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. Author manuscript; available in PMC: 2016 Jul 25.
Published in final edited form as: Stat Med. 2015 Nov 3;35(11):1848–1865. doi: 10.1002/sim.6785

Bayesian Penalized Spline Models for the Analysis of Spatio-Temporal Count Data

Cici Bauer a,*, Jon Wakefield b, Håvard Rue c, Steve Self d, Zijian Feng e, Yu Wang e
PMCID: PMC4959802  NIHMSID: NIHMS802612  PMID: 26530705

Abstract

In recent years, the availability of infectious disease counts in time and space has increased, and consequently there has been renewed interest in model formulation for such data. In this paper, we describe a model that was motivated by the need to analyze hand, foot and mouth disease (HFMD) surveillance data in China. The data are aggregated by geographical areas and by week, with the aims of the analysis being to gain insight into the space-time dynamics and to make short-term prediction to implement public health campaigns in those areas with a large predicted disease burden. The model we develop decomposes disease risk into marginal spatial and temporal components, and a space-time interaction piece. The latter is the crucial element, and we use a tensor product spline model with a Markov random field prior on the coefficients of the basis functions. The model can be formulated as a Gaussian Markov random field and so fast computation can be carried out using the integrated nested Laplace approximation (INLA) approach. A simulation study shows that the model can pick up complex space-time structure and our analysis of HFMD data in the central north region of China provides new insights into the dynamics of the disease.

Keywords: Bayesian spatio-temporal analysis, Gaussian Markov random field, INLA, infectious diseases, penalized splines, surveillance count data

1. Introduction

In recent years, the modeling of infectious disease data in space and time has gained increasing popularity. The aims of such modeling are many fold but gaining insights into space-time dynamics, which may give etiological insights, is a common objective. Modeling may also allow one to predict the potential effects of health interventions such as the introduction of vaccination. Many examples of such infectious disease data arise from government disease surveillance systems, and in many situations are in the form of spatially and temporally aggregated disease counts. In this paper, the motivating example was hand, foot and mouth disease (HFMD) surveillance data collected and monitored by the Chinese Center for Disease Control and Prevention. The dataset comprises weekly number of HFMD cases in 341 prefectures between 2009 and 2010. In this paper we develop a Bayesian space-time modeling framework for infectious disease count data, and achieve significant computational savings using the integrated nested Laplace approximation (INLA) technique. The goals of such modeling are to utilize the surveillance data to understand the space-time dynamics of the infectious disease outbreaks, and to provide predictions of future disease counts to aid in policy formation.

The literature on space-time models is ample, with many proposals based on disease mapping models. For example, in (1) and (2) the risk evolution over time is assumed to take a parametric form. A different approach is taken by (3) and (4), where the risks for every time period are modeled as time-independent spatial random effects with a time-specific precision parameter for the spatial variability. In the context of modeling multiple diseases, (5) proposes a temporally independent multivariate conditional autoregressive (MCAR) model for lung cancer mortality, with data on both males and females being studied. In (6), the authors propose a shared compartment model where the underlying risk for each disease is separated into a shared and a disease-specific component. Many of these space-time models were developed for non-infectious diseases such as cancers.

Disease mapping models developed for non-infectious diseases have been used, often for descriptive purposes, for infectious disease data, see for example, (7) for flu and (8) for meningococcal disease. In a series of papers (9; 10; 11; 12) a framework for the analysis of infectious disease surveillance counts has been developed. In its simplest form, the models can be viewed as Poisson branching process models with immigration, with mean risk being decomposed into an endemic component and an epidemic component. These models have been applied to a variety of diseases and have also used vaccination coverage as a covariate (13; 14). (15) modeled weekly counts of HFMD in prefectures of China using a similar negative binomial model with epidemic and endemic components. In a different approach, time series susceptible-infected-recovered (SIR) models have been developed (16; 17). The fitting of SIR models (and their variants) is challenging unless the populations under consideration are small, however, since the number of individuals in each of the compartments is imputed under the above Bayesian approach to inference (18). A variety of alternative approaches have been suggested for situations in which populations are not small, but all are computationally expensive (19; 20). Approximate Bayesian Computation (ABC) methods have been proposed as an alternative (21; 22).

