Evaluation of real time methods for epidemic forecasting 3

Figure 3.1: Predictions of the number of infected subjects in the populationper day using the particle lter method for the deterministic SIR model.The shaded area indicates a series of s

In this section we display the output of predictions for the average number

of ILIs per GP for the pandemic and the parameters associated with eachmodel used

We have performed the particle lter routine on deterministic SIR modeland stochastic SIR model respectively However, the estimate of the pos-terior distribution for the states is unsatisfactory for the deterministic SIRmodel as shown in Figure 3.1, as the peak of the pandemic is not captured.Consequently, the predictions for the deterministic SIR model presented inthis section is formed by using three iterations of MCMC followed by threeiterations of importance sampling We use importance sampling on Richardsmodel to obtain predictions

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Figure 3.1: Predictions of the number of infected subjects in the populationper day using the particle lter method for the deterministic SIR model.The shaded area indicates a series of successive regions spanned by 95%,93%, 91%, , 0% credible intervals in increasing intensity.

3.1 Parameters and predictions for each model

The scatterplot for the parameters in the Richards model is shown in Figure3.2, with the size of points proportional to their respective likelihoods Theprediction on the number of infected subjects per GP each day throughout

lected from 25th June 2009 to 24th September 2009 The shaded area in redindicates a series of successive regions spanned by 95%, 93%, 91%, , 0%credible intervals of the predicted number of infected subjects per GP eachday in increasing intensity, and the empirical number of infected subjects per

GP from the data is plotted in black

The corresponding scatterplot for parameters and plot of predicted ber of infected subjects per GP are also shown for deterministic SIR model(Figure 3.4 and Figure 3.5) and stochastic SIR model (Figure 3.6 and Figure3.7)

num-Figure 3.2: Scatterplot of parameters for Richards model, the area of thepoints are proportional to their corresponding weight.

Figure 3.3: Predictions of the number of infected subjects per GP per dayusing the Richards model with credible intervals shown in red Black lineindicates observed data.

Figure 3.4: Scatterplot of parameters for deterministic SIR model, thearea of the points are proportional to their corresponding weight.

Figure 3.5: Predictions of the number of infected subjects per GP per dayusing the deterministic SIR model with credible intervals shown in red.Black line indicates observed data.

Figure 3.6: Scatterplot of parameters for stochastic SIR model, the area

of the points are proportional to their corresponding weight

Figure 3.7: Predictions of the number of infected subjects per GP per dayusing the stochastic SIR model with credible intervals shown in red Blackline indicates observed data.

In order to assess the predictive ability of the models, we also generatepredictions on number of infected subjects per GP for the future for eachmodel assuming that we know only a subset of the data Using data from

days 114 (8th July), days 121 (15th July), days 128 (22th July), days135 (29th July), days 142 (5th August), days 156 (19th August), days170 (2nd November) and days 184 (16th November), we generate a pre-diction up to day 120 (22nd October) Figures 3.8, 3.9 and 3.10 shows thepredictions for Richards model, deterministic SIR model and stochastic SIRmodel respectively.

Figure 3.8: Predictions using data subset (black solid line) on number ofInfected subjects per GP by Richards model shown in 95% credible interval(red) Future data (black dotted line) shown does not contribute to theposterior distribution.

Figure 3.9: Predictions using data subset (black solid line) on number of fected subjects per GP by deterministic SIR model shown in 95% credibleinterval (red) Future data (black dotted line) shown does not contribute tothe posterior distribution.

Figure 3.10: Predictions using data subset (black solid line) on number ofInfected subjects per GP by stochastic SIR model shown in 95% credibleinterval (red) Future data (black dotted line) shown does not contribute tothe posterior distribution.

To assess how the length of dataset aects the predictive performances of themodels, we calculate the following values for each of the predictions using theeight data subsets mentioned in the previous section for all three models to

to compare the forecasts with the actual observations:

The ME and PE are signed measures of error which indicate whetherthe forecasts are biased, specically, whether they are disproportionatelypositive or negative Conversely, the MAE and MSE disregard the direction

of the bias, they measure the average magnitude of functions the errors inthe forecasts The MSE determines the quality of forecasts based on theirvariance and bias

The output is plotted in Figure 3.11 below, with predictive performancesfor Richards model, deterministic SIR model and stochastic SIR model indi-cated in blue, red and magenta The values are calculated by excluding the

period used to make the predictions, therefore calculations starts from time

from days 114, days 121, days 128, days 135, days 142, days 156, days170 and days 184 respectively

Figure 3.11: Predictive performances for Richards model, deterministic SIRmodel and stochastic SIR model

We examine the eects of prior in this section, by comparing the outputobtained using informative and non-informative priors respectively All ouroutput in the previous section is generated using informative prior, as listed inthe rst column of Table 3.1 We generated output based on non-informativeprior shown in the second column of Table 3.1 for Richards model (Figure3.12), deterministic SIR model (Figure 3.13) and stochastic SIR model (Fig-ure 3.14).

Table 3.1: Informative and non-informative priors for each for Richards

model, deterministic SIR model and stochastic SIR model

Figure 3.12: Predictive performances for Richards model using informative priors shown with 95% credible interval (red) Black line in-dicates observed data.

Figure 3.13: Predictive performances for deterministic SIR model usingnon-informative priors shown with 95% credible interval (red) Black lineindicates observed data.

Figure 3.14: Predictive performances for stochastic SIR model using informative priors shown with 95% credible interval (red) Black line indi-cates observed data.