Dynamic COVID risk assessment accounting for community virus exposure from a spatial-temporal transmission model

1 Introduction . The coronavirus disease 2019 ( COVID-19 ) has created several surges of pandemic globally since early 2020 and continues to be a major public health threat . COVID-19 related hospitalizations and 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . deaths have caused immense burdens to the health systems [ 20 , 21 ] . Therefore , it is crucial to study the community-level risk factors affecting the transmission of the disease and the individual-level risk factors of severe COVID-19 outcomes . We propose a spatial-temporal model for COVID-19 transmissions and a weighted semiparametric time-varying coefficient model for hospitalizations that can integrate multiple levels of information and perform causal inference . Various models have been proposed to forecast the trend of COVID-19 transmissions [ 32 , 4 , 18 , 27 , 14 ] based on susceptible-exposed-infectious-recovered ( SEIR ) model , Gaussian process , or agentbased modeling . Their main goals are on the prediction or forecast accuracy . On the other hand , socioeconomic disparities and spatial variations have been observed in COVID-19 transmissions and individual outcomes [ 12 , 34 ] . Spatial correlations have been detected for infectious disease transmissions where the disease incidence tends to occur in spatial clusters especially during the initial stages of an outbreak [ 25 , 5 ] . However , some of these previous forecast models analyze the individual-area temporal trends separately , and do not account for socioeconomic disparity effects . Our first goal is to propose a joint model for the temporal and spatial patterns across all areas , which allows explaining what accounts for the spatial variability and the spread of infectious disease , and would be more efficient than modeling two trends and for different areas separately . Specifically , the temporal dependence is modeled by taking a convolution of past infection numbers during a transmissible time interval following the framework in [ 27 ] . The latent pre-symptomatic disease transmission phase is captured . To allow the spatial dependence , we construct a spatial conditional autoregressive ( CAR ) model under the Gaussian process to account for the correlation in the infection rates among neighborhood areas and improve the estimation for areas with fewer cases . Area-specific socioeconomic factors are included to account for the heterogeneity in infection rates across regions . For efficient estimation , the proposed temporal and spatial dependence structures are united under a single objective function linked by the infection rates . By examining the spatial disparities and community-level risk factors , our model can facilitate policy decisions and better allocation of healthcare resources to control the disease transmission . To assess an individual ’ s risk of severe COVID outcomes after diagnosis , risk factors such as demographics and pre-existing medical conditions haven been identified to partially account for the heterogeneity in patients ’ risks [ 22 , 29 , 16 , 33 ] . These existing risk assessment models are solely based on individual-level data . However , community-level risk factors such as neighborhood poverty have been shown to be associated with hospitalization rate [ 26 ] . Another important community risk factor is the background risk of virus exposure in the neighborhood measured as the total number of diagnosed and pre-symptomatic infected individuals . This latent virus exposure can be estimated from the proposed spatial-temporal disease transmission model . To fully capture individual risks , we need to consider all these community-level factors . Furthermore , there are multiple challenges to construct a valid risk assessment model . Firstly , sample selection bias from multiple sources can be present . For example , it is shown that several neighborhoods in New York City ( NYC ) with high case rates also had high testing rates [ 26 ] . As a result , residents from areas with high case rate are more likely to be tested positive and thus more likely to be included in the study sample consisting of only diagnosed patients . A serious challenge is that such selection bias may induce spurious association