In recent years, spline models have become increasingly popular in the space-time disease mapping setting, see for example (23). However, such models have mostly been applied to provide a flexible way of modeling the temporal trend, see for example (24). A notable exception is (25) who describes a broader class of regression spline models with spatial-temporal relative risks being modeled as spatially varying or randomly varying regression B-splines. (26) and (27) both consider space-time splines models, though do not include Markov random field (MRF) prior penalty terms, which is the cornerstone of the approach we describe in this paper. (28) compares a traditional space-time conditional autoregressive (CAR) model and a CAR-Pspline model in the context of small area predictions, with inference made using penalized quasi-likelihood (PQL).

In this paper, the approach we take to modeling the space-time spread of infectious disease data is to decompose the relative risk of disease into three components: a large-scale temporal trend, a large-scale spatial trend and a space-time interaction. The latter is the crucial element, and we use a tensor product B-spline model with a Markov random field (MRF) prior on the coefficients of the basis function. Such a model formulation allows the computation to be carried out using the integrated nested Laplace approximation (INLA) technique, resulting in significant computational savings when compared to a classical MCMC approach. Our approach therefore has the advantage of fast computation and easy implementation using R and existing packages.

The outline of this paper is as follows. We begin with a motivating example that concerns the China hand-foot-mouth (HFMD) surveillance data in Section 2. In Section 3 we describe the Bayesian spatial-temporal models that we propose. In Section 4, we demonstrate the performance of our proposed models via a simulation study. The simulated data is designed to mimic the epidemic center movements in space and time typically observed in infectious disease surveillance data. We return to the HFMD data in Section 5 and describe the results after applying the models to the dataset. We conclude the paper with a discussion in Section 6. The supplementary materials contain more technical details and additional supporting information.

2. Motivating Example

In this section, we provide details of the China HFMD surveillance data analyzed in this paper. HFMD is an acute contagious viral infection that has caused large-scale outbreaks in Asia during the past decade (29). It is caused by enterovirus pathogens and usually involves mild or moderate symptoms such as fever, oral ulcer or rashes on the hand and foot. HFMD is a disease often seen in children and in a small fraction of cases there is severe illness with neurological problems, and even death. Little is known about the etiology of the enterovirus, the factors associated with its spread, or an effective means of public health intervention. Therefore understanding the dynamics of HFMD patterns of spread can greatly benefit authorities charged with policy making to control this infectious disease.

Enterovirus-related HFMD, with the first large-scale epidemic outbreak in 2008 in China, has been included as one of the 39 notifiable infectious diseases in the Chinese Center for Disease Control and Prevention (CCDC) disease surveillance system. Each reported case from the CCDC surveillance system includes information on the person’s current home address, gender, age and the symptom onset date. Therefore, the disease surveillance system from China provides an extensive data resource for space-time modeling. More information about these data can be found in (15).

In this paper, we analyze data from the central north region of China from 2009 and 2010; this region is shown in relation to the whole of China in the supplementary materials. The central north region consists of 59 prefectures spread in five provinces and one direct-controlled municipality (i.e., Tianjin, Hebei, Henan, Shandong, Shanxi and Beijing). The total population in the region is estimated to be 318,022,505 in 2009/2010. Within the region, 418,949 and 478,238 HFMD cases were reported in 2009 and 2010, respectively. Here we aggregate the number of HFMD cases by week and by prefecture, see Figure 1(a). The temporal trend is clear: the epidemic starts around March, reaches its peak in May/June and gradually dies down towards the winter. This is the same pattern observed in both years, though the time that the epidemic reaches its peak seems to be later in 2010 than in 2009.

Figure 1.

Figure 1

Summaries of the central north region of China HFMD data from 2009–2010: (a) weekly numbers of cases, (b) weekly expected numbers, (c) marginal (across time) SMR, (d) centroids of prefectures (red dots) and location of spline bases (blue crosses), after scaling the study region.

Based on the population composition in each prefecture, we calculate weekly expected numbers of counts adjusting for age (with age bands 0–0.9, 1–2.9, 3–5.9, 6–9.9 and ≥10 years) and gender, using internal standardization (30). These expected numbers are shown in Figure 1(b), with a wide range from 18 to 364.We then calculate the marginal standardized morbidity ratios (SMRs) across time. These SMRs are the ratios of the total counts to the expected counts (adjusted for the confounders age and gender) over 2009–2010. A thematic map of the resulting SMRs is presented in Figure 1(c). Though subject to sampling variability, the spatial variation in the SMR is clearly seen in the map (an animation of the weekly log SMRs can be found at http://www.stat.brown.edu/cbauer/). However, the temporal pattern in the SMRs is unclear apart from a general south-to-north movement of the area with high SMRs. These initial examinations clearly suggest a model with spatial and temporal components is needed for analyzing the China HFMD data.

