https://orcid.org/0000-0002-5017-7395
Figures
Abstract
The impact of the COVID-19 pandemic on University students has been a topic of fiery debate and of public health research. This study demonstrates the use of a combination of spatiotemporal epidemiological models to describe the trends in COVID-19 positive cases on spatial, temporal and spatiotemporal scales. In addition, this study proposes new epidemiological metrics to describe the connectivity between observed positivity; an analogous metric to the R number in conventional epidemiology. The proposed indices, Rspatial, Rspatiotemporal and Rscaling will aim to improve the characterisation of the spread of infectious disease beyond that of the COVID-19 framework and as a result inform relevant public health policy. Apart from demonstrating the application of the novel epidemiological indices, the key findings in this study are: firstly, there were some Intermediate Zones in Edinburgh with noticeably high levels of COVID-19 positivity, and that the first outbreak during the study period was observed in Dalry and Fountainbridge. Secondly, the estimation of the distance over which the COVID-19 counts at the halls of residence are spatially correlated (or related to each other) was found to be 0.19km (0.13km to 0.27km) and is denoted by the index, Rspatial. This estimate is useful for public health policy in this setting, especially with contact tracing. Thirdly, the study indicates that the association between the surrounding community level of COVID-19 positivity (Intermediate Zones in Edinburgh) and that of the University of Edinburgh’s halls of residence was not statistically significant. Fourthly, this study reveals that relatively high levels of COVID-19 positivity were observed for halls for which higher COVID-19 fines were issued (Spearman’s correlation coefficient = 0.34), and separately, for halls which were non-ensuite relatively to those which were not (Spearman’s correlation coefficient = 0.16). Finally, Intermediate Zones with the highest positivity were associated with student residences that experienced relatively high COVID-19 positivity (Spearman’s correlation coefficient = 0.27).
Citation: Laxton MR, Nightingale G, Lindgren F, Sivakumaran A, Othieno R (2023) Extending the R number by applying hyperparameters of Log Gaussian Cox process models in an epidemiological context to provide insights into COVID-19 positivity in the City of Edinburgh and in students residing at Edinburgh University. PLoS ONE 18(11): e0291348. https://doi.org/10.1371/journal.pone.0291348
Editor: Kaspar Staub, University of Zurich, SWITZERLAND
Received: October 11, 2022; Accepted: August 29, 2023; Published: November 21, 2023
Copyright: © 2023 Laxton et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The shapefiles used for some of the figures has been uploaded and the source for these shapefiles is as follows: https://spatialdata.gov.scot/geonetwork/srv/eng/catalog.search#/metadata/389787c0-697d-4824-9ca9-9ce8cb79d6f5 The above source is publicly available. (i) The covariate data (COVID-19 summaries per Edinburgh Intermediate zones) is within the Supporting information files (an Excel file). (ii) The information on COVID-19 cases at the University of Edinburgh in particular can be obtained by contacting: Candice Schmid (safety.covid19@ed.ac.uk) Occupational Hygiene and Projects Manager Chartered Safety and Health Practitioner, Occupational Hygiene Unit Health and Safety Department University of Edinburgh (iii) Information on the halls of residence at the University of Edinburgh can be found here: Brenda Piercy (brenda.piercy@ed.ac.uk) Head of Operations – Self-Catered Accommodation, Catering and Events The University of Edinburgh St Leonard’s Hall 18 Holyrood Park Road Edinburgh Data regarding the University of Edinburgh is managed by the University and requests for access can be made directly to the above contacts.
Funding: Funding was obtained by GN from the Edinburgh Futures Institute only for the analysis of the data. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
The observation of infectious disease incidence in space is frequently accompanied by spatial correlation. The analyses of count data such as the counts of disease per measurement station can provide an indication of the central tendency and spread of the observed counts and the quantiles associated with the data. Choropleths provide an indication of the spatial context by illustrating the spatial proximity of the observed counts.
In this study we present three novel epidemiological indices, R spatial, R spatiotemporal, and R scaling. These are all based on hyperparameters in a Log Gaussian Cox process model. This section includes a discussion of Log Gaussian Cox process models and the importance of each index. In later sections we will show how the proposed indices can be applied to the context of COVID-19 positivity in University halls of residence.
Literature review
Spatiotemporal analytical approaches.
Various approaches to investigating spatiotemporal epidemiological trends exist [1–3]. Overall, the approaches highlighted by the aforementioned authors for spatial modeling include spatial interpolation, spatial statistical modeling and spatiotemporal statistical modeling. The most common methods are cast in a Bayesian framework and are namely, spatial interpolation methods and GLMM spatiotemporal models.
Spatial interpolation [4] involves estimating the value of a variable at geographic aggregations based on observed values. This method employs smoothing techniques to address the observation of extreme values in some of the geographic aggregations. Cast in a Bayesian context, this method allows for inputting expert, prior information.
The GLMM spatiotemporal models [5] observed were all Poisson based and contained the facility for the inclusion of random effects.
The BYM models [6], also Poisson based, are commonly used to estimate disease incidence/mortality in conjunction with disease/mortality mapping is an example of the abovementioned class of models. These models incorporate random effects as well, but unlike the Log Gaussian Cox Processes which we propose, these models are only specific to aggregated spatial data and not point pattern based data.
The proposed models which are Log Gaussian Cox Processes (or models) are already existing spatiotemporal models. These are commonly used in ecology [7], geography [8], and climate related research [9]. The uniqueness of our approach is to apply all of the model parameters to the epidemiological setting. So, in addition to estimating the effect of a given covariate (the traditional approach) on disease incidence—we take this further and apply the hyperparameters of the model to the epidemiological context. Typically, in spatial epidemiology aggregated data is used and incidence rates are estimated (estimates) using the BYM models. Our approach is to use both point and aggregate data—hence our marked point patterns as the data.
Extending the R number.