between risk factor and COVID outcomes and thus misleads interpretation and decision [ 15 ] . Secondly , considering the dynamic nature of the pandemic , the effect of the risk factors may not be the same over time . For instance , the number of infected subjects varies substantially throughout the outbreak [ 26 ] and the drivers of the variation ( e.g. , racial disparity ) may be different early on in the pandemic versus a later period . Hospital capacity is another important time-varying factor that influences a patient ’ s chance of being admitted , especially during the initial outbreak [ 11 ] . However , most existing studies do not consider the selection bias inherited in their study samples [ 15 ] and ignore the time-varying effects of the risk factors due to the evolving dynamics of pandemic . Lastly , most of existing risk assessment models lack rigorous calibration and validation procedures [ 33 ] . Our second goal is to assess individual risk of severe COVID-19 outcomes accounting for community virus exposure burden and selection bias by a weighted spatial-temporal model . Specifically , we integrate individual-level risk factor data and neighborhood environmental risk factors including the latent time-varying neighborhood virus exposure to better account for patients ’ heterogeneous risks . Two sets of subject-specific weights are constructed to mitigate the selection biases so that after weighting the study sample could reflect the target population of interest and are not overly selected from neighborhoods with high infection rates . Furthermore , we build semiparametric models with time-varying coefficients to allow time-dependent effects of hospital capacity and community factors ( e.g. , number of infectious COVID-19 patients predicted from the spatial-temporal transmission model ) . The method is applied to model COVID-19 transmission in NYC and assess individual risks using electronic health records ( EHRs ) of PCR confirmed COVID-19 patients diagnosed at New York Presbyterian Hospital ( NYPH ) . Validation and calibration are conducted to evaluate the proposed method . Our model assists medical decision making and clinical management by integrating risk factors at multiple levels to test which factors significantly influence individual ’ s severe health outcomes after COVID-19 infection . 2 Method . 2.1 Community-level transmission model accounting for spatial and temporal correlation . Model structure We propose a generative model that accounts for both temporal and spatial correlation of disease transmission while allowing heterogeneous infection rates across different areas accounted for by area-specific characteristics ( e.g. , distribution of minority population and social distancing measures in an area ) . We follow the framework of the survival convolution model proposed in [ 27 ] to account for the disease transmission during a pre-symptomatic phase . We let ai ( t ) denote the effective infection rate for the ith area ( i = 1 , · · · , n ) on day t defined as ai ( t ) = Ni ( t ) Mi ( t ) , ( 1 ) whereMi ( t ) is the number of infected subjects who remain in the transmission chain and can transmit virus to others ( including those who are pre-symptomatic or asymptomatic ) for the ith area on day t , and Ni ( t ) is the number of newly infected subjects for the ith area on day t. Note that Mi ( t ) and Ni ( t ) are both latent processes , and they are more accurate measures of the underlying virus exposure than the daily reported number of diagnosed cases . Let S ( k ) denote the proportion of individuals remaining infectious and who can transmit disease after k days of being infected , then Mi ( t ) and Ni ( t ) are related as Mi ( t ) = Ci∑ k=1 Ni ( t− k ) S ( k + 1 ) , ( 2 ) where Ci = min ( t− ti0 , C̃ ) , ti0 is the unknown day when the first subject is infected in area i , and C̃ is the maximum incubation period ( i.e. , 14-21 days for COVID-19 [ 19 ] ) . We let ti0 be C̃ days prior to the first reported diagnosis of COVID-19 case . We assume that subjects under quarantine will not be in the transmission chain , and thus ai ( t ) reflects the effective transmission rate either due to quarantine or out of infectious period of SARS-COV-2 virus . It follows that the expected number of diagnosed subjects out of transmission chain in area i on day t , denoted as Yi ( t ) , can be calculated based on the latent processes as Yi ( t ) = Ci∑ k=1 Ni ( t− k ) [ S ( k ) − S ( k + 1 ) ] . ( 3 ) Note the number of diagnosed patients is the only quantity that is observe . To obtain other quantities , given ti0 and Ni ( ti0 ) = 1 , the expected number of newly infected cases Ni ( t ) and diagnosed cases Yi ( t ) can be updated sequentially based on ( 2 ) , ( 1 ) , and ( 3 ) if the infection rates ai ( t ) and the survival function S ( · ) are known . Here we model S ( · ) as the normalized survival function of the exponential distribution , i.e. , S ( k ) = ( e−k/δ − e−C̃/δ ) / ( 1− e−C̃/δ ) , and we set its mean δ = 5.2 following [ 19 ] . More importantly , we will model the time-varying infection rate ai ( t ) borrowing strength from the neighborhood areas and area-specific time-invariant and time-varying characteristics , e.g. , demographics , social vulnerability index , and mobility . Let λi ( t ) denote the relative rate of infection in region i compared to a “ baseline '' expected rate at day t − 1 averaged across regions denoted by ā ( t − 1 ) , i.e. , λi ( t ) = ai ( t ) /ā ( t − 1 ) , where ā ( t − 1 ) = 1n ∑n i=1 ai ( t− 1 ) , and λi ( 0 ) = 1 . We denote the baseline average expected number of infections as Ei ( t ) = Mi ( t ) ā ( t− 1 ) , and assume the actual number of new infections has a mean of Mi ( t ) ai ( t ) , or equivalently a mean of Ei ( t ) λi ( t ) . We model the logarithm of the relative infection rate , Zi ( t ) = log ( λi ( t ) ) , by a spatial conditional autoregressive ( CAR ) model to account for correlation between regions . Let Z ( t ) = ( Z1 ( t ) , · · · , Zn ( t ) ) T for time points t = 0 , · · · , T . Similar to the small area estimation problem [ 10 ] where the disease rates for areas with smaller at-risk populations are estimated less accurately , here when estimating ai ( t ) in ( 1 ) , areas with smaller infectious population Mi ( t ) are subject to more estimation variability . Therefore , to borrow strength from areas with more infectious subjects , we assume Z ( t ) follows a Gaussian process that exhibits spatial dependence as Z ( t ) ∼MVN ( XTt βt , Σt ) , ( 4 ) where Xt is a matrix of covariates ( e.g. , demographics , social vulnerability index , mobility , and include a column of constants of one ) . The covariance matrix Σt should reflect heterogeneous variances of estimated infection rates at different locations due to differential baseline number of infectious subjects Mi ( t ) . In addition , the infection rates estimated from smaller populations are more variable and should use more pooling than areas with larger populations . To accommodate these factors , following the spatial rate model used for disease mapping [ 9 , 10 ] and a spatial CAR model [ 10 ] , under the condition that I− ρtH is positive definite , we specify Σt = τ 2 t ( I− ρtH ) −1∆ , where ∆ = diag ( 1 E1 ( t ) , · · · , 1 En ( t ) ) . ( 5 ) Furthermore , H = ( hij ) has zeros as the diagonal terms , and the off-diagonal terms are specified as hij = [ Ej ( t ) /Ei ( t ) ] 1/2 if j ∈ G ( i ) , and 0 elsewhere . where G ( i ) indicates the set of areas that share borders with the ith area ( i.e. , the neighborhood of area i ) . In other words , we only borrow information from the neighborhood areas to improve the estimation , without pooling over irrelevant areas . With the covariance matrix specified in ( 5 ) , we account for the larger variability of small areas with lower expected infection numbers Ei ( t ) ( or equivalent Mi ( t ) ) . Under this parametrization , it can be shown that ρt represents the spatial partial correlation at time t between neighborhood counties ( i , j ) given other regions , i.e. , corr ( Zi ( t ) , Zj ( t ) |Zk ( t ) , k 6= i , j ) = ρt . This correlation is invariant to the neighborhood structure of counties ( i.e. , does not depend on H ) , which is desirable . We provide more explanation of the model structure in the Supplementary material . We will estimate a separate model at each time point t. We present an illustrate diagram for the model architecture in Supplementary Figure A.1 . Estimation For estimation , we will combine the loss function for the reported daily new cases Ri ( t ) to the expected number based on ( 3 ) and the likelihood of the log relative infection rates λi ( t ) as n∑ i=1 T∑ t=ti0 { √ Ri ( t ) − √ Yi ( t ) } 2 − T∑ t=t̃ logl ( Z ( t ) ; ξt ) + λ T∑ t=t̃+1 ||ξt − ξt−1||1 , ( 6 ) where l ( Zt ; ξt ) is the likelihood under the multivariate Gaussian distribution in ( 4 ) with ξt denoting the parameters of ( βt , ρt , τ 2 t ) , and ||.