3. Methods

In this section, we describe the Bayesian spatial-temporal models that we propose. We start by describing the overall model framework, followed by details on each of the model components.

3.1. The Poisson Likelihood

Let Yit denote the observed disease counts and Eit the expected counts, which account for the known confounders such as age and gender, in area i and in week t, i = 1, …, I; t = 1, …, T. In the HFMD example, the population is treated as constant at each time point so that Eit = Ei. The constant population is based on the assumption that the changes in population size over time are relatively small and so can be ignored. The expected count Ei is calculated using internal standardization, which takes into account of the differences in age-gender populations between areas.

The data model is a Poisson distribution with relative risk μit

Yit|μit~Poisson(Eiμit). (1)

Our overall strategy is to decompose the relative risk into the following components

log(μit)=α+zitTβ+fS(xi)+fT(t)+fST(xi,t) (2)

where α is the intercept, zit is a vector of area-level covariates, and xi = (x1i, x2i) is the centroid of area i. In this formulation, fS(xi) is the spatial trend, fT(t) is the temporal trend and fST(xi, t) is the spatial-temporal interaction. In a Bayesian framework, the spatial and temporal dependency is incorporated into the model through the prior. We now describe the priors used for the spatial trend, the temporal trend, and the spatial-temporal interaction in (2) respectively.

3.1.1. Temporal trend

The temporal component is modeled as the sum of a large-scale component, and noise terms with and without temporal structure

fT(t)=xtTδ+εt+γt.

The large-scale component xtTδ can be modeled using parametric or semi-parametric models, which we have assumed can be represented via a linear basis. A popular example of the latter are spline models, see (24) and (25). As an alternative, seasonal variability in the data can be modeled via frequency domain models. For example, we can include a set of harmonic functions

xtTδ=s=1S[as sin(ωst)+bs cos(ωst)],s=1,,T/2 (3)

where S is the number of harmonics, δ = {as, bs, s = 1, …, S} and ωs are the Fourier frequencies with ωs = 2πs/52 for weekly data. The row vector xt is of length 2S with the sth pair of entries consisting of [sin(ωst), cos(ωst)]. The unstructured temporal term may be modeled as εt~iidN(0,τε1) while γt, which picks up short-term temporal structure, may be given a conditional autoregressive prior. In this paper we use a second-order random walk prior denoted as RW2. This model is a local stochastic smoothing model and may be represented as

p(γ|τγ)exp [τγ2γTATγ],

where the T × T matrix AT is of rank T − 2 and has the form

AT=[1212541146411464114641146411452121], (4)

with the blank spaces corresponding to zeros. We use τ through out the paper for the precision parameter, which is the inverse of the variance. The subscript of τ indicates the random variable with which the precision parameter is associated. Further technical details on this model, including the conditional mean and variance, are provided in the supplementary materials.

3.1.2. Spatial trend

The spatial trend is modeled as

fS(xi)=β1x1i+β2x2i+υi+ui. (5)

The covariates x1i and x2i represent the latitude and longitude of the centroid of area i and β1 and β2 are the associated coefficients. The unstructured spatial terms υi is assigned a N(0,τυ1) prior, and the structured spatial term ui is assigned an intrinsic conditional autoregressive (ICAR) prior. Hence, we are using the classic (31) convolution model. The joint distribution of the structured spatial term ui is

π(u|τu)exp [12uTASu], (6)

where AS has entries

ASij={mii=j,1i~j,0otherwise, (7)

with i ~ j indicating that areas i and j are neighbors defined as areas sharing the same boundary, and mi the number of neighbors for area i. Further details of the ICAR model are relegated to the supplementary materials.

3.1.3. Spatial-temporal interaction

Models with only spatial and temporal trends are usually insufficient for modeling the variation in temporal dynamics between regions, particularly for infectious diseases. Hence, the spatial-temporal interaction term we define is crucial, as this term captures the space-time dynamics of the infectious disease spread. In a disease mapping context, for example, (32) and (33), it has been suggested that a space-time interaction term could be constructed by combining the spatial random effects u and the temporal random effects γ. However, such a form contains a large number of random effects, IT to be exact, which is not a parsimonious representation. In addition, the interaction term adds great complexity to the model and makes the implementation for Bayesian inference, usually via Markov chain Monte Carlo (MCMC), very difficult for data with many areas and/or time points.