In conventional epidemiology the reproduction number is described as [10] is: “the expected number of new infections produced by a single (typical/average) infectious individual, when introduced into a totally susceptible population”. Note that the basic reproduction number R0, is applicable to situations where there is no immunity [11]. The importance of the R0 in public health was highlighted during the COVID-19 pandemic. Globally, governments and public health decision makers used R0 as part of the body of evidence to inform COVID-19 public health policy. In particular R0 will help policymakers decide whether a given community will be impacted by the disease at hand [12] and also which fraction of the community is estimated to be vaccinated as a protective measure. When R0 is greater than 1, the implication is that the disease will spread. There are various methods for the estimation of R0 [10, 13] and these are underpinned by ordinary differential equations [14]. The SIR modeling approach for example, is one such method, and involves the division of the population at hand (N), into three compartments; S (those individuals who are susceptible to the disease), I(those who are infectious), and R (those who have recovered or have been removed). Various statistical software, such as the R can facilitate the computation necessary. Our proposed indices will act as a supplement to the basic reproduction number, R0 because they will provide spatial insights into the spread of the disease at hand. R0 gives an indication of the spread of the disease between individuals (under the conditions previously discussed), whilst our proposed indices give an indication of the distance over which cases of infections are correlated. Together these metrics will help the public health decision maker in terms of assessing the risk of a community to infections, and assessing the distance over which cases can be linked with each other. The latter objective might be useful in establishing localized lockdowns. Similar to R0, our new indices can be affixed with a subscript 0 when the situation is such that there is no immunity in the community. We propose the use of credible intervals when these new indices are used/published. R spatial—an indication of the distance over which cases are correlated with each other and serve as an estimation of the extent of the spread of the virus within halls of residence (a spatial R number) R spatiotemporal- indicates the correlation in the spatial distribution of COVID-19 positivity as the timeline progresses. The positive estimate of this index indicates that as the months progress from September towards December the COVID19 positivity will be correlated to the previous month. R scaling—defines the strength and direction of the interaction between density of university halls of residence and residence COVID-19 levels. For the chosen model, R scaling is estimated to have a posterior mean value of 1.91, with 95% CI between 0.61 and 3.28. This suggests that areas with higher densities of student halls are likely to have higher COVID-19 levels than areas with lower densities of halls. This index can be useful in comparative studies where the numerical value of R scaling for different universities for example, can be compared and inform related policy. Overall, this index provides an indication of the trend (higher vs. lower) of disease cases relative to the density of spatial units (university halls or residential buildings etc).
Study setting
In terms of COVID-19 incidence at university halls of residence, research so far has indicated that the influx of first year students might be an important factor in terms of the spread of the disease, most likely due to living in halls of residences [15]. It should be mentioned however, that the University community from which this research has its basis is the University of Bristol. In the Bristol study, the use of stochastic transmission models [15] has suggested that first year university students were the main drivers of the transmission of COVID-19. Additionally, these researchers suggest that the most effective intervention for reducing transmission is to limit or reduce face to face teaching. Interestingly, using SEIR models, researchers [16] have discussed the successful control of COVID-19 transmission in an American university, Boston University, despite an accelerating in the rate of transmission in the surrounding community, the City of Boston.
The above-mentioned studies highlight the importance of examining the association of student influx on COVID-19 spread. They do not however, infer that there is a link between the influx of first year students on COVID-19 spread for all or most universities in general. The current study will investigate the situation for one University, the University of Edinburgh where 17184 students were enrolled in the autumn of 2020.
An analysis of the observed counts per measurement station and simultaneously, that of the unobserved spatial correlation, represented using a Gaussian Random Field, necessitates a modeling approach which incorporates both components. This study seeks to model the observed counts of COVID-19 positivity in students living in close proximity in University of Edinburgh halls of residence whilst accounting for the spatial correlation of the levels of positivity per hall and for the spatial correlation of the location of the hall. This approach is potentially valuable to public health policy and University health and safety policy since the model provides an estimate of the spatial range at which the levels of positivity per hall are correlated (or connected) as well as the spatial correlation of the hall locations. This information can be used to inform decisions on implementing localized lockdowns, for example.
The models will also investigate whether the spatial correlation of COVID-19 positivity varies by residence category, that is, ensuite or non-ensuite categories. The number of rooms per hall is used as an offset in the models to account for the density of students. A secondary goal for this study is to examine the association between the positivity levels at halls of residence and that of the surrounding community. In the case of Edinburgh University, the surrounding community would be represented by the Intermediate Zones where each hall/residence is represented by. Note that Intermediate Zones are stable geographies which fall between Data Zones and Local Authority geographies.
The proximity of services such as supermarkets, shopping centres, gyms, and restaurants to halls of residence where students have recently taken up residence may serve as a pathway of epidemiological interaction (epiaction) between halls of residence and the community. The extent of the epiaction would depend on the zone of influence [17] of the service, the activity (social interaction and shared amenities) within the residences and the positivity levels in neighbouring Intermediate Zones.
Importantly epiaction can be symmetric or asymmetric. The use of multiple analytical tools in tandem is the ideal approach to deciphering the direction of epiaction in this scenario. The inclusion of temporal variables such as intervention measures would add to the usefulness of this model. Overall, universities bring together students that come from a variety of settings locally, nationally and internationally—this makes them a potential vector for transmission of infection to and from the places of their origin. The dynamics associated with student teaching, studying and dining arrangements are also potentially important factors that may facilitate easy transmission of infection. Understanding and risk assessing these dynamics are important in the control of COVID-19 in these institutions of higher learning.
Research questions
The study timeframe is September 2020 to December 2020 and the study research questions are outlined below: (i) Are there any spatial patterns of COVID-19 positivity in the halls of Edinburgh University and the surrounding communities? (ii) Are there any differences in living in ensuite/non-ensuite on student positivity levels (Edinburgh University), and separately, in halls which had relatively higher/lower fines issued? (iii) What are the temporal positivity trends (in the presence of interventions such as mandatory face covering use, limitations in size of gatherings, and closure of hospitality venues) in the surrounding communities? (iv) What is the relationship between the COVID-19 positivity at halls of Edinburgh University and the community, and (v) What are the public health implications of the research findings?
