Addendum: Wang et al. A Bayesian Approach to Real-Time Monitoring and Forecasting of Chinese Foodborne Diseases. Int. J. Environ. Res. Public Health, 2018, 15(8):1740; doi:10.3390/ijerph15081740

Abstract

[...]of the first paragraph in Section 3, the authors would like to update the last two sentences of the paragraph and add some citations. [...]the authors would like to update “proposed in this paper” to “described in this paper”. 1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China 2School of Public Health, Center of Statistical Science, Peking University, Beijing 100871, China 3School of Mathematical Sciences, Center of Statistical Science, Peking University, Beijing 100871, China 4China National Center for Food Safety Risk Assessment, Beijing 100022, China *Author to whom correspondence should be addressed.

The authors wish to update the Abstract and Section 3 in their paper published in the International Journal of Environmental Research and Public Health (IJERPH) [1].

They would like to rewrite the abstract as follows:

Abstract: Foodborne diseases have a big impact on public health and are often underreported. This is because a lot of patients delay treatment when they suffer from foodborne diseases. In Hunan Province (China), a total of 21,226 confirmed foodborne disease cases were reported from 1 March 2015 to 28 February 2016 by the Foodborne Surveillance Database (FSD) of the China National Centre for Food Safety Risk Assessment (CFSA). The purpose of this study was to make use of the daily number of visiting patients to forecast the daily true number of patients. Our main contribution is that we take the reporting delays into consideration and apply a Bayesian hierarchical model for this forecast problem. The data shows that there were 21,226 confirmed cases reported among 21,866 visiting patients, a proportion as high as 97%. Given this observation, the Bayesian hierarchical model was established to predict the daily true number of patients using the number of visiting patients. We use several scoring rules to assess the performance of different nowcasting procedures. We conclude that Bayesian nowcasting with consideration of right truncation of the reporting delays has a good performance for short-term forecasting and could effectively predict the epidemic trends of foodborne diseases. Meanwhile, this approach could provide a methodological basis for future foodborne disease monitoring and control strategies, which are crucial for public health.

In the end of the first paragraph in Section 3, the authors would like to update the last two sentences of the paragraph and add some citations. The revised sentences are as follows:

In this paper, we apply a Bayesian nowcasting model proposed by Höhle and an der Heiden [2] to forecast the daily total number of cases. Thanks to Salmon et al. [3], who provided a convenient R package “surveillance”, the inference for the model could be easily performed. The R package surveillance also contains a few other nowcasting methods that we also tried and did comparisons with using the scoring rules implemented in the package. The results are shown in Section 4. Below we review the model.

The authors revised Section 3.2 to describe the inference approach proposed by Höhle and an der Heiden (in Section 3.2 of [2]) in some greater detail. This part is now as follows:

For the convenience of the reader, we describe the inference approach proposed by Höhle and an der Heiden (in Section 3.2 of [2]) in some greater detail.

Define _pd as the (time-homogeneous) probability that a case will have a reporting delay of d days. The _pd ’s satisfy the following equation: _{∑d=0D pd}=1 . Following Kalbfleisch and Lawless’s previous work [4] and Zeger et al.’s previous work [5], we assume that the occurrence time of cases follows an underlying inhomogeneous Poisson process. A reasonable data generating process for the daily number of cases is thus as follows:

_Nt|_λt∼Po(_λt)(_nt,D,_nt,D−1,⋯,_nt,0)′|_Nt,p∼MN(_Nt,(_pD,_pD−1,⋯,_p0)′)

where Po(λ) denotes the Poisson distribution with expectation λ>0 and MN(N,p) denotes the multinomial distribution with size parameter N and probability vector p . Nowcasting for a given time T can thus be divided into steps of determining the _λt ’s, estimating the unknown delay distribution (i.e., the _pd ’s), and finally predicting the unobserved _nt,d ’s in order to compute the total _Nt . As T increases, and if the assumption about a time-homogeneous delay distribution is acceptable, the available data make it possible to estimate the delay distribution better and better, and hence the quality of the predictions near T improves with time.