Motivated by (33), we propose an alternative approach to constructing the interaction term. We model the space-time interaction surface with a bivariate spline, and place a GMRF prior on the coefficients associated with the basis functions. Specifically, we assume

fST(xi,t)=k=1KbktBik,k=1,,K, (8)

where Bik is a tensor product of two univariate cubic B-spline functions, one in each spatial dimension, so that Bik is a function of xi. For notational convenience we index the bivariate basis functions by k. The number of basis functions K will typically be far less than the number of areas I, and so the number of latent random effects is reduced. Because these basis functions are constructed as a tensor product, they can be visualized as being laid out on a rectangular grid. Figure 2(b) presents such visualization using a simulated data set, for a total of 36 basis functions (6 for each spatial coordinate). The advantages of choosing a cubic B-spline include the computation efficiency and the relative insensitivity to the choice of basis numbers and locations (34; 35). More details of the tensor product basis functions can be found in the supplementary materials.

Figure 2.

Figure 2

Simulated Data: (a) true values of the basis functions bkt versus time; (b) locations of the basis coefficients, color coded as in (a); the grey dots are the locations of the observed counts; (c) number of cases aggregated over time (with the size of the circles being proportional to the number of cases); (d) number of cases, aggregated over space, plotted by week.

The spatial-temporal structure of the interaction is imposed through the prior on the basis coefficients and can take one of the following four types, as with the disease mapping approach of (33)

Type I: A product of independent normal priors so that there is no spatial or temporal structure in the spline coefficients

π(b|τb)exp [τb2k=1Kt=1Tbkt2].

This type of interaction assumes that, after accounting for spatial and temporal main effects, the residuals do not have structure in space and time. This model is the easiest to implement but the least appealing in a biological sense.

Type II: A second-order random walk (RW2) prior on the temporal structure with no spatial structure, so that we have a product of K temporal smoothing priors

π(b|τb)exp [τb2k=1Kt=3T(bkt2bk,t1+bk,t2)2].

Hence, the interactions have temporal structure but the bases are independent across space.

Type III: An ICAR prior in space but with no temporal structure, so that we have a product of T independent spatial models

π(b|τb)exp [τb2t=1Tk~k(bktbkt)2].

The notation k ~ k′ indicates that bases k and k′ are first order neighbors, hence each interior basis has four neighbors. This model assumes that at time t, the bases that are close to each other are more likely to be similar, but across time the T sets are independent.

Type IV: A Kronecker product of RW2 and ICAR priors

π(b|τb)exp [τb2t=3Tk~k{(bkt2bk,t1+bk,t2)(bkt2bk,t1+bk,t2)}2].

This model assumes that the basis coefficient bkt at location xk and at time t depends on those from the two previous time periods bk,t−1 and bk,t−2, as well as those from the neighboring basis xk, k′ ~ k. In addition, bkt depends on those coefficients in the neighboring bases in the two previous time points, i.e. bk′,t−1 and bk′,t−2, for k′ ~ k. Therefore estimation of the coefficients borrows information in both space and time.

In addition to reducing the dimension of random effects, our proposed method has the advantage that, even with complicated structures, after writing (b11, …, bK1, b12, …, bK2, …, b1T, …, bKT)′ as b, all four types of interactions have a unified representation as GMRF models with the general form

π(b|τb)exp [τb2bTASTb].

The structure matrix AST of the spline coefficients is then the Kronecker product of the structure matrix for the space AS and the temporal structure matrix AT, i.e. AST = ASAT. The supplementary materials give details on the form of AST for each interaction model. Such a representation also brings a flexibility to the modeling, as one can select different spatial and temporal structures, depending on the application.

One other advantage of our proposed models is in the computation. The GMRFs presentation of the spatial-temporal dynamics makes the computation far more convenient. In particular, this model can be fitted using the fast and flexible integrated nested Laplace approximation (INLA) technique (36). INLA has been demonstrated to provide accurate inference in a number of applications (36; 37; 38; 39; 40) and, compared to the more traditional MCMC method, is orders of magnitude faster for the model described above. In the supplementary materials we provide the WinBUGS code to implement our proposed models using MCMC, as well as the code that uses the R package INLA. We also show in the supplementary materials a numerical comparison between MCMC and INLA.

4. Simulation Study

4.1. Data Generation

In this section we describe a simulation study to investigate the performance of our proposed models. The key novelty of our proposed models lies upon the spatial-temporal interaction so we focus on assessing this aspect and fit the Poisson model (1), with the relative risk

log(μit)=α+fST(xi,t), (9)

where fST is described by (8).