In the news
Thousands of students returned in person to further education in the UK /Scotland in September 2020. In Edinburgh, and the rest of Scotland, COVID-19 testing of student returners was limited to PCR testing of symptomatic cases only. Other UK universities such as Cambridge University introduced asymptomatic pooled testing of all students from autumn 2020 [18]. In this scenario, individuals tested do not display COVID-19 symptoms. Note that COVID-19 can be transmitted from infected individuals who are themselves asymptomatic.
In Scotland, outbreaks at universities were reported soon after their reopening at the start of the 2020/21 academic year, for example in Glasgow [19], Napier University [20], and in Edinburgh. In Sept 2020, large COVID-19 outbreaks [21] were reported at the majority of universities and further education institutions (The BBC, September 2020). These outbreaks were managed by Incident Management Teams (IMT) set up and chaired by consultants in Public Health from the local Health Board working jointly with key staff from the local universities.
University COVID-19 outbreaks and policies were, amongst other things, a popular topic of discussion during the autumn of 2020 [22, 23]. There were also expressions of concern regarding the level of occupancy of university halls with one newspaper (The Guardian, October 2020) publishing the headline “UK university student halls too full to be safe, experts warn”. In this report, an individual is quoted as saying “reopening a university is like dumping a cruise ship in the middle of town and letting passengers off”.
The independent UK Scientific Advisory Group for Emergencies (SAGE) [24] group advised that the virus was transmitted within residential halls and via face-to-face teaching. This group recommended online instruction. Issues arose regarding students returning to their homes for the holidays, with some advising that they should remain at the university halls to minimise onward transmission into their local communities [25]. In terms of students needing to isolate, policies were set up to allow students to isolate at their homes [25, 26]. The government reached an agreement to set up policies (at the end of the academic term) to allow students who were required to isolate due to a positive test, to return to their local residence and isolate in their homes instead of remaining in halls of residence. Some universities also viewed the challenges presented by the pandemic as an opportunity to rethink the future of higher education including methods for delivery of teaching [27, 28]. In terms of students returning to universities [29], various modes of testing were used; for example, mass testing in St. Andrews [30]. At the University of Edinburgh, lateral flow testing of students began in November 2020. This followed the Scottish Government’s “test before you travel” policy on November 17th, 2020 (personal communication, University of Edinburgh). In total 28511 tests were conducted between 30th November 2020 and 30th June 2021.
In terms of seeding of infections in Scotland as a whole, some researchers indicated that new infections in Scotland during the second wave was linked to a variant discovered in Spain [31]. The UK’s travel policy in the summer of 2020 was described as being flawed as ‘the virus moves when people move’ [31]. Studies suggest that traveling outside of the UK led to reseeding of infections [32, 33]. Genomic epidemiology reveals multiple introductions of SARS-CoV-2 from mainland Europe into Scotland [32].
Finally, some described the possibility of outbreaks at University residences as an “Accident waiting to happen” due to the report that many Scottish universities did not reduce capacity [23] of student numbers in their halls of residence in order to increase social distancing.
Key local control measures are outlined in Table 1 in the Appendix.
Tables 2 and 3 show the number of rooms at the University of Edinburgh halls of residence during the study period, and enrolment per month during the study timeframe respectively.
Method
The proposed methods include data visualization (choropleths, heatmaps) and point process models, which are ideal for point pattern data where locations of observed events are provided. The spatial distribution of residences is modelled using a Log Gaussian Cox Process (LGCP) model, which is doubly stochastic by nature -comprising of a Poisson point process, and a spatial random field [34, 35]. The proposed suite of 8 nested models are marked point process models, extending the LGCP model to jointly model the locations of residences (the point pattern) alongside the count of COVID-19 positivity (a property measured at the residence locations, called a ‘mark’) with two dependent likelihoods. Apart from the parameters associated with the covariates in the models, the parameters of the random fields representing residence distribution and COVID-19 count distribution are important in that they provide an estimate of the spatial range over which the observed counts are correlated and the effect of residence clusters on COVID-19 positivity. We propose to demonstrate how the parameters associated with the random fields can be useful epidemiological indices. These indices, Rspatial, Rspatiotemporal, and Rscaling represent the modeled posterior estimate of the range of the spatial field, temporal correlation and scaling parameter respectively.
Finally, ethics approval was given by the University of Edinburgh’s Ethics Committee (in Nursing Studies) on 12th November 2021 with an ID: NUST011s.
Data
The data for this study has been provided by the University of Edinburgh’s Accommodation, and Health and Safety divisions. Data has been provided on the number of students who tested positive by PCR for COVID-19 positivity per university owned residences over 8 months starting in September 2020. Data for 51 residences was obtained. This study focuses on the months September to December 2020. Information on the halls of residence (such as ensuite status) was provided for the residences within halls. The location (latitude and longitude) of each hall of residence was obtained from Google maps. The data has been provided at an aggregated level so individuals cannot identify themselves in the analyses. Finally, data on the number of COVID-19 fines issued per hall of residence during the study period was also provided by the Univeristy. Note that these fines were issued by the University’s Accommodation, Catering and Events (ACE) team. For this study, the data is stored on a dedicated laptop and after this study the data will be destroyed. The intended output of the study includes:
- Visual illustrations of the time series of COVID-19 positivity in the City of Edinburgh and separately, in Intermediate Zones within the City of Edinburgh,
- Visual illustrations of the spatial trend of COVID-19 positivity in Intermediate Zones within the City of Edinburgh,
- Visual illustrations of the estimated spatial correlation of COVID-19 positivity in the University of Edinburgh halls of residence,
- Modeled estimates of the spatial range and standard deviation of the spatial random field underlying the data, and an
- Interpretation of the estimated spatial fields and the implications for public health policy and University health and safety policy.
