基于贝叶斯深度学习方法的上海新冠肺炎病例时空预测和不确定性量化.pdf

应用概率统计第 40 卷第 2 期2024 年 4 月Chinese Journal of Applied Probability and StatisticsApr.,2024,Vol.40,No.2,pp.298-322doi:10.3969/j.issn.1001-4268.2024.02.006Spatio-Temporal Forecasting and Uncertainty Quantificationof COVID-19 Cases in Shanghai via a Bayesian DeepLearning ApproachZHOU Shirong1,2TANG Yincai1WANG Pingping3ZHUANG Liangliang4XU Jiawei1(1KLATASDS-MOE,School of Statistics,East China Normal University,Shanghai,200062,China)(2College of Mathematics and Physics,Wenzhou University,Wenzhou,325035,China)(3School of Economics,Nanjing University of Finance and Economics,Nanjing,210023,China)(4School of Statistics and Mathematics,Zhejiang Gongshang University,Hangzhou,310018,China)Abstract:The outbreak of COVID-19 in Shanghai in the spring of 2022 had a serious impact onthe society,economy,and daily life of residents.The spread of COVID-19 often exhibits complexnon-linear dynamics influenced by environment,demographics,medical conditions,frequency ofnucleic acid or antigen testing,epidemic control strategies,etc.Long-short term memory(LSTM)models with complex network structures and extensive training are widely adopted to learn andpredict the spreading of epidemic.However,such a model neither explains the uncertainty in data,nor takes the influence of various covariates and heterogeneities into account.Therefore,a two-stage LSTM nested generalized Poisson regression(LNGPR)model is proposed in this paper toanalyze COVID-19 infectious data in Shanghai outbroke in the Spring of 2022.In the first stage,a multi-layer LSTM network is trained to learn district-specific infectious data,then the trainedLSTM is used to fit and predict the number of symptomatic COVID-19 infections.In the secondstage,the predicted number of cases is modeled by a generalized Poisson regression model undera hierarchical Bayesian framework,in which the logarithm of the relative risks is modeled as alinear function of covariates and random effects with spatio-temporal heterogeneities.Facilitatedby a deep learning approach,the spatio-temporal generalized Poisson regression model can forecastand quantifies uncertainty of the number of daily new symptomatic infections.Furthermore,thepredictions based on the proposed Bayesian deep learning approach performs better than thosebased on LSTM method in virtue of borrowing strength from covariates,and spatial and temporalheterogeneity.Keywords:COVID-19;LSTM;Poisson regression model;integrated nested Laplace approxima-tion(INLA)2020 Mathematics Subject Classification:62F15Citation:ZHOU S R,TANG Y C,WANG P P,et al.Spatio-temporal forecasting and uncertaintyquantification of COVID-19 cases in Shanghai via a Bayesian deep learning approachJ.ChineseJ Appl Probab Statist,2024,40(2):298322.The research was supported by the National Natural Science Foundation of China(Grant Nos.12171432,11671303,12271168)and the 111 Project of China(Grant No.B14019).Corresponding author,E-mail:.Received December 8,2023.Revised January 26,2024.No.2ZHOU S.R.,et al.:Spatio-Temporal Inference of COVID-19 Cases in Shanghai2991IntroductionIn March 2022,a new wave of COVID-19 outbroke in Shanghai,which has had a greatimpact on society,economy,and residents daily life.At the initial phase of the outbreak,residents suffered from shortage of food and medical treatment for common diseases,publictransport in city suspended and production stagnated.The two-month citywide lockdownhave has caused havoc on supply chain,leading to a dramatic drop in economy.In orderto conduct effective containment strategies as responses to the varying pandemic,policy-makers have imposed multi-level restrictions corresponding to different phases,such like“precise clearing”,“no crossing over Huangpu River”,plenary static management with“stay-at-home”orders,“three-zone division”,etc.Hence,plausible,dynamic,and legibleprediction and quantification of the epidemic changes is practically required for precisedecision-making.The modeling effort,which aims to portray the epidemic trajectory alongwith assessing the effectiveness of containment strategies implemented in different phases,plays a vital role for the gradual resumption of social and economic activities.Particu-larly,following Chinas general policy of“zero-COVID”,it is also of great significance forShanghai residents to restore the confidence in beating offobstacles with the constructiveconclusions drawn by statistical models.Different statistical approaches have been widely developed or implemented for mod-eling the daily new COVID-19 cases,which give support to the designation of furthervaccination policies and quarantine policies,for instance,the personalized vaccinationprograms based on the probability of COVID-19 transmission in different populations,precise physical isolation measures based on the spatial distribution of COVID-19 in astudy area,and setting the size of isolation hospitals based on the trend of COVID-19changing over time.According to the comprehensive work by Cao