The study region is a unit square and we select two equal-distance inner knots in each spatial dimension to construct the tensor product cubic B-splines bases, so that K = 62 = 36.We assume T = 24 discrete observation times. The simulation is constructed to mimic the movements of epidemic centers over time. We first obtain fixed values of the spline basis coefficients bkt, using gamma functions and one sine function of time t, and then use these bkt to construct the underlying risk surface. The functions for obtaining the values of bkt are treated as unknown at the analysis. Figure 2(a) plots these coefficients versus time with the curves being color coded by their locations, which are shown in panel (b). The values of the coefficients are chosen so that higher values are given to those in the bottom left corner at early weeks of the observation period, those in the top left corner at middle weeks, and those in the top right corner at later weeks. We choose the expected number of cases Ei to be between 5 and 50 and assume they are constant over the study period. The observed counts Yit are then simulated at I = 150 randomly selected locations, which are indicated by the points in Figure 2(b). These point locations are used to represent areal centroids. The MRF prior is with respect to the bases, and so for the interaction term this is the only place that requires specification of a neighborhood structure. The simulated weekly counts Yit range from 0 to 1063. In Figures 2(c) and (d) the number of cases over space and time are displayed. The aggregated counts in panel (c) show large spatial variation, ranging from 2148 to 9541. The weekly observed counts aggregated over space in panel (d) show two peaks, which is similar to that observed in the China HFMD data. The supplementary materials include more details on the simulation, such as the map of the realized Ei, and the bubble plots of the simulated number of cases at selected time points which give more details on how epidemic centers move over time.

4.2. Results of the simulation study

We fit the spatial-temporal interaction model (9) with the four interaction priors described in Section 3.1.3. A Gamma(1, 0.98) distribution is used as the prior for the precision parameter τb. Results of the parameter estimates are given in the supplementary materials, along with a sensitivity analysis with different priors for the precision parameter. Time series plots of the estimated basis coefficients are shown in Figure 3, with one for each interaction type. The true values of the coefficients are superimposed in the graph using the color scheme of Figure 2, and the grey lines denote the posterior median estimates. In the Type I interaction model where the coefficients have no spatial or temporal structure, the grey lines are very jagged and substantially deviate from the truth. In the Type II model where a RW2 model is used for the temporal structure in the interaction, the estimated coefficients are much smoother over time and closer to the truth. In the Type III model, the estimates are similar to those observed in the Type I model. Because the majority of the variation in the simulated data is temporal, smoothing across space alone cannot improve the estimation greatly. Finally, the Type IV model improves on the Type III fit and provides the estimates that are the closest to the true values. We now assess the model fits more formally.

Figure 3.

Figure 3

Estimated basis coefficients with four different priors in the simulation study: true values are colored as in Figure 2, while the grey lines are the estimates.

The overall measure of the model fit may be assessed using the conditional predictive ordinate (CPO) (41; 42) which represents a particular form of cross-validation measure. The CPO of observation it is

CPOit=Pr(yit|yit,θ)π(θ|yit)dθ

where yit represent the data with observation yit removed. We may calculate i=1It=1Tlog CPOit and models with larger values are preferred. (43) discuss how CPO values may be calculated efficiently with INLA, without the need to fit the model I × T times. For the simulated data, the sum of the log CPO values under the four types of interactions (scaled by the maximum, which is the Type II model) are: −429, 0, −106 and −20. The CPO values from the Type II and IV models are considerably higher than those from the Type I and III models, so that the former are preferred to the latter, which is not surprising given the strong temporal structure in the data.

Examining the change of the spline coefficients provides some knowledge of the temporal trend in the risk. However, we would also like to obtain the estimated log relative risks at the observed locations. We select four areas with different expected numbers (E = 5, 15, 30, 50) and examine the model fits. In Figure 4(a) we give the true log relative risk for these four areas while in panel (b), we plot the log SMRs. It is clear that areas with lower expected numbers have larger variability in the log SMRs. In panels (c)–(f) we show the fits from the four types of interaction models. In these plots the estimated log relative risks using our models are clearly more accurate than the log SMRs. Informally comparing the four types of interaction models using these figures, the fits from the Type II and IV interaction models are closest to the truth.

Figure 4.

Figure 4

Comparison of the log relative risk in four selected areas.