Models
Here, we aim to answer the research questions detailed earlier (in the section on Research Questions) by fitting a series of 8 nested marked point process models to data on the counts of COVID-19 positivity by university hall of residence between September 2020 and December 2020. Details of the models are provided in Table 4.
Mark likelihood.
Models 1 and 2 are purely spatial models, which follow the form:
(1)
Where, y(s) represents the total count of COVID-19 cases observed at a residence at location s (during the study period).
Models 3–8 incorporate a temporal correlation structure, and model the data in space and time. Models 3–8 follow the form:
(2)
Where, y(s, t) represents the total count of COVID-19 cases observed at a residence at location s and time t (during the study period). The structure of all 8 models is outlined in Table 5.
In all 8 models (Table 5), α1 represents an intercept term; ψ is a spatially structured random field that also appears in the linear predictor for the spatial pattern; and κ is a scaling parameter. In Models 1–2, ω denotes a spatially structured random field that is unique to the counts of COVID-19 cases per residence (i.e. the ‘marks’ of the point pattern). Both random fields (ψ and ω) follow a Matérn covariance structure with spatial range parameter ρ and standard deviation σ. In Models 3–8, the ω field is spatiotemporally structured, incorporating an AR1 process with correlation parameter ρt.
In the section on Novel epidemiological Indices, we introduce 3 novel epidemiological indices, derived from the parameters of random effects in the models. We refer to the mean posterior estimate of the spatial range (ρ) of the mark field (ω(s) and ω(s, t)) as Rspatial; the mean posterior estimate of the temporal correlation parameter (ρt) from the AR1 process of the spatiotemporally structured random field (ω(s, t)) as Rspatiotemporal; and the mean posterior estimate of the scaling parameter (κ) as Rscaling.
In Model 2, xi(s) represents the value at location s of the covariate quantifying the number of COVID-19 cases in the community per Intermediate Zone. A spatiotemporal version of this covariate is included in Models 4–8.
In Models 5 and 8, xj(s) represents the value at location s of the covariate quantifying the proportion of total rooms with ensuite facilities per residence.
In Model 6, xl(s, t) represents the value at location s and time t of the covariate quantifying the number of fines received by each residence for breach of COVID-19 regulations.
In Models 7 and 8, xm(s, t) represents the value at location s and time t of the binomial covariate indicating whether or not the residence is part of one of the largest halls (unnamed) on campus.
The regression coefficients (βi, βl, βj, and βm) of covariates are estimated in the models.
Prior specifications
In this study, PC-priors are specified to inform the Matérn covariance structure of the Gaussian Random Fields ψ and ω. PC-priors penalise complexity away from a simpler base model, so ensure that the effects of complex model components (such as Gaussian Random Fields) are only included when this is necessitated by the data [36]. In the case of the Matérn field, a PC-prior is specified on the joint density of the spatial range (ρ) and marginal standard deviation (σ2). The spatial range gives an indication of the spatial span of the covariance structure, and the standard deviation provides the degree of spatial variability. The hyper-parameters (R and S) for this prior are set when using inlabru, the R software package used for this analysis [37, 38]. This is done indirectly by specifying the lower tail quantile and probability for the range, and the upper tail quantile and probability for the standard deviation.
In the R code for the analysis, the user must specify:
prior.range = c(ρ0,pρ)
Where ρ0 corresponds to the lower tail quantile for the range, and pρ corresponds to the probability that the actual range value (ρ) is less than ρ0. This can be written as:
(4)
Similarly, for the standard deviation, the user must specify:
prior.sigma = c(σ0,pσ)
Where σ0 corresponds to the upper tail quantile for the standard deviation, and pσ corresponds to the probability that the actual standard deviation (σ) is greater than σ0. This can be written as:
(5)
This means that users are able to inform both the locations of the tails and their certainty about this. For the models fitted in this paper, the priors on the parameters for both random fields (ψ and ω) were P(ρ < 0.2) = 0.01 and P(σ > 1) = 0.01, where the lower tail location for the range parameter is given in units of km.
For the Matérn field, PC-priors penalise complexity away from the base model in which the random field is spatially constant (ρ = ∞) and has zero variance (σ = 0). This means the effects of a field with the user-specified priors on the parameters above are only included if this is supported by the data.
Results
Temporal trends
Here we provide an indication of the trend of COVID-19 positivity in the City of Edinburgh between September 1st 2020 to October 31st 2021. This is provided against the backdrop of the chronology of COVID-19 public health interventions in Scotland (See Figs 1 and 2) with Intermediate Zones of relatively higher positivity highlighted. From Fig 1 we note that public health policies in Scotland included a 77-day lockdown (restricting social and spatial interaction of residents) and later the use of face coverings and were mostly, layered interventions [15]. In addition, university students were encouraged (by email) to avoid socializing in pubs and restaurants during the weekend of September 26th, 2020. This timeframe is denoted as “++” in Fig 2. Note that the university was closed during the spring term.
The policies are indexed as follows: a. Ban on visiting other households; b. Uni students advised to socialise amongst acc.grp.; c. Police patrols to enforce rules; d. Student guidance issued for visiting parents; e. More student guidance on visiting parents; f. Bars/restaurnts/pubs close early till Oct 25; g. Restrictions on Glasgow prison; h. Care home rules relaxed; i. Restrictions on hospitality businesses extended; j. Digital Christmas advised; k. Five tier Scottish COVID-19 system announced; l. Halloween discouraged; m. Face coverings advised for senior pupils and teachers.
The symbols NDR, MES, and DF on the y-axis represent the Intermediate Zones, Newington, and Dalkeith Road, Marchmont East and Sciennes, and Dalry and Fountainbridge. Note that Zones with NA as value were filtered off.