and Liu1,the mod-eling of COVID-19 is mainly based on three categories of methods:ordinary differentialequation(ODE),deep learning(DL),and regression.A well-known and exceptionallypopular ODE based model in epidemiology monitoring is the compartmental model thatdepicts the transition of individuals between several stages of conditions.The susceptible-infected-recovered(SIR)compartmental model is the basic model for epidemiological dy-namic systems,which could be easily applied for the prediction of the transmission ofCOVID-19,see 2 and 3.From the perspective of statistics,the SIR model can be inter-preted as a three-state Markov chain model.The dynamic behavior of the compartmentalmodel relies on solving the system of ordinary differential equations which interpret theanalytic trajectory of the infectious disease.Compared with SIR,susceptible-exposed-300Chinese Journal of Applied Probability and StatisticsVol.40infected-recovered(SEIR)compartmental model,which is an extended model of SIR withan additional compartment E,especially contributes to the flexibility of the infectiousperiod.For the study of COVID-19,the additional compartment of E is indispensable,considering high transmissibility of COVID with non-negligible incubation period.Thus,the SEIR model reveals the preponderance in the COVID-19 modeling literatures.A-mong studies that focused on the first wave of COVID-19 epidemic,Zhao et al.4modeledthe epidemic trends of COVID-19 at the early stage and estimated the transmission rateof COVID-19,R0,based on the data of Wuhan,China from 10 January to 24 January2020.Tang et al.5proposed a deterministic SEIR compartmental model for COVID-19spreading.Wu et al.6employed a typical SEIR compartmental model to infer the num-ber of infected cases in Wuhan from the data on the number of cases that internationallyexported from Wuhan.The regression-based model is commonly used in disease mapping to explore asso-ciates between relative risk and available covariates,which allows potential temporal andspatial autocorrelation after accounting for covariates.Although it is possible to incor-porate various covariates and spatio-temporal random effects into the ODE based model,by setting the transmission rate from S to I to be a function of available covariatesand random effects7,the inference is somehow tricky and intractable.Besides,the basicSIR model could hardly handle complex epidemic changes after the initial outbreak sinceit remains the illustration of a symmetric exponential growth and fails to capture noise.Even if the simulation learning can achieve approximation inference for this problem,itremains challenging to incorporate the spatio-temporal interaction effects into the ODE-based model.Thus,compared to ODE based model,the regression based model is moresuitable for conducting the spatio-temporal analysis for various COVID-19 scenarios.Inthe regression based model,the area-specific relative risk is an important quantity thatconnecting the interested COVID-19 cases with various covariates and spatio-temporalrandom effects8.The estimated relative risk can help us examine the spatial distribu-tion of COVID-19 in that region,predict the trend of temporal and spatial variabilityof COVID-19,and identify the key covariates affecting the spread of COVID-199.Theregression based model has been successfully utilized to model the trend of COVID on var-ious covariates and different temporal and spatial random effects.For example,a Poissonhierarchical Bayesian model was proposed by Briz-Red on and Serrano-Aroca10,to ex-plore the influence of various meteorological(e.g.temperatures)and non-meteorologicalfactors(e.g.population density,demographic structure,number of travellers and com-panies)on the cumulative number of COVID-19 cases in the Spanish provinces.TheyNo.2ZHOU S.R.,et al.:Spatio-Temporal Inference of COVID-19 Cases in Shanghai301found that meteorological factors had less effect on the increase in the cumulative numberof COVID-19 cases in Spanish provinces.To provide guidance for medical interventionsand allocation of scarce resources for COVID-19 in Africa,Gayawan et al.11developeda zero-inflated Poisson model to predict the trend of COVID-19 in Africa.They foundthat neighboring countries typically have similar trends in incidence,with higher inci-dence rates in West African countries.The distribution of the daily COVID-19 cases is ofheavily right-skewed shape in Spain,Italy and Germany and spatio-temporal analysis wasconducted by Jalilian and Mateu 12,through a generalized Poisson hierarchical model.Considering the effect of population movement on COVID-19,Niraula et al.13proposeda new spatial weight to calculate the trend of population change over time in different riskareas based on the movement data of the moving risk.Both ODE and regression based models predict trends of interesting COVID-19 quan-tities through specific parametric assumptions,however DL based methods are able tolearn directly from historical data and make