1. Calculation of excess influenza mortality for small geographic regions Al Ozonoff, Jacqueline Ashba, Paola Sebastiani Boston University School of Public Health, Department of Biostatistics

2. Abstract influenza exhibits strong seasonal patterns of morbidity and mortality The study of influenza as a public health problem has focused in part on “excess mortality” as a measure of the mortality burden attributed to influenza. The traditional calculation of excess mortality relies on estimating a seasonal baseline, and then summing incident deaths above this baseline. Model parameter estimation is thus an essential component to achieve precise and accurate estimates of excess mortality.

3. Abstract While this is a straightforward procedure for data aggregated at the national level, the problem is more challenging when considering smaller geographic regions such as individual cities. HMMs allow for more efficient use of data to estimate model parameters, which in turn provides improved accuracy and precision during the estimation process.

4. 1. Introduction There has been recent concern widely expressed over the potential for a pandemic caused by a novel strain of influenza. Surveillance of influenza mortality is one component in the national strategy to prepare for such an event, or a similar event of importance to the public health. However, influenza data are of notoriously bad quality, and utilization of these data on small spatial scales is challenging. Efforts to apply modern statistical methods to these data may address some of these challenges.

5. 2.1 Serfling’s method Since the mid-1960s, the Centers for Disease Control and Prevention (CDC) influenza surveillance programs have used a cyclic regression model to determine epidemic influenza ac tivity and excess mortality attributed to influenza. The method determines the epidemic threshold. Serfling’s method uses five years of weekly influenza mortality data to fit a periodic regression model containing terms for intercept, linear trend, and a pair of harmonic terms to capture the underlying sinusoidal behavior of seasonal influenza.

6. 2.1 Serfling’s method Serfling’s model ：

7. 2.1 Serfling’s method The model estimates the non-epidemic seasonal baseline, so weeks that typically show high influenza activity are excluded to avoid biased parameter estimates. Current CDC practice is to exclude the months of October through April. After parameters of the model have been estimated, the upper limit of a confidence band around the sinusoid is used to determine the epidemic threshold for that time period.

8. 2.1 Serfling’s method CDC uses this model prospectively, meaning each week the model can be refitted and a new threshold determined. This provides a continually updated determination of the state of epidemic influenza in the U.S. which is timely up to a two to three week reporting lag.

9. 2.2 HMM Intuitively: Hidden Markov models (HMMs) use a hidden (latent, unobserved) discrete random variable, the state of the system at time. Observed variables are then modeled, conditional upon the hidden state. knowledge of the state implies knowledge of the stochastic distribution of the observed random variable. HMMs allow both likelihood and Bayesian approaches to parameter estimation. We have chosen Bayesian Inference Using Gibbs Sampling (BUGS), with open-source implementation in the statistical system R via the OpenBUGS project.

10. 3.1 122 Cities Mortality Program 122 selected cities voluntarily report deaths each week attributed to pneumonia or influenza, representing roughly 25% of the U.S. population. P&I is used as a proxy measure for influenza activity, and is generally accepted as a reasonably good measure of influenza mortality. Since reasonably complete morbidity data are not available at the national level, P&I mortality data represent the best available data in terms of completeness and similarity to actual influenza activity.

11. 3.2 City level mortality data Surveillance data from the 122 Cities program are most commonly analyzed at the national or regional levels. the median across all cities of average weekly P&I mortality is four deaths per week Since excess mortality is calculated as additional deaths above seasonal baseline, it is difficult to use this measure of influenza activity at such a small geographic scale. This is necessary using a traditional model such as Serfling’s method. HMMs allow model formulations that may include parameters common to several states (i.e. Both epidemic and non- epidemic periods).

12. 4. Model formulation HMMs may offer improved efficiency when estimating seasonal parameters. For example, a typical two-state HMM for influenza surveillance might use a single parameter to reflect a shift in mean upon transition to an epidemic state, while keeping baseline (seasonal) parameters constant for both states:

13. 5. Evaluation model goodness-of-fit using root mean squared error (RMSE) ， selecting a six year (312 week) period of influenza data

14. 6. Result