Fig 1 shows the evolution of COVID-19 positivity in Edinburgh against the backdrop of public health policy. In the study timeframe, three prominent peaks are observed in the figure. The first peak in late September 2020 is observed at Dalry and Fountainbridge, whilst the second (relatively larger peak) is observed at Marchmont East and Sciennes. The third peak (highest) is at Newington and Dalkeith in early October 2020. In addition to observing distinct peaks, it was observed that the trend for the Intermediate Zone, Old Town, Princes Street and Leith Street, plateaued for approximately 21 days (longer than the abovementioned peaks) before declining.
The heatmap in Fig 2 provides an indication of the temporal trend between the study period with separate rows for each Intermediate Zone. The legend for this heatmap is suppressed to avoid any statistical disclosures. The highest count range is green, the medium range is yellow, and the lowest, red. From the heatmap the comparatively higher positivity is observed in 2021, whilst in 2020, a sustained “spark” is observed in October 2020 with a progression in positivity towards the end of the year. That spark is represented by the Intermediate Zone, Newington, and Dalkeith Road. Earlier sparks of shorter duration were observed as well, such as in Dalry and Fountainbridge and increasing for Marchmont East and Sciennes.
Spatial trends
Figs 3 and 4 depict the spatiotemporal trends of COVID-19 positivity (counts) in 51 University of Edinburgh owned student residences, and the spatiotemporal trends in COVID-19 positivity (7-day average) across Intermediate Zones in the City of Edinburgh respectively.
In Figs 3 and 4 it is clear that there were flare-ups in COVID-19 positivity in October 2020 in both the Intermediate Zones in the City of Edinburgh and within the University of Edinburgh owned student residences. Importantly we note that the relatively higher positivity levels are observed to be spatially clustered in similar locations. In particular, the Intermediate Zone with the highest level of positivity in October 2020 is Newington, and Dalkeith Road—this is the immediate spatial neighbourhood (community) of the University of Edinburgh student residences which experienced the highest levels of positivity in the aforementioned month.
Modeling
Model discrimination.
We considered a suite of 8 nested marked point process models. Based on the model diagnostics and principles of parsimony, the model with the lowest Widely applicable information criterion (WAIC) [39] value is Model 4, indicating that this model provides the best balance between goodness of fit and model simplicity (Table 6). Model 8 has a similar WAIC (higher than model 4 by 7 WAIC points). Based on Spiegelhalter’s advice on DIC [40], the models 4 and 8 can be considered different from each other. Model 4 contains a component to account for spatiotemporal correlation structures in the data, and a covariate for the community COVID-19 level. It is important to note here that the meaning of scores such as WAIC in this modelling context remains largely unexplored, and that modelling aims and biological interpretability should also be considered in the model selection process.
Posterior mean and 95% credible intervals for: scaling parameter/Rscaling (κ); spatial range (Rspatial) and SD of the random fields (ω and ψ); temporal correlation/(Rspatiotemporal) of the AR1 process (ρt); and regression coefficients(βi,βj,βl,βm,) of covariates: number of COVID-19 cases in the community per Intermediate Zone (xi); proportion of total rooms with ensuite facilities per residence (xj); number of fines received by each residence for breach of COVID-19 regulations (xl); and whether or not the residence is part of one of the largest halls (unnamed) on campus (xm). Model run times (seconds), Watanabe–Akaike information criterion (WAIC) scores, and log conditional predictive ordinate (log-CPO) scores are also given. All values given are rounded to 2 decimal places.
Covariates.
In all 8 models, all of the 95% credible intervals for the covariate effects crossed 0, indicating that none of the covariates had a statistically significant effect on the response (Table 6).
The community level covariate was hypothesised to be of particular importance due to its coverage in the media. However, for the 6 spatiotemporal models considered here, the credible intervals for this covariate effect were very large, meaning that no statistically significant inferences could be made about the effect of this covariate. In the spatial model which included this covariate (Model 2), the posterior mean regression coefficient is positive, suggesting that as the COVID-19 positivity increases in the community, there is an expected increase in the COVID-19 positivity at the university halls of residence. While the 95% credible interval for βi in Model 2 is smaller than that of the spatiotemporal models, it does cross 0, so this inference is not supported by statistical significance.
Novel epidemiological indices.
We use the terms Rspatial, Rspatiotemporal and Rscaling to represent the mean posterior estimates of the mark field (ω) spatial range (ρ), the temporal correlation (ρt), and the scaling parameter (κ), respectively. These are the three indices that we propose in this study to describe epidemiological spread. The posterior estimates of these values from the selected model (Model 4) are Rspatial = 0.19 [0.13, 0.27], Rspatiotemporal = 0.48 [0.37, 0.59] and Rscaling = 1.91 [0.61, 3.28], respectively (Table 6).
Rspatial is given in the spatial units of the input data (here, km). Thus, we can interpret the range in connectivity in COVID-19 levels between university halls of residence in Edinburgh as an average of 0.19km, with 95% CI between 0.13km and 0.27km. The mean nearest neighbour distance between the halls of residence considered in this study is 0.23km, with the distances between halls ranging from 0.003km to 5.061km.
The value for Rspatiotemporal is bounded between -1 and 1, where -1 suggests a strong negative correlation, 1 suggests a strong positive correlation, and 0 suggests no correlation. The posterior estimate for Rspatiotemporal in the chosen model suggests a positive correlation in COVID-19 levels in halls of residences over time. This means that the level of COVID-19 in a halls of residence in a given month is impacted by the COVID-19 level in that spatial area during the previous month. Because the parameter comes from a spatiotemporal field, space and time are also ‘tied together’ here, so this should be interpreted as the spatial structure of the field being correlated across time. So, if there are high levels in one area and low in another in a given month, the next month is likely to follow a similar spatial structure. Fig 5 demonstrates the correlation in the spatial structure of the mark random field (ω(s, t)) from Model 4 over time.
Colour scale is given in low-high intensity as we are interested in relative differences across space and not absolute values.