a prediction.As discussed by Hu et al.14,DL can be seen as a non-parametric approach that can flexibly carve out very complexand irregular data via extensive learning.The COVID-19 data,such as the daily numberof new symptomatic infections,the cumulative number of deaths,and the daily numberof recoveries,collected in a particular region can be viewed as a time series.Time seriesdata can be learned and predicted based on recurrent neural networks(RNNs)that is oneof the popular DL methods.RNNs are a class of neural networks with short-term memorycapabilities.A neuron in a RNN can receive information not only from other neuronsbut also from themselves,which can be represented in a network and finally results ina network of structure with loops.However,if the input sequence is long,this will leadto an issue of gradient explosion and disappearance known as the long-range dependenceproblem.This problem can be well solved by long-short term memory(LSTM)networks.The LSTM network is still an RNN essentially,but with a change in internal structure.The LSTM controls the transmission system by gating the state,remembering what needsto be remembered in the long term and forgetting the unimportant information.Sincethe COVID-19 outbreak,a number of scholars have used LSTM based networks to predicttrends over time through various COVID-19 indicators of interest.For example,a Deep-LSTM ensemble framework given by Shastri et al.15,can accurately forecast COVID-19confirmed cases and death cases in India.A similar LSTM based framework was proposedby Chimmula and Zhang16to forecast COVID-19 cases of Canada,Italy and USA,andcompared transmission rates in these countries.A rescent study by Krba s et al.17mod-eled the confirmed COVID-19 cases of several European countries by LSTM,RNN,and302Chinese Journal of Applied Probability and StatisticsVol.40the traditional autoregressive integrated moving average(ARIMA),respectively.Theyfound that LSTM approach performs best compared with other two models.Althoughthe LSTM based approach can capture COVID-19 complex trends,it is challenging toaccount for various temporal or spatial heterogeneities.The LSTM network implicitlyforecasts trend of COVID-19 across different areas,but the black-box nature of the LSTMmakes it lack of interpretation for correlation between inputs and outputs.The main objective of this study is to model COVID-19 pandemic in Shanghai in theSpring 2022 based on a novel Bayesian deep learning approach.Particularly,we aim todepict and forecast the on-going epidemic trajectory precisely by incorporating covariatesand spatio-temporal heterogeneity.The model includes two stages:firstly,we forecastthe daily new symptomatic infections of each district in Shanghai with a LSTM approach;secondly,uncertainty quantification is conducted by plugging the LSTM based predictionsinto a generalized Poisson regression model under the hierarchical Bayesian framework,which is called LSTM nested generalized Poisson regression(LNGPR)model.In theproposed model,the relative risk is utilized to connect LSTM predictions with the availablecovariates and various spatio-temporal random effects via an additive structure.As theproposed model follows the structure of latent Gaussian model,it can be evaluated fastlyand accurately via the integral nested Laplace approach(INLA)method.Model selectionis also conducted through comparing some criteria of different models with temporal,spatial or spatio-temporal interaction effect.The final optimal model not only improvesthe forecasting accuracy,but also provides credible interval for the predictions via Bayesianapproach.From the perspective of statistics,the prediction obtained is prominent forpolicymakers to assess relative risk and conduct effective mitigation at appropriate spatialscales.With the proposed Bayesian deep learning framework,we come to some helpfulconclusions.Firstly,people aged 1559 and 65+tend to accelerate the relative risk,probably because the people aged between 15 and 59 are of greater demand for commuting,leading to higher chance of infection,while people aged 65+have more underlying diseases,thus they are less resistant.Secondly,the adolescents,under the age of 15,the effect onrelative risk is not significant,which may be due to the fact that this group has betterprotective measures.Finally,air quality and migrant population have some effect onrelative risk,but the effect is weak.The structure of this paper is organized as follows.Section 1 offers a brief descriptionof the background.Section 2 lists multiple types of data applied for our study.Section 3elaborates the detailed methodologies of LNGPR.Section 4 demonstrates results for thedata analysis and prediction with geographical visualizations.Conclusions and discussionsNo.2ZHOU S.R.,et al.:Spatio-Temporal Inference of COVID-19 Cases in Shanghai303are given in Section 5.2Data2.1Study AreaShanghai is located in the east of China,between 1205212212east longitude and30403153north latitude.By the en