967. Using a subset of this data, we created this  graph to illustrate how the number of measles cases varied over time in each city (y and x axes are GPS coordinates).
Based on this visual representation of the data, it appears that several cities may act as pathogen sources from which the disease spreads to surrounding cities. We plan to reconstruct this spatial contact network for the measles data and, in doing so, introduce a method for inferring such networks for other disease outbreaks.
In addition, we will apply techniques such as time series embedding , wavelet transformation , and Fourier transformation  in order to tease out any cyclic behavior in the data and determine the time-scale of infection outbreaks. Removing this cyclic behavior from the data should also help us uncover any chaotic dynamics of the system.
- First, we will use embedding, wavelet, and Fourier transform to determine the cyclic behavior of the time series. Then we will filter out any cyclic behavior to determine if chaotic dynamics of the system exist.
- We will use Granger Causality and Directed Transfer Function measures to determine the causal links between cities. We will then compare our causality measures to a null model of the network which will be constructed by randomizing links between the 60 cities in our data set. This will enable us to construct the spatial contact network for the measles cases under study.
- This measles data set has been used by Grenfell and others to construct predictive models of disease transmission that include both temporal and spatial dynamics. One recent model proposed by Xia, Bjornstad, and Grenfell  claims to be comprehensive and predictive of the measles outbreak from 1944-67. Using this as a null model for the spatial contact network, we will compare the results from our causality predictions to determine how well Grenfell's model can capture network dynamics.
- Use other methods of inferring causality, such as a mutual information approach, to determine spatial contact network. Compare to results using Granger Causality and DTF.
- Explore how the network changes over time - it is likely that the spatial contact network will have changed dramatically from 1944 to 1967. It would be interesting to formalize a method for determining the evolution of this network over time.
- Disease outbreaks will exhibit cyclic behavior most likely at several different time scales. Seasonal variation in disease transmission rates is well-known so this is expected. Chaos may or may not be present.
- We will formalize a method for inferring spatial contact networks from time-series data. In seems likely that in the network of measles outbreaks, large cities such as London may be sources of infection. We may also see interesting dynamics between distant cities which will hopefully correlate with known travel patterns.
- Our analysis will either confirm or refute the accuracy of other disease transmission models.
- Xia, Bjornstad, and Grenfell. 2004. 
- Grenfell's spatial/temporal model of disease outbreak
- Kaminski, Ding, Truccolo, Bressler. 2001. 
- Granger causality and Directed Transfer Functions for inferring causality
Cyclic Dynamics and Chaos
Perform embedding on time series Perform wavelet transform on series
- Perform Fourier transform
- Infer cyclic dynamics and filter these out of the data to determine chaos
Inferring spatial contact network
- Implement Granger Causality and DTF on data set
- Gauge accuracy of causality results, e.g. plot against city size - large cities should influence smaller ones
- Generate randomized null model
- Implement Grendell's gravity model and use it to obtain null contact network
- Compare results from GC and DTF to null models to determine significance
- Determine contact network