The third index, Rscaling, can be interpreted as defining the strength and direction of the interaction between density of university halls of residence (represented by the point field ψ) and residence COVID-19 levels. For the chosen model, Rscaling is estimated to have a mean value of 1.91, with 95% CI between 0.61 and 3.28 (Table 6). This suggests that University halls of residence in areas with higher densities of University hall of residence are likely to have higher COVID-19 levels than areas with lower densities of halls.
Exploratory analyses.
Our exploratory analyses indicate that residences for which there were relatively more fines issued also reported higher COVID-19 positivity (Spearman’s correlation coefficient = 0.34). Similarly, residences with non-ensuite-type accommodation reported higher COVID-19 positivity (Spearman’s correlation coefficient = 0.16). For the communities associated with each residence, it was noted that the Intermediate Zones with the highest positivity were associated with student residences that experienced relatively high COVID-19 positivity (Spearman’s correlation coefficient = 0.27).
Prior sensitivity.
In order to assess the sensitivity of posterior estimates of the Gaussian Random Field parameters to prior specifications, a further 8 models were fitted. These were variations on Model 4 (selected as the most appropriate model using WAIC) in which the priors for the upper tail of the marginal standard deviation and lower tail of the range of the mark random field (ω) were altered. Details of the prior specifications used can be found in Table 7 and the outputs, in Table 8.
Posterior mean and 95% credible intervals for: scaling parameter Rscaling (κ); spatial range (Rspatial) and SD of the random fields (ω and ψ); temporal correlation (Rspatiotemporal) of the AR1 process (ρt); and regression coefficient (βi) of covariate: number of COVID-19 cases in the community per Intermediate Zone (xi). Model run times (seconds), Watanabe–Akaike information criterion (WAIC) scores, and log conditional predictive ordinate (log-CPO) scores are also given. All values given are rounded to 2 decimal places. Model 4F failed to run and so outputs for this model are not included in the table.
The prior sensitivity tests indicate that the estimates of the parameters are not affected significantly by the additional variances used. If significant changes were observed, this would be interpreted as sensitivity of the estimates to the prior information/understanding of the variables at hand.
Discussion
The advantages of the proposed approach are that (i) we interpret the Random fields of a Log Gaussian Cox Point Process to inform disease spread. Traditionally in epidemiological models, the estimation of the impact of covariates is of focus. In our study we focus on both the estimation of the impact of covariates, and the estimation of the hyperparameters of the random field. The hyperparameters of the random field are our proposed indices. As discussed earlier on, these can be adapted for the situation when there is no immunity, and adapted to facilitate comparison between spatial settings. (ii) we use both point and aggregate data—hence our marked point patterns as the data. Typically, in spatial epidemiology (reference) aggregated data is used and incidence rates are estimated (estimates) using the BYM models for example.
Recall the research questions (i) Are there any spatial patterns of COVID-19 positivity in the halls of Edinburgh University and the surrounding communities? (ii) Are there any differences in ensuite/non-ensuite positivity levels (Edinburgh University), and separately, in halls which had relatively higher/lower fines issued? (iii) What are the temporal positivity trends (in the presence of interventions such as mandatory face covering use, limitations in size of gatherings, and closure of hospitality venues) in the surrounding communities? (iv) What is the relationship between the COVID-19 positivity at halls of Edinburgh University and the community, and (v) What are the public health implications of the research findings?
We will address each research question in turn and relate the corresponding findings to public health implications.
Research question 1
We use the terms Rspatial and Rspatiotemporal to represent the estimate of the mark field (ω) range and the temporal correlation (ρt) respectively. These are the two of the indices that we propose in this study to describe epidemiological spread. We will now discuss the practical interpretation of these estimates in relation to COVID-19 (and in general, infectious disease) spread.
Rspatial.
The estimation of the range of the spatial field will give an indication of the distance over which cases are correlated with each other and serve as an estimation of the extent of the spread of the virus within halls of residence (a spatial R number). In the case of Model 4, the posterior parameter estimate is 0.19 [0.13,0.27]. This indicates that the distance over which the COVID-19 counts are spatially correlated (or related to each other) is 0.19km and the associated credible interval is 0.13km to 0.27km.
Rspatiotemporal.
In general, this index would give an indication of how correlated the spatial distribution of the disease under consideration is over time. Positive values for example, indicate positive correlation. The posterior estimate of the spatiotemporal correlation is 0.48 [0.37,0.59]. This indicates the correlation in the spatial distribution of COVID-19 positivity as the timeline progresses. The positive estimate of this index indicates that as the months progress from September towards December the COVID-19 positivity will be correlated to the previous month.
In terms of the spatiotemporal correlation, the practical application is that the model estimate gives us an indication of how correlated the COVID-19 positivity in one month is to that of another. In this study, we only considered 3 months. This parameter will be more important where there are more timepoints—such as 12 months where mass student vaccination took place within this timeframe. We hope to address this in a future study.
Rscaling.
This index defines the strength and direction of the interaction between density of university halls of residence and residence COVID-19 levels. For the chosen model, Rscaling is estimated to have a posterior mean value of 1.91, with 95% CI between 0.61 and 3.28. This suggests that areas with higher densities of student halls are likely to have higher COVID-19 levels than areas with lower densities of halls. This index can be useful in comparative studies where the numerical value of Rscaling for different universities for example, can be compared and inform related policy.
Research questions 2 and 4
Exploratory analyses.
Our exploratory analyses indicate that residences for which there were relatively more fines issues also reported higher COVID-19 positivity (Spearman’s correlation coefficient = 0.34). Residences with non-ensuite-type accommodation reported higher COVID-19 positivity (Spearman’s correlation coefficient = 0.16), and for the communities associated with each residence, it was noted that the Intermediate Zones with the highest positivity were associated with student residences that experienced relatively high COVID-19 positivity (Spearman’s correlation coefficient = 0.27). These are all exploratory and a larger dataset would be needed to explore these preliminary findings. The practical significance of some of these observations should not be ignored however. For example, accommodations with shared facilities (unlike ensuite residences) would be likely to facilitate more interaction between students and hence increase the probability of them being exposed to infectious disease such as COVID-19.