||1 is the L1 norm . Note in ( 6 ) , t̃ denotes the first day of diagnosed case across all areas . In other words , we align the areas according to the relative stage of disease transmission in modeling the spatial dependence . Here the first term in ( 6 ) is to evaluate the prediction performance from the latent disease transmission process where the square root transformation is a variance stabilization transformation for count data ; the second term is to ensure the spatial smoothness between the areas and account for the heterogeneous infection rates by areaspecific factors such as socioeconomics and mobility ; and the last term is the fused lasso penalty to ensure the smoothness of the estimated parameters over time . Note that the fused lasso penalty will consequently encourage smoothness in the daily infection rates over time , which reduces variability and is consistent with observed data . Furthermore , to avoid the high computational burden involved in maximizing the likelihood of the high-dimensional multivariate Gaussian process Zt , we consider an alternative approach based on optimizing conditional pseudo-likelihood often used to learn Gaussian graphical models . For given Xt and all Zjt ’ s with j 6= i , [ 6 ] and [ 24 ] showed that the multivariate Gaussian model ( 4 ) with covariance matrix of the form ( 5 ) implies the conditional normal distributions with Zi ( t ) |Zj ( t ) , j ∈ G ( i ) ∼ N ( θit , τ2it ) , θit = µit + ∑ j∈G ( i ) ρthij ( Zj ( t ) − µjt ) ) , ( 7 ) where µit = E ( Zi ( t ) ) = XTitβt and τ 2 it = τ 2 t /Ei ( t ) . Then the objective function ( 6 ) becomes n∑ i=1 T∑ t=ti0 { √ Ri ( t ) − √ Yi ( t ) } 2 + n∑ i=1 T∑ t=t̃ logτit + ( 2τ2it ) −1 [ Zi ( t ) − θit ] 2 +λ T∑ t=t̃+1 ||ξt− ξt−1||1 , so that the joint likelihood reduces to products of area-wise conditional likelihood and optimization is much easier . To obtain solution , we treat Zi ( t ) and ξt−1 as parameters and optimize the objective function by gradient descent implemented by PyTorch [ 23 ] on local and cluster CPUs . Under this model , we can estimate the time-varying between-area correlations of the infection rate and the effects from time-dependent community area-level covariates on the infection rates . With available parameters , the log-relative infection rates θit can be estimated from ( 7 ) and forecast can be provided based on assumptions of the future epidemic trend ( e.g. , assuming similar trend over next d weeks as previous d weeks ) .

Summary: This work proposes (a) a community-transmission model, and (b) an individual risk-assessment model for COVID-19. The model is general enough to be applicable to any infectious disease. The model for (b) is enriched by inferred variables from (a) by (i) using them as covariates and (ii) addressing the selection bias in the data used for (b).   ### community-transmission model  First, this work links the temporal model for forecasting the number of infectious cases with a spatial model to account for disease propagation between regions.  The connection is made via modelling the relative infection rate as a Gaussian Process parameterized by a covariance matrix that depends on the expected number of infectious cases in the neighbouring regions.  The simulation results in the Appendix demonstrate that the model is able to uncover the true underlying parameters.  This spatiotemporal model is then fitted to the reported number of daily COVID-19 cases in NYC from March to July, and the results show a substantial spatial correlation in disease propagation.   ### individual risk-assessment model  Finally, an individual risk-assessment model is constructed using the inferred variables from the above fitted spatiotemporal model. The data used to build this model is from two hospitals in New York.  The authors addressed the issue of selection bias in the data via subject-specific weights by making the sample representative of true underlying prevalence and spatial population. The results seem to suggest a superior fit to the data compared to the unweighted model.