Latent Graphical Models of Multivariate Count Time Series

Conventional mathematical models of infectious diseases frequently overlook the spatial spread of the disease concentrating only on local transmission. However, spatial propagation of various diseases have been noted between geographical regions mainly due to the movement of infectious individuals from one region to another. In this work, we propose generalized linear models to study the graph of dependencies between multiple infection count time series from neighbouring regions. Due to the inherent theoretical and computational difficulties in inferring traditional partial correlation and causality graphs for such multiple count time series data, weakened concepts of correlation and causality of appropriate latent variables are introduced to simplify computation. In order to estimate these latent graphs with tunable sparsity, a novel Monte Carlo expectation and maximization algorithm is used to iteratively maximize an appropriate regularized likelihood function, and asymptotic convergence is established. In addition to simulated data, the algorithm is applied on observed weekly dengue disease counts from each region of an Indian city. The interdependence of various regions in the proliferation of the disease is characterized by the edges of the inferred latent graphs. It is observed that some regions act as epicentres of dengue spread even though their disease counts are relatively low.