The abovementioned observations are important for policy and highlight the potential importance of social gatherings and the role of non-ensuite rooms in the dynamics of infectious disease spread.
Covariates.
Four covariates were considered in this study: impact of the community, the number of fines associated with each residence, the type of accommodation (ensuite vs. non-ensuite), and whether or not the residences were part of one of the largest halls (unnamed) on campus. A statistically non-significant model estimate was obtained for each covariate. What does this suggest to policy? The statistically non-significant estimates cannot be relied upon for informing policy regarding the spread of COVID-19 amongst university halls of residence. In a future study, we plan to conduct a larger scale study which involves multiple universities in the City of Edinburgh over a longer time period. The inclusion of more data will allow us to investigate these relationships and epiaction even further.
Research question 3
On inspection of the temporal trends and heatmaps we note that various Intermediate Zones exhibit high levels of COVID-19 positivity and in some cases, the timing of the positivity peak varies between Intermediate Zones. In this study the Dalry and Fountainbridge Intermediate Zone COVID-19 positivity appears to peak earliest, and this raises questions as to the cause of this peak. Some researchers suggest this peak is due to an outbreak at one of the universities in Edinburgh. This outbreak was followed by police enforcement of targeted policy.
Subsequent outbreaks were observed along the timeline. In most cases the outbreaks were quelled within a relatively short timeframe (in days), however the outbreak for the Intermediate Zone Old Town, Princes Street and Leith Street, during September 29th 2020 and 20th October 2020, took a relatively longer period (21 days) to decline before increasing once more. The abovementioned observations present questions to public health policy with regards to protecting communities.
The timing of the advice to students to socialize with students from their halls of accommodation coincides with the timeframe where the positivity rates are decreasing for Dalry and Fountainbridge and increasing for Marchmont East and Sciennes and Newington and Dalkeith. It is not possible to decipher the impact of the advice to students though since there are multiple dynamics at play. We can say however, that as the rates decreased in Dalry and Fountainbridge, they increased in Marchmont East and Sciennes and Newington and Dalkeith during the last two weeks of September 2020.
The fact that the effects of COVID-19 interventions are typically observed 2 weeks later suggests that it is plausible that the decreasing trends for various Intermediate Zones in early October could be due to the advice to students. The advice was for one weekend though, so it is likely that the resurgence of positivity increases in those zones could be due to resumption of mixing. The policy interventions, “a”, “b”, “c”, “d”, and “+” were all concentrated in September 2020, an indication of the dynamic response to the challenge of the rising levels of positivity.
Overall, we have shown how the use of selected hyperparameters of Log Gaussian Cox Process models can be used as epidemiological indices and can help improve the description of disease spread. In particular, we have shown how the spatial dimension can be incorporated in epidemiological indices and we have demonstrated this in a case study, alongside other important supporting analyses.
Supporting information
S1 Fig. Exploratory analyses.
Exploratory analyses were used to determine trends in COVID-19 positivity against community COVID-19 levels, number of fines received by halls of residence, and proportion of ensuite rooms.
https://doi.org/10.1371/journal.pone.0291348.s001
(PNG)
S1 File. R Code for prior sensitivity specifications—models 1-8.
https://doi.org/10.1371/journal.pone.0291348.s002
(PDF)
S2 File. R Code for prior sensitivity specifications—models 4A—4I.
https://doi.org/10.1371/journal.pone.0291348.s003
(PDF)
S2 Data. Shapefiles for edinburgh intermediate zones.
https://doi.org/10.1371/journal.pone.0291348.s005
(ZIP)
Acknowledgments
We would like to thank the University of Edinburgh for providing us the records of COVID-19 positive cases per hall of residence, and enrollment summaries for the study time period. We are also grateful for the other general characteristics provided by the University on each hall.
References
- 1. Ezra Gayawan, Awe Olushina O, Oseni Bamidele M, Uzochukwu Ikemefuna C, Adekunle Adeshina, Samuel Gbemisola, et al. The spatio-temporal epidemic dynamics of COVID-19 outbreak in Africa. Epidemiology & Infection, 148:e212, 2020.
- 2. Everton Emanuel Campos de Lima, Gayawan Ezra, Baptista Emerson Augusto, and Queiroz Bernardo Lanza. Spatial pattern of COVID-19 deaths and infections in small areas of Brazil. PLoS One, 16(2):e0246808, 2021.
- 3. Nazia Nushrat, Butt Zahid Ahmad, Bedard Melanie Lyn, Tang Wang-Choi, Sehar Hibah, and Law Jane. Methods used in the spatial and spatiotemporal analysis of COVID-19 epidemiology: A systematic review. International Journal of Environmental Research and Public Health, 19(14):8267, 2022.
- 4. Achaiah Nithya C, Subbarajasetty Sindhu B, and Shetty Rajesh M. R0 and re of covid-19: can we predict when the pandemic outbreak will be contained? Indian journal of critical care medicine: peer-reviewed, official publication of Indian Society of Critical Care Medicine, 24(11):1125, 2020.
- 5. Raymundo Carlos Eduardo, Oliveira Marcella Cini, Tatiana de Araujo Eleuterio, Suzana Rosa André, Marcele Gonçalves da Silva, Eny Regina da Silva Queiroz, et al. Spatial analysis of COVID-19 incidence and the sociodemographic context in Brazil. PLoS One, 16(3):e0247794, 2021. pmid:33647044
- 6. Besag Julian, York Jeremy, and Mollié Annie. Bayesian image restoration, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics, 43:1–20, 1991.
- 7. Kuronen Mikko, Särkkä Aila, Vihola Matti, and Myllymäki Mari. Hierarchical Log Gaussian Cox process for regeneration in uneven-aged forests. Environmental and Ecological Statistics, 29(1):185–205, 2022.