Finally we examine the prediction performance of our proposed models. We take the data until time T and predict the number of cases within q week ahead, i.e., yit for t = T + 1, …, T* + q. The prediction performance is evaluated using the mean squared error of prediction (MSEP) given by

MSEP=1IqT+1T*+qi=1I(ŷityit)2. (10)

The results for q = 1 and q = 2, i.e. predicting 1 and 2 time periods ahead, are given in Table 1. We choose T* = 6 to give a minimum that has sufficient observed data for prediction. For each T and q, the MSEP is reported for the Type I–IV models, with the smallest MSEP indicated in bold. Type II model predominately outperform the other types by giving the smaller MSEP, with 15 out of 18 weeks for q = 1 and 11 out of 17 weeks for q = 2. The Type IV model gives the smallest MSEP in all other weeks for q = 1 and three of the remaining weeks for q = 2. The predictions become worse for all models in weeks 19–22, which is the period in which the second epidemic peak occurs. This is somewhat unsurprising: without additional information of the disease etiology or drivers of the outbreaks, prediction of the time when an epidemic “kicks off” is generally very difficult in infectious diseases, particularly based on the surveillance data alone. Nevertheless, when the surveillance data is the only source to understand the space-time dynamics of the infectious disease process, including the temporal component is more crucial than the spatial one, which is emphasized by the better performance of the Type II and Type IV models.

Table 1.

The model is fitted to the simulated observations up until T, and predictions are made for q = 1 and q = 2 weeks ahead. For each T* and q, the mean squared error of prediction (MSEP) is reported for the Type I–IV models, with the smallest MSEP indicated in bold.

q = 1 q = 2
T Type I Type II Type III Type IV Type I Type II Type III Type IV
6 857 683 911 687 504 804 527 819
7 153 36 142 43 135 44 107 67
8 115 35 73 45 128 44 83 61
9 135 26 90 39 134 42 96 64
10 134 31 100 46 197 45 173 67
11 260 30 247 36 675 74 694 107
12 1088 71 1144 75 1649 91 1729 78
13 2209 781 2311 663 2688 6433 2823 5449
14 3170 395 3331 443 3086 2140 3263 2364
15 2998 336 3188 364 2619 1253 2820 1346
16 2237 201 2444 210 2578 473 2808 503
17 2918 82 3163 85 4258 149 4535 141
18 5595 193 5895 198 7501 1391 7831 1510
19 9411 361 9756 331 10917 2314 11275 2166
20 12424 426 12779 435 11451 2757 11792 2825
21 10474 784 10793 799 8845 2477 9131 2526
22 7216 173 7460 167 5582 471 5793 458
23 3954 120 4125 122 - - - -

5. Application to HFMD Data in China

We now return to the HFMD data in the central north region of China over years 2009–2010. We assume the Poisson model (1) with μit given by (2). The study region is rescaled to a [0, 1] × [0, 1] square but retain the ratio [max(xi2) − min(xi2)]/[max(xi1) − min(xi1)]. We include the population density per square kilometer as a covariate, denoted as zi, with this variable being normalized to have mean 0 and standard deviation 1. We manually remove those basis functions that lie outside of the study region. The locations of K = 16 bases used in the analysis are shown in Figure 1(d).

The model is implemented using INLA, and parameter point and interval estimates are presented in Table 2. The running time for the four models are 307, 501, 289 and 429 seconds, respectively, on a MacBook Pro with 2.6 GHz Intel Core i7 processor. In the Type I and III models, we find that population density has a significant positive association, which is consistent with the fecal-oral transmission route of HFMD since areas in which children are in closer proximity are likely to have greater rates of infection. Estimates of the structured and unstructured temporal components γt and εt are very similar in the four models, and therefore we only show the results from the Type IV interaction model in Figure 5. The overall temporal pattern that is common to all areas is captured in the structured temporal random effect γ and is shown in Figure 5(a). The unstructured temporal random effects εt, shown in Figure 5(b), reflects the residual temporal variability after accounting for the structured RW2 temporal effect γt. No clear pattern is observed and the range of the estimated εt is narrow (between −0.10 and 0.075). The estimated structured spatial random effect ui ranges from −0.067 to 0.085, and the unstructured spatial random effect υi ranges from −1.1 to 0.95 so that the latter dominates. Maps of the median estimates are shown in Figure 5(c) and (d). These random effects are log relative risks of disease and so the spatial terms are not of great epidemiological significance. The sum of the log CPO values, again scaled by the Type II model, are: −7672, 0, −8075 and −3894, strongly suggesting the Type II model is most appropriate for the HFMD data.

