Figure 3. Computation of causality field, Approach 1 Plates A and B: Incidence data from neighboring counties in Alabama, US.

Plates C and D: Transformation to difference-series, $i\mathrm{.}e\mathrm{.}$, change in the number of reported cases between weeks. We imposed a binary quantization, with positive changes mapping to ‘1,’ and negative changes mapping to ‘0.’ From a pair of such symbol streams, we computed the direction-specific coefficients of Granger causality (see Supplement). For each county, we obtained a coefficient for each of its neighbors, which captured the degree of influence flowing outward to its respective neighbors (Plate L). We computed the expected outgoing influence by considering these coefficients as representative of the vector lengths from the centroid of the originating county to centroids of its neighbors. Viewed across the continental US, we then observed the emergence of clearly discernible paths outlining the ‘causality field’ (Plate G). The long streamlines shown are highly significant, with the probability of chance occurrence due to accidental alignment of component stitched vectors less than ${\mathrm{10}}^{\mathrm{-}\mathrm{185}}$; while each individual relationship has a chance occurrence probability of $\mathrm{\sim }\mathrm{6}\mathrm{\%}$ (Plates E and F). Plate H: Spatially-averaged travel patterns (see text in Materials and methods) and the sink distribution between expected travel patterns. These patterns (Plate H), along with the inferred causality field (Plate I), match up closely, with sinks showing up largely in the Southern US, explaining the central role played there. In Plate H, the size of the blue circles indicate the percentage of movement streamlines (computed by interpreting the locally averaged movement directions as a vector field) that sink to those locations. In Plate I, the size of the red circles indicate the percentage of causality streamlines that sink to the indicated locations. We note that $\mathrm{\sim }\mathrm{75}\mathrm{\%}$ of the movement streamlines sink in counties belonging to the Southern states, which matches up well with the sinks of the causality streamlines. In Plates J and K show spatial analysis results for two different infections (HIV and E. coli, respectively) and which exhibit very different causality fields, negating the possibility of boundary effects.