- 8. D’Angelo Nicoletta, Siino Marianna, D’Alessandro Antonino, and Adelfio Giada. Local spatial Log-Gaussian Cox processes for seismic data. AStA Advances in Statistical Analysis, 106(4):633–671, 2022.
- 9. Serra Laura, Saez Marc, Mateu Jorge, Varga Diego, Juan Pablo, Carlos Díaz-Ávalos, et al. Spatio-temporal Log-Gaussian Cox processes for modelling wildfire occurrence: the case of Catalonia, 1994–2008. Environmental and Ecological Statistics, 21:531–563, 2014.
- 10. Boonpatcharanon Sawitree, Heffernan Jane M, and Jankowski Hanna. Estimating the basic reproduction number at the beginning of an outbreak. PLOS ONE, 17(6):e0269306, 2022.
- 11.
Elisabeth Mahase. Covid-19: What is the R number?, 2020.
- 12. Samath Dharmaratne, Supun Sudaraka, Ishanya Abeyagunawardena, Kasun Manchanayake, Mahen Kothalawala, and Wasantha Gunathunga. Estimation of the basic reproduction number (R0) for the novel coronavirus disease in Sri Lanka. Virology Journal, 17:1–7, 2020.
- 13. White Laura F, Moser Carlee B, Thompson Robin N, and Pagano Marcello. Statistical estimation of the reproductive number from case notification data. American Journal of Epidemiology, 190(4):611–620, 2021.
- 14. Castro RR, Santos RSC, Sousa GJB, Pinheiro YT, Martins RRIM, Pereira MLD, and RAR Silva. Spatial dynamics of the COVID-19 pandemic in Brazil. Epidemiology & Infection, 149, 2021.
- 15. Brooks-Pollock E, Christensen H, Trickey A, Hemani G, Nixon E, Thomas AC, Turner K, Finn A, Hickman M, Relton C, and Dannon L. High COVID-19 transmission potential associated with re-opening universities can be mitigated with layered interventions. Nat Commun., 12(1), 2021.
- 16. Hamer DH, White LF, Jenkins HE, Gill CJ, Landsberg HE, Clapperich C, Bulekova K, Platt J, Decarie L, Gilmore W, Pilkington M, MacDowell TL, Faria MA, Densmore D, Landaverde L, Li W, Rose T, Burgay SP, Miller C, Doucette-Stamm L, et al. Assessment of a COVID-19 control plan on an urban university campus during a second wave of the pandemic. JAMA Netw Open, 4(6), 2021.
- 17. Nightingale G. Applying zones of influence to social distancing in schools. Royal Statistical Society Significance Magazine, 2020.
- 18.
MyScience. Asymptomatic screening and genome sequencing help Cambridge understand spread of SARS-CoV-2 among its students. MyScience, 2021.
- 19.
W Hazell. Glasgow University covid outbreak: ‘we were told it was safe to come—now we’re locked in our rooms almost like prison’. inews, 2020.
- 20.
The BBC. Coronavirus in Scotland: 11 cases linked to student flats. The BBC, 2020.
- 21.
The BBC. Scottish University outbreaks’should have been predicted’. The BBC, 2020.
- 22.
M Blackall and A Mistin.’broken and defeated’: UK university students on the impact of Covid rules. The Guardian, 2020.
- 23.
C Clements. Covid in Scotland: University outbreaks were ‘accident waiting to happen’. The BBC, 2020.
- 24.
SAGE. Principles for managing SARS-CoV-2 transmission associated with higher education, 3 september 2020. UK Government, 2020.
- 25.
A Fazackerley. UK university student halls too full to be safe, experts warn. The Guardian, 2020.
- 26.
The BBC. Covid in Scotland:’struggling’ students told they can return home. The BBC, 2020.
- 27.
T Kupe and G Wangenge-Ouma. Post COVID-19: Opportunity for universities to have a rethink. The BBC, 2020.
- 28.
M Phakeng, A Habib, and T Kupe. Three South African vice-chancellors paint a post-COVID picture for universities. The Conversation, 2020.
- 29.
The BBC. Covid in Scotland: Staggered return for university students. The BBC, 2020.
- 30.
J Harrison. Coronavirus Scotland: mass testing centre set up at St Andrews university. The Herald, 2020.
- 31.
N Davis. Coronavirus strain from Spain accounts for most UK cases–study. The Guardian, 2020.
- 32.
A da Silva Filipe, J.G. Shepherd, and T. et al. Williams. Genomic epidemiology reveals multiple introductions of SARS-CoV-2 from mainland Europe into Scotland. Nat Microbiol, 2020.
- 33.
SJ Lycett, J Hughes, MP McHugh, ADS Filipe, R Dewar, L Lu, et al.’epidemic waves of COVID-19 in Scotland: A genomic perspective on the impact of the introduction and relaxation of lockdown on SARS-CoV-2’. Cold Spring Harbor Laboratory Press, 2021.
- 34. Lindgren F, Bolin D, and Rue H. The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running. Scandinavian Journal of Statistics, 2022.
- 35. Møller Jesper, Syversveen Anne Randi, and Waagepetersen Rasmus Plenge. Log Gaussian Cox Processes. Scandinavian Journal of Statistics, 25(3):451–482, 1998.
- 36. Simpson Daniel, Rue Håvard, Riebler Andrea, Thiago G. Martins, and Sigrunn H. Sørbye. Penalising model component complexity: A principled, practical approach to constructing priors. Statistical Science, 32(1):1–28, 2017.
- 37. Bachl Fabian E., Lindgren Finn, Borchers David L., and Illian Janine B. inlabru: An R package for Bayesian spatial modelling from ecological survey data. Methods in Ecology and Evolution, 10(6):760–766, 2019.
- 38.
R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2021.
- 39. Sumio Watanabe. A widely applicable Bayesian information criterion. The Journal of Machine Learning Research, 14(1):867–897, 2013.
- 40. Spiegelhalter David J, Best Nicola G, Carlin Bradley P, and Van DerLinde Angelika. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4):583–639, 2002.