Table 2.

Parameter estimates (median and 95% interval) from four types of interaction models, using weekly central north HFMD data in China 2009–2010.

Parameter Type I Type II Type III Type IV
α −1.08 (−1.26, −0.88) −0.97 (−1.45, −0.48) −0.84 (−1.31, −0.37) −0.96 (−1.45, −0.46)
β 0.30 (0.14, 0.45) 0.15 (−0.12, 0.41) 0.33 (0.13, 0.52) 0.16 (−0.11, 0.42)

σγ 0.15 (0.12, 0.20) 0.17 (0.13, 0.21) 0.15 (0.12, 0.19) 0.15 (0.12, 0.19)
σϕ 0.06 (0.04, 0.09) 0.06 (0.04, 0.09) 0.06 (0.04, 0.09) 0.06 (0.04, 0.09)
σu 0.13 (0.06, 0.46) 0.19 (0.06, 1.31) 0.13 (0.06, 0.50) 0.24 (0.07, 2.50)
συ 0.51(0.43, 0.63) 0.45 (0.36, 0.57) 0.51 (0.42, 0.63) 0.44 (0.35, 0.56)

σb 4.01(3.87, 4.16) 0.50 (0.47, 0.53) 8.09 (7.78, 8.42) 0.83 (0.79, 0.89)

Figure 5.

Figure 5

Estimated temporal component γ and ϕ, and spatial component u and υ from the Type IV interaction model with the China central north HFMD data.

An animation of the term α^+γ^t+k=1Kb^ktBik can be found at http://www.stat.brown.edu/cbauer/, which provides insight into the broad-scale space-time dynamics of HFMD. Compared to the log SMR maps, the movement of the epidemic center is much clearer as the regions with elevated relative risks are easier to identify. We see two local epidemics that start, respectively, in the north and south-west areas around week 22, and gradually move towards each other. This movement is not evident in the log SMR maps.

Next we take the first T weeks of HFMD surveillance data and perform the q-week ahead prediction of total HFMD counts, by area. The performance of the prediction is again assessed via the MSEP in (10). For comparison the prediction is also carried out using the epidemic/endemic model of (9) implemented within the R surveillance package. Specifically, we use the form yitit ~ Poisson(θit) with

θit=λiARyi,t1+λiNEjiwjiyj,t1+EiλitEN (11)

where λiAR=α0AR+biAR represents the autoregressive “self-area” component with random effects biAR,λiNE=α0NE+biNE is an autoregressive “neighboring area” component (defined as sharing a common boundary) with random effects biAR, and

λitEN=α0EN+biEN+βzi+s=1S(γs sin [st522π]+δs sin [st522π])

is an endemic component with the random effects biEN and S = 1. All the random effects are assumed independently and normally distributed. Results with T* starting at week 90 are presented in Table 3. More comprehensive experiments with T* going back as early as week 34 can be found in the supplementary materials. We attempted analyses with T* smaller than 34 but the epidemic/endemic model encountered convergence issues. Compared to the epidemic/endemic model, out of the 14 weeks our models give smaller MSEP in 8 weeks for q = 1 and 7 weeks for q = 2. Among the four types of interaction models, the Type II and IV models give smaller MSEP in all weeks. Adding spatial structure can aid the prediction, as seen in the smaller MSEP in the Type III model as compared to the Type I model. However, since most of the variation in the data comes from the temporal trend, it is the temporal structure components included in the Type II and IV models that are most important for prediction.

Table 3.

Results of q-weeks ahead prediction using China central north weekly HFMD data from 2009 to 2010. The figures are MSEP, as given by 10 and are for the Type I–IV models, as well as the epidemic/endemic (epi/end) model 11. For each q, figures in bold are the row minimum MSEP.

