Evaluation of real time methods for epidemic forecasting 4

Chapter 4Discussion In this study, we aimed to model the inuenza A-H1N12009 epidemic in Singapore and to be able to perform predictions of number of individuals presenting ILIs per GP us

Chapter 4

Discussion

In this study, we aimed to model the inuenza A-H1N1(2009) epidemic in Singapore and to be able to perform predictions of number of individuals presenting ILIs per GP using presently available data The results in Section 3.1 illustrate that the three models and methods that we proposed can be used for predictive purposes Of these, tting the stochastic SIR model using particle lter performed the best in terms of the prediction, as shown when superimposed with the daily observed data (Figure 3.7)

The predictions from the deterministic SIR model using MCMC together with importance sampling is satisfactory (Figure 3.5), as the model captures the general underlying dynamics of the disease, although the number of fected individuals is underestimated at the peak of the pandemic The in-accuracy in the estimated infected individuals might stem from the proposal distribution we used in the importance sampling algorithm, as the algorithm requires a proposal distribution which is close to the posterior distribution

to return good estimates (Christian and Casella, 2001) However, it should

69

be noted that by running short MCMC chains to obtain a rough estimate

of the location and scale of the posterior distribution of the parameters of the deterministic SIR model, and then using these to build a multivariate normal proposal distribution for importance sampling, the prediction results are improved compared to performing only importance sampling

We observed that importance sampler on the Richards model (Figure 3.3) does not perform as well as the deterministic SIR and stochastic SIR models, for instance, the weekly peaks in the observed data at the end of the surveillance period is not captured in the predictions This shows that the Richards model proposed in this study might not be as representative of the disease dynamics as the SIR model We postulate this failure is because

it is not compartmental in nature Instead of dividing the population into dierent compartments throughout the pandemic and having several proba-bilities to govern the transitions between compartments, the Richards model has only key epidemiological parameters to describe the overall trend for the pandemic (Hsieh et al., 2010)

The scatterplot (Figure 3.6) shows that the weights associated with the parameters are generally well distributed, indicating that the algorithm worked well without weight degeneracy at the end of the iterations The scatter-plots for the Richards model (Figure 3.2) and the deterministic SIR model (Figure 3.4) also show that the importance sampling algorithms return well distributed weights for the parameters

From Section 3.2, we have chosen some commonly used functions of the

71 estimator to compare how well the predictions fare across dierent data sub-sets used to perform the predictions and also across dierent models The result in Figure 3.11 shows that in most cases, the stochastic SIR model performs consistently the best as the absolute values it takes for most of the calculated functions are the lowest among the three models, with the exception of prediction using days 17 This result agrees with the graphical output from Section 3.1 which shows that particle lter on stochastic SIR model performs the best in generating predictions for the disease

Furthermore, from Figure 3.11 we can also see that, mostly, the absolute values for the ME, MAE, and MSE decrease as the number of days used for prediction grows Data subsets with larger number of days used for prediction perform better than data subset of shorter period because more information

is incorporated in the algorithm to produce more accurate predictions for the future

In Section 3.3, we compare the Richards model output generated using informative priors (Figure 3.3) with that generated using non-informative priors (Figure 3.12) and observe that apart from the underestimation at the

rst peak on the last week of June 2009, the output from non-informative priors actually performs very well Since the amount of information the priors carry does not aect the predictive performance of the Richards model, non-informative priors should be used in future epidemics with similar information content to permit accurate prediction

For the deterministic SIR model, informative priors (Figure 3.5) and

non-informative priors (Figure 3.13) return similar results, possibly because the data play a bigger part in providing information for the algorithm to produce the posterior distribution For the stochastic SIR model, informative priors (Figure 3.7) perform much better than non-informative priors (Figure 3.14) The output from non-informative priors show an overestimation at the rst peak on the last week of June 2009, but is gradually corrected by mid July, however, the peak at the end of July is not captured Therefore, the informa-tive priors, which are constructed with input from previous study or expert opinion, is important in generating good predictions for the stochastic SIR model

We acknowledge that although the SIR model is adequate in explaining the disease dynamics in this context, other models, for instance, models which account for spatial spread (Mollison and Kuulasmaa, 1985), age-structure (Schenzle, 1984) or herd immunity (Starr et al., 1997), might provide more details on the evolution of the disease The assumptions imposed on the models, for example, homogeneous mixing between infected individuals and susceptible individuals, might not be an exact and realistic description of the disease dynamics in real life (Burr and Chowell, 2006) Nevertheless, the simple SIR model provides insights on how the dierent factors interact in the pandemic and forms a basis on which more assumptions details can be added to provide more sophisticated results and predictions

The dataset used in this study was obtained from a network of general practitioners or family doctors who submitted observations of ARIs among

73 their patients on a voluntary basis, and therefore the data might not be representative of the actual numbers of infected individuals in the whole population of Singapore However, the simple surveillance system serves its intended purpose in disease tracking and forecasting in real-time (Ong et al., 2010), and estimates for the local outbreak size in terms of percentage of infected in the population from the deterministic model has a 95% credible interval of (5%, 20%) and that of the stochastic model has a 95% credi-ble interval of (7%, 21%), are consistent with estimates from independent serological studies (Chen et al., 2010; Lee et al., 2011)

Other limitations include the fact that we cannot fully account for the nature of patients in our study Studies (Hancock et al., 2009; Chi et al., 2010) show that the elderly are less susceptible to an epidemic because of pre-existing cross-reactive antibodies, however, we do not have access to past serological information to be incorporated into the predictions It might be more appropriate to incorporate age-structure in the models for this reason,

so that we can have more insight into the dierent eect of the pandemic on individuals in dierent age groups

We conclude from this study that stochastic SIR model with particle

lter is the most eective among our models and can be extended to per-form predictions for future pandemic outbreaks or other similar diseases in real-time Model selection and averaging can also be done within the frame-work of particle lter by including model as an element of the parameter set (Doucet et al., 2000) The deterministic SIR model performs well but

requires more computational time compared to stochastic SIR model Al-though the Richards model does not return better predictions compared to compartmental models in our study, a two-phased Richards model which dis-tinguishes the turning point at the end of exponential growth and the turning point where the cumulative cases increases again has been shown to perform better than the simple Richards model (Hsieh et al., 2010) when applied to inuenza A-H1N1(2009) observations in Canada, and hence might be the topic of future investigations in the local setting

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