<i>S</i>^<i>ν</i><i>EIR</i> with enKS vs. straw man forecast for the 2013–2014 U.S. ILI data.

<p>The M-distance between U.S. 2013–2014 ILI data and the two forecasts is plotted. The M-distance between the forecast and ILI data is calculated for each epidemiological week until the end of the influenza season. The M-distances at week 36 uses the forecast observations from week 36 of 2013 to week 20 of 2014 and the ILI data from week 36 of 2013 to week 20 of 2014. The M-distances plotted for the straw man prediction use sample covariances and means calculated from 300 time series draws of the straw man forecast. Due to the lack of causal relations included in the straw man model this measure of accuracy is significantly lower in the early season for the straw man prediction. This figure shows that the data assimilation forecast has a noticeably smaller M-distance, and therefore is quantitatively better, than the straw man model for the early influenza season. Once the influenza season peaks the success of the forecast breaks down due to model error. It is interesting to note that due to the enKS data assimilation our <i>S</i>^<i>ν</i><i>EIR</i> forecast seems to attempt self-correction, i.e. the M-distance is increasing and then decreases.</p

<i>S</i>^<i>ν</i><i>EIR</i> peak quantiles for 2013–2014 U.S. ILI.

<p>50% and 90% credible interval estimates of the influenza season peak are plotted along with the median. Forecasts for the size of the ILI peak were widely varying in the 90% credible interval. This could possibly be reduced by the elimination of high peak outliers such as the 2009 H1N1 emergence and through adjustment of low forecasts in our prior. However, even with these draw backs the 50% credible region has a width of only 1%–2%.</p

Histogram of the marginal distribution for the average recovery time, measured in days.

<p>The rate parameter in our <i>S</i>^<i>ν</i><i>EIR</i> model, <i>γ</i>, is the inverse of this average time. We see that this distribution is concentrated over 6–7 days and skewed toward longer incubation times. The prior distribution for <i>γ</i> is more concentrated than the distributions for <i>θ</i> and <i>β</i>₀ which means that the ILI data determine this parameter more exactly.</p

Defining a maximal influenza season.

<p>We highlight the weeks corresponding to our maximal influenza season over which we parameterize our forecast. Since our model does not include re-infection or loss of immunity we can only hope to forecast one pre-defined season at a time.</p

Histogram of the marginal distribution for the average incubation time, measured in days.

<p>The rate parameter in our <i>S</i>^<i>ν</i><i>EIR</i> model, <i>θ</i>, is then the inverse of this average time. We see that this distribution is concentrated over 3–6 days and skewed toward longer incubation times.</p

U.S. ILI prior forecast for 2013–2014 flu season.

<p>This figure shows the prior forecast along with the 2013–2014 ILI data. Note the potential for an early and late peaking influenza season. The red line represents the median forecast from 300 samples of the prior. The dark blue and light blue regions represent the 50% and 90% credible regions centered around this median, respectively. Credible intervals were also generated from 300 samples of the prior.</p

The transmission rate function <i>β</i>(<i>t</i>;<i>β</i>₀, <i>α</i>, <i>c</i>, <i>w</i>).

<p>The transmission function is chosen to be a smooth, five times differentiable, bump function ranging between <i>β</i>₀(1+<i>α</i>) at the peak of flu transmission and <i>β</i>₀(1−<i>α</i>) at the low point. This is done to account for seasonality in our model. The parameters <i>c</i> and <i>w</i> control the center (peak of elevated flu transmission) and width (duration of elevated flu transmission).</p

<i>S</i>^<i>ν</i><i>EIR</i> start week quantiles for 2013–2014 U.S. ILI.

<p>50% and 90% credible interval estimates of the influenza season start week are plotted along with the median. Each week, as new ILI data become available the forecast is revised. This causes the uncertainty in our forecast to diminish. However, due to the model’s inability to maintain an elevated ILI level past the peak, we see that late in the flu season, the model adjusts by pushing the start week later into the season. This causes an overestimation of the start week that worsens as the season progresses. In practice, once the start week has been observed the <i>forecast</i> would be fixed. However, adjustment of the model parameterization using the enKS would continue to affect the model simulation start date.</p

Histogram of the marginal distribution for the duration of heightened transmissibility.

<p>The parameter <i>w</i> is represented in weeks. A value <i>w</i> = 14 corresponds to 16 weeks of elevated transmission. We see that this distribution is concentrated over 14–20 weeks and skewed toward longer periods of elevated transmission.</p

Percent improvement in M-distance using <i>S</i>^<i>ν</i><i>EIR</i> with enKS to forecast 2013–2014 U.S. ILI data.

<p>The percent improvement in the M-distance for the U.S. 2013–2014 ILI forecast using the data assimilative method compared to the straw man method is shown. Here we see that up to a month before the peak of ILI the use of a mathematical influenza model with data assimilation provides up to a 20% improvement in the forecast with a minimum of a 10% improvement. However, this improvement is quickly degraded due to model bias close to the peak. It is notable that after the peak is observed the data assimilation attempts to correct the model but can not make up for the <i>S</i>^<i>ν</i><i>EIR</i> model’s strong tendency to a zero infected state after the peak.</p