Consider a fixed time T and define _pT=(_pT,D,_pT,D−1,⋯,_pT,1)′ as the probability vector denoting that a case is reported with a delay of d days given the observed incomplete information at time T , i.e., the set of _nt,d, where t+d≤T . We choose as prior distribution the generalised Dirichlet distribution GD(α,β) with fixed constants α=(_α0,_α1,⋯,_αD)′ and β=(_β0,_β1,⋯,_βD)′ . Now we use Property 3 in the Web appendix of [2] that shows that the posterior of p under right-truncated multinomial sampling is again a GD distribution with parameters _αT*,_βT* given by

_αT,i*=_αi+∑τ=0T−D+i_{nτ,D−iβT,i*}=_βi+∑τ=0T−D+i(_Nτ,τ+D−i−_nτ,D−i),i∈{0,⋯,D−1}.

hence, for a given T we can assume the following model hierarchy for the time points t∈{T−D,⋯,T} :

_pT∼GD(_αT*,_βT*)_λt∼Ga(_aλ,_bλ)_Nt|_λt∼Po(_λt)_Nt,T|_Nt,_pT∼Bin(_Nt,_qT,T−t),

where _qT,d=_{∑δ=0d pT,δ} is the proportion reported within a delay of d days. We denote by Ga(_aλ,_bλ) the gamma distribution with parameters _aλ>0,_bλ>0 . For this hierarchical model, the marginal distribution of _Nt is a negative binomial distribution with the following mean and variance:

E(_Nt)=_μλ=_{aλ bλ}Var(_Nt)=_μλ+_μλ2aλ.

To estimate _Nt given the observed counts _nt,d at time T , we have to perform two steps: (1) update the delay distribution _qT and (2) update the prediction for _Nt : (1)

For the given T we compute _αT*,_βT* as stated above. We then draw for k=1,…,K random vectors _pT(k)∼GD(^α* _T,_βT*) by the algorithm of Wong [6] and calculate

_qT,d(k)=∑δ=0d_pT,δ(k). (2)

Given the updated delay distribution _qT(k) and the observed counts _nt,d , we can now update the prediction of _Nt,t=T−D,⋯,T .

For n∈{0,1,2,⋯} we approximate by Monte Carlo sampling

f(_Nt=n|_Nt,T)≈1K∑k=1K_fn,t(k),t∈{T−D,⋯,T}

An application of Bayes theorem provides _fn,t(k)=_f̃n,t(k)/_ct(k) , where _ct(k)=_{∑n=0∞ f̃n,t(k)} is the normalization constant and

_f̃n,t(k)=f(_Nt,T|_Nt=n,_qT,T−t(k))f(_Nt=n|_λt)f(_λt)

for all t∈{T−D,⋯,T} . The factors of the last equation can be evaluated using the distributional assumptions of the model hierarchy. For numerical convenience we do not sum over the entire support {0,1,2,⋯} to get the normalization, but instead approximate

_ct(k)≈∑n=0_{Nmaxf̃n,t(k)},

where _Nmax is chosen sufficiently large.

Finally, the authors would like to update “proposed in this paper” to “described in this paper”.

The changes do not affect the results. The manuscript will be updated and the original will remain online on the article webpage, with a reference to this addendum.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

References

1. Wang, X.; Zhou, M.; Jia, J.; Geng, Z.; Xiao, G. A Bayesian Approach to Real-Time Monitoring and Forecasting of Chinese Foodborne Diseases. Int. J. Environ. Res. Public Health 2018, 15, 1740.

2. Höhle, M.; An der Heiden, M. Bayesian nowcasting during the STEC O104:H4 outbreak in Germany, 2011. Biometrics 2014, 70, 993-1002. [Green Version]

3. Salmon, M.; Schumacher, D.; Höhle, M. Monitoring Count Time Series in R: Aberration Detection in Public Health Surveillance. J. Stat. Softw. 2016, 70, 1-35.

4. Kalbfleisch, J.D.; Lawless, J.F. Inference Based on Retrospective Ascertainment: An Analysis of the Data on Transfusion-Related AIDS. J. Am. Stat. Assoc. 1989, 84, 360-372.

5. Zeger, S.L.; See, L.C.; Diggle, P.J. Statistical methods for monitoring the AIDS epidemic. Stat. Med. 1989, 8, 3-21.

6. Wong, T.T. Generalized Dirichlet distribution in Bayesian analysis. Appl. Math. Comput. 1998, 97, 165-181.