Low-Order Autoregressive Models in Early Detection of Epidemic Outbreaks and Explosive Behaviors in Economic and Financial Time Series

In our SUGI 2006 presentation, we suggested using low-order autoregressive models, AR(1) and AR(2), in biosurveillance and outbreak detection (PROC ARIMA, SAS/ETS ® ). Our suggestion was based on empirical data. Here we propose a strong theoretical ground for this. Based on the workhorse model in epidemiology, a classic susceptible-infectious-recovered (SIR) model, we arrive at AR(1) models of epidemics. In this approach we need to estimate only one parameter, the first-order autoregressive coefficient. Its least squares estimate has a very simple epidemiological meaning. Note that in the vast majority of applications, AR and ARMA are used as purely empirical models, with no specific substance matter meaning for coefficients. The value of our first-order autoregressive coefficient less than one corresponds to a stationary, noepidemic regime. If the parameter is greater than one, we have an explosive case (an outbreak of epidemic). When the coefficient is equal to one, we have the so-called Unit Root case. Based on the observed data in a chosen time window, the least squares estimates and confidence intervals allow us to decide which case is more appropriate. We discuss also the question of bias correction of our estimates. After performing purely temporal analysis, we can proceed to the spatial step with logistic or Poisson regressions as in our SUGI 2006 paper. The approach described above can also be used in describing explosive behaviors of economic and financial time series (e.g., stock market bubbles).