q = 1 q = 2
T Type I Type II Type III Type IV Epi/End model Type I Type II Type III Type IV Epi/End model
90 6204 (0.93) 1563 (0.86) 4758 (1.00) 1421 (0.92) 1310 4837 1614 3711 1488 2736
91 3129 (0.88) 2492 (0.85) 2704 (0.95) 3263 (0.83) 873 2589 5453 2184 7263 1535
92 1702 (0.92) 947 (0.81) 1524 (0.97) 893 (0.85) 3900 2783 1111 2253 1087 3093
93 4161 (0.90) 1774 (0.80) 3523 (1.00) 1602 (0.80) 397 5255 3051 4510 2774 583
94 3576 (0.93) 1071 (0.83) 2993 (0.97) 984 (0.85) 1670 4495 1909 3768 1774 2716
95 5481 (0.86) 749 (0.83) 4288 (0.97) 755 (0.83) 1655 5693 1834 4755 1366 2174
96 5449 (0.88) 2058 (0.73) 5217 (0.98) 2216 (0.76) 1746 5321 3352 5468 3789 1811
97 5723 (0.88) 1643 (0.76) 4907 (0.97) 1630 (0.78) 992 4765 4118 4202 4135 1268
98 3940 (0.92) 1757 (0.76) 3569 (0.97) 1813 (0.78) 836 3473 2040 3110 2113 1017
99 3380 (0.88) 839 (0.66) 2895 (0.97) 849 (0.71) 927 2590 955 2237 971 1621
100 1390 (0.90) 727 (0.75) 1352 (0.95) 815 (0.76) 1121 1353 878 1244 1090 1843
101 853 (0.92) 600 (64) 831 (0.96) 573 (0.68) 1811 790 602 789 556 2106
102 744 (0.88) 507 (0.69) 729 (0.95) 509 (0.66) 841 773 529 756 538 1060
103 924 (0.88) 498 (0.68) 810 (0.90) 512 (0.66) 302 - - - - -

In addition to the q-week ahead prediction we also investigate the performance of our proposed models for cumulative prediction, where we take the first T* weeks of the observations and predict the total HFMD counts for the remaining weeks in 2009 and 2010, by area. As can be seen in Table 4, in all weeks, our models give better prediction than epi/end model, except in the last week, where the scenario reverts to the 1-week ahead prediction. The results suggest that our proposed models perform as well as the epidemic/endemic model for short-term prediction, and outperform the competing model when prediction is carried out over longer time period.

Table 4.

Results of cumulative prediction using China central north weekly HFMD data from 2009 to 2010. The figures are MSEP, as given by 10 and are for the Type I–IV models, as well as the epidemic/endemic (epi/end) model 11. For each q, figures in bold are row minimum MSEP.

T Type I Type II Type III Type IV Epi/End model
99 1527 1852 1372 1915 3514
100 1695 1086 1322 1590 3601
101 784 629 785 571 2679
102 773 529 756 538 1061
103 924 498 810 512 302

Residuals plots of these models can be found in the supplementary materials. There is still considerable variation in the residuals from all models and some systematic variability showing that not all of the signal is being adequately modeled.

6. Discussion and Future Work

In this paper we have described a model for space-time infectious disease count data. The model bridges the gap between disease mapping models for chronic diseases and the more traditional SIR type models that are difficult to fit when the populations are large and individual-level information such as time of infection is not available. A number of aspects deserve further discussion.

The choice of prior distributions on the random effects variances in the interaction models is challenging because of the conditional interpretation, which changes across the four model types. An obvious approach is to use simulation, in tandem with substantive knowledge on the range of relative risks. For example, we may believe that the 95% range of the residual relative risks is [0.1, 10], which means that the range of the spline contribution is ∑k bktBik is [log(0.1), log(10)]. Experimentation with different prior choices via simulation can be carried out to achieve this aim. This is the approach that was taken to set the priors in our analysis, and the details can be found in the supplementary materials. From a practical point of view, our approach is computationally efficient since it is based on INLA. The improvement in computation time when compared with MCMC is considerable.

For the China central north HFMD data our model has allowed the following insights. In each year the epidemic moves from two centers, a major one in the south and a smaller one in the north. Population density is also seen to be an important aggregate level predictor, the credible intervals for the Type I and III models do not contain zero, but the intervals of the Type II and IV models contain zero, because the more flexible temporal model has absorbed some of the association with population density. This highlights the differences between models for prediction and those with which the identification of important covariates is sought. Without the etiology knowledge of the disease, prediction of future counts is difficult, in particular using the surveillance data alone. For HFMD disease, most of the variation in the surveillance data comes from the temporal trend. Hence, it is crucial to include the temporal structure in models for prediction.

The epidemic/endemic model is very appealing since it has autoregressive components, which allows aggregate level transmission to be estimated, and a good implementation. We would not suggest that our model be used as a replacement. Rather, we think the approaches are complimentary. The model we have described has the appealing feature of producing a smoothly time-varying relative risk surface and this can greatly help in gaining insight into the space-time dynamics.

Supplementary